Advanced quantum mechanics

Bassano Vacchini

Università degli Studi di Milano

INFN Sezione di Milano

bassano.vacchini@mi.infn.it
http:/www.mi.infn.it/~vacchini

Lehrschemel für offene quanten systemen

Vorlesungsskript Version as of: December 17, 2018

Table of contents
I Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Quantum mechanics as quantum probability . . . . . . . . . . . . . . . . . . . . . . . . . 7
Classical probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Quantum probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Trace class, Hilbert-Schmidt and compact operators . . . . . . . . . . . . 15
Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Generalized probability measures . . . . . . . . . . . . . . . . . . . . . . . . 29
Dispersion-free states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Maximal entropy state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Positive operator-valued measure . . . . . . . . . . . . . . . . . . . . . . . . 36
Naimark’s dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Position observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Position and momentum observable . . . . . . . . . . . . . . . . . . . . . . 50
4 Transformations of states and observables . . . . . . . . . . . . . . . . . . . . . . . . . 54
Complete positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Operations and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Operations and state preparation . . . . . . . . . . . . . . . . . . . . . . . . 63
Mathematical characterization of operations . . . . . . . . . . . . . . . . . 64
Trace distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Von Neumann instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

II Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Open quantum systems as composite systems . . . . . . . . . . . . . . . . . . . . . . . 85
Reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Bipartite states and entanglement . . . . . . . . . . . . . . . . . . . . . . . 88
Schmidt decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Pure entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3
4 Table of contents

Mixed entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Positive maps and entanglement detection . . . . . . . . . . . . . . . . . . 93
6 Quantum maps and their representation . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Parametrization of quantum maps . . . . . . . . . . . . . . . . . . . . . . . 95
Linear operators on Hilbert-Schmidt space . . . . . . . . . . . . . . . . . . 95
Canonical basis and Bloch basis representation . . . . . . . . . . . . . . . 97
Map basis representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Choi-Jamiolkowski isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 103
7 General dynamics of open quantum systems . . . . . . . . . . . . . . . . . . . . . . . 107
Quantum dynamical semigroups . . . . . . . . . . . . . . . . . . . . . . . 107
Structure of the generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Pure dephasing spin-boson model . . . . . . . . . . . . . . . . . . . . . . . 114
Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Standard projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Correlated projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Projected equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 123
Nakajima-Zwanzig master equation . . . . . . . . . . . . . . . . . . . . . 124
8 Weak coupling master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
General properties of the master equation . . . . . . . . . . . . . . . . . 130
Quantum optical master equation . . . . . . . . . . . . . . . . . . . . . . 133
Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Coherent driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Resonant case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
NMR equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Wiener-Khintchine theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Multi-time correlation functions . . . . . . . . . . . . . . . . . . . . . . . . 147
Resonance fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

III Erweiterungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

1 Lectures splitting . . . . . . . ........ ...... ............... . . . . 161
2 Deep structure of quantum mechanics: Gleason, Kochen-Specker, Bell and other
(in)famous researchers. . . . . . ........ ...... ............... . . . . 161
3 Various results . . . . . . . . . ........ ...... ............... . . . . 162
4 Newly raised questions . . . . ........ ...... ............... . . . . 162
5 Points to be solved soon . . . ........ ...... ............... . . . . 162

@ B. Vacchini - Unimi & INFN

1 Quantum mechanics as quantum probability 7

“These are my principles. If you don’t like them I have others.”

(Groucho Marx)

1 Quantum mechanics as quantum probability

Quantum mechanics is a probability theory for the description of statistical experiments,
also valid at a microscopic level. To highlight this fact we first address the general for-
mulation of a statistical theory, which is relevant for the description of physical systems
in which the outcome of experiments has to be described by a probability distribution.
The general structure of a statistical theory can be outlined considering two basic
sets, the convex set of states S and the set of observables B, so that for any p ∈ S and
any X ∈ B one can construct a probability measure µX p defined on the σ-algebra of the
Borel sets over R, let us denote it by B(R), such that for any M ∈ B(R) the number
µXp (M ) provides the probability that X takes values in M if the state is p. The whole
analysis of course relies on the premise that the statistical experiments are reproducible,
and the frequencies with which the different outcomes appear are well defined. This
structure based on three basic elements, the set of states and observables and a proba-
bility formula to combine each state with each observable to provide a probability distri-
bution, is realized in two different mathematical frameworks in the classical and
quantum case. It is interesting to notice that the formalization of the two theories, by
Kolmogorov and von Neumann respectively, took place around the same time.
◦

Classical probability
In the classical case, as realized by Kolmogorov, probability theory is naturally
embedded within abstract measure theory. Just to recall, in abstract measure theory
one has:

• a set Ω and a σ-algebra F (that is a collection of subsets of Ω closed under com-

plementation and countable union)

• a measure µ defined on F and taking values in R+ ∪ {∞} such that

− µ(∅) = 0
!
− σ-additivity holds: µ(∪iAi) = i µ(Ai) provided Ai ∩ A j = ∅ for ∀i=j.

◦
The triple (Ω, F , µ) is called a measure space, and given a real measurable function
f , that is a function such that f −1(M ) is a measurable
" set in F for all open sets M ∈
B(R), one can consider integrals of the form A f dµ for A ∈ F . If we further ask the
measure to be normalized in the sense that µ(Ω) = 1, we call it a probability measure.
Denoting such measure with p we say that the triple (Ω, F , p) is a probability space. As
suggested by Kolmogorov this is the natural framework to describe a probability theory
in the classical setting, so that a probability model is fixed by specifying a measure
space, the space of elementary events, a σ-algebra on this space characterizing the mean-
ingful events, and a probability measure on it.
8 Table of contents

In the Kolmogorov model of classical probability theory one starts from a measurable
space and considers the following Kolmogorov construction of states and observables to
give a statistical theory. The set of states S is identified with the convex set of proba-
bility measures on F and the set of observables B is identified with the set of measur-
able functions seen as random variables. The probability formula which provides the dis-
tribution for the possible values of the random variable X given the state p is given by
#
µ p (B) =
X
dp
X −1(B)
= p(X −1(B)).
In order to evaluate the mean values we have by definition
#
⟨X ⟩ p = x dµX
p (x)
# R

= x dp(X −1(x))
# R

= X(y)dp(y).
Ω
Note that the first equality is always meaningful in a statistical theory, while the last
expression calls for an underlying event space, which is there only in the classical case,
and whenever available can be the most convenient for calculations.
We can thus compactly write
#
⟨X ⟩ p = Xdp,
Ω
similarly for higher moments and in general for a measurable function of the observable
#
⟨f (X)⟩p = f (X)dp.
Ω

Consider a point particle. The measure space is the usual phase-space, the proba-
bility measure can be expressed via a probability density f (x, p), i.e. a positive and
normalized element of L1(R3 × R3), and observables are described as random variables
given by real functions X(x, p) in L∞(R3 × R3).
◦
Exploiting the canonical duality relation between L1 and its dual L∞ mean values
are given by #
⟨X ⟩ f = d3xd3pX(x, p)f (x, p).
R3 ×R3

Any observable taking values in R defines a probability measure on this space according
to the formula #
µ f (M ) =
X
d3xd3p f (x, p)
X −1(M )

where M is a Borel set in the outcome space R of the observable. In particular the
expectation value of any observable, i.e. random variable, is obtained from the very
same probability density. Considering the case of the position observable X(x, p) = x
one can check that the probability measure µx f can be expressed through the density
corresponding to the marginal
#
f x(x) = d3p f (x, p).
R3

@ B. Vacchini - Unimi & INFN

1 Quantum mechanics as quantum probability 9

◦
Let us make a few remarks on the logic underlying these models. We call events the
elements of the σ-algebra F over which µ is defined, and note that the structure of the
σ-algebra naturally provides a Boolean algebra, so that one has the basic elements of
classical logic. Indeed if the events A and B are in F one can also consider the events
A ∪ B, A ∩ B and AC , while ∅ and Ω can be identified with the impossible and certain
event respectively.
◦
The notion of inclusion, union and intersection then allow to consider on the σ-
algebra a natural structure of lattice.
◦
Indeed a lattice is a set in which one has partial order, join and meet operations (∨
and ∧ respectively) which are associative and commutative. The lattice is said to be
complete if it has a minimum and a maximum element (0 and 1), orthocomplemented if
for any element of the lattice a one finds a complement a ′ such that a ∨ a ′ = 1, and dis-
tributive if join and meet are distributive the one with respect to the other

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

A lattice with these properties is said to be a Boolean algebra. As shown above a σ-

algebra of subsets of a set Ω is a Boolean algebra, and conversely it can be shown that
any Boolean algebra is isomorphic to a lattice of subsets of a set. The outcomes of trials
or experiments naturally have a lattice structure, but distributivity as we shall see is a
crucial property, satisfied by classical events but not by quantum events.
We are now in the position to point to three features of classical probabilistic model
which are not shared by quantum probability.
In the classical case the logic of events identified with the subsets of the σ-algebra
provides as shown a Boolean algebra, which implies in particular the distributivity of the
meet and join operation. This fact allows to consider the decomposition of an event in
terms of elementary events. This is no more true in quantum mechanics, as one sees
considering the following suggestive example. Let us introduce the following incompat-
ible experimental outcomes within a double slit experiment: 1) a=interference pattern;
2) b=passage through slit one; 3) b ′=passage through slit two. One then has

a ∧ (b ∨ b ′) =
/ (a ∧ b) ∨ (a ∧ b ′),

since the l.h.s. is equal to a, but since a is incompatible with both b and b ′ the r.h.s. is
equal to the empty set. The point is that in the quantum case a natural lattice of events
is given by the set of orthogonal projections on a given Hilbert space P(H), which
despite being complete and orthocomplemented, is not distributive. One can directly
check violation of distributivity by a suitable choice of projectors. This fact can be
rephrased stating that in quantum mechanics there are no elementary events.
◦
◦
◦
10 Table of contents

In the classical case the convex set of states is in particular a simplex, that is each
state can be uniquely demixed in terms of extreme points of the set. We recall that the
extreme points of a convex set are those elements which cannot be expressed as non
trivial convex combination of other elements in the set. E.g. in the phase-space example
introduced above extreme elements are given by measures concentrated in one point.
This is no more true in quantum mechanics, as we will show by means of example later
on. Another feature of the classical case no long recovered in the quantum setting is
given by the fact that the set of random variables over a measure space is a commuta-
tive algebra under point wise multiplication, while as well known quantum observables
do generally not commute.
◦
To highlight these facts we consider another example complementary to the previous
one.
We consider a classical statistical N level system, so that we take Ω = {1, ..., N } and
the σ-algebra F is given by the power set of Ω. The generic state can be identified with
a probability vector, so that
$ N
&
%
S = p = (p1, ..., pN ) | pi ! 0, pi = 1 .
i=1

Note that introducing the extreme points of this convex set as (ei) j = δij , so that e.g.
e1 = (1, 0, ..., 0), for each state one has the convex decomposition
N
%
p = piei ,
i=1

whose uniqueness follows from the fact that the extreme points also provide a linear
basis.
Real random variables corresponding to observables can be identified with N -tuples
of real numbers

B = {x = (x1, ..., xN ) | xi ∈ R},

and their commutativity under pointwise multiplication is immediately apparent. The

probability formula then simply reads

p({xi }) = pi.
µx

The mean values according to their definition are given by

N
%
⟨x⟩ p = p({xi })
xi µx
i=1
N
%
= xi pi
i=1

Let us now consider a quantum N -level system, whose Hilbert space is CN . The set of
states is

S = {N × N hermitian matrices S | S ! 0, Tr S = 1},

while for the observables

B = {N × N hermitian matrices }.

@ B. Vacchini - Unimi & INFN

1 Quantum mechanics as quantum probability 11

Exploiting the expression of a hermitian matrix X

N
%
X = x jE j ,
j =1

where {x j } are the eigenvalues, and {Ej } provide an orthogonal resolution of the iden-
tity, one can express the probability formula as

S (xi) = Tr S Ei ,
µX
which provides a probability distribution over the spectrum of the possible values
attained by the observable X. In quantum mechanics this is assumed to be the proba-
bility distribution of X provided the system is in the state S. The mean values can be
obtained according to their definition
N
%
⟨X ⟩S = S ({x j })
x j µX
j=1
N
%
= x j Tr S E j
j=1
= Tr S X ,
thus recovering a simple formula in which the random variable X directly appears.
◦
Within this quantum probabilistic model one recovers the classical probabilistic
model in dimension N considering suitable subsets of states and observables. This shows
that a classical probabilistic model can be embedded in a quantum one, but not vicev-
ersa. To this aim consider the subset of S given by the states diagonal in a given basis
⎧⎛ ⎞. ⎫
⎨ p1 ... 0 .. %N ⎬
Sdiag = ⎝ ··· ··· ··· ⎠.. pi ! 0, pi = 1 ,
⎩ ⎭
0 ... pN . i=1

and note that this is a convex subset. On the same footing consider the subset of B
⎧⎛ ⎞. ⎫
⎨ x1 ... 0 .. ⎬
Bdiag = ⎝ ··· ··· ··· ⎠.. xi ∈ R ,
⎩ ⎭
0 ... xN .
which is closed under matrix multiplication. In this case the quantum probability for-
mula reduces to the classical one

S (xi) = Tr S Ei
µX
= pi
= µx
p({xi })

and similarly for the mean values

N
%
⟨X ⟩S = S ({x j })
x j µX
j=1
%N
= xj pj
j=1
%N
= p({xi }).
x j µx
j=1
12 Table of contents

◦
Quantum probability
As a starting point we stress the fact that quantum mechanics arises in order to pre-
dict the outcomes of statistical experiments. Thus, despite the original evidence to look
for quantum mechanics was in well-known but more elaborated experiments such as the
study of black body radiation and atomic spectra, having in mind to identify the basic
building blocks of the theory one is led to consider the most simple statistical experi-
ments, in terms of which all other experiments can be formulated. Such experiments are
given by single particle statistical experiments in which a single microsystem is prepared
by a preparation apparatus which feeds without feedback a registration apparatus
according to the simple scheme depicted below as Ludwig’s kisten

[preparation apparatus] " [registration apparatus].

The experiment is of statistical nature in that only the frequencies and not the single
outcomes are reproducible. Note that in this spirit the state of the system is the mathe-
matical representative for the concrete preparation through which a certain system has
been obtained.
The basic question to be answered is how to correctly obtain the statistics of the
outcome of the experiment. That is to say, in analogy with the classical case, which
mathematical objects describe the preparation procedure (state) and the registration
procedure (observable), together with how we combine them (probability formula) to
obtain the probability measure on the outcomes of the experiment. Note that actually
these mathematical objects will characterize whole equivalence classes. We will point to
these mathematical objects in a bottom up approach, starting from the usual formalism
of quantum mechanics.
◦
◦

2 States
In the standard presentation of quantum mechanics one is given the Hilbert space of the
system H, and states are described by vectors ψ ∈ H with unit norm. Given the Hilbert
space one also has the lattice of projection operators, which as discussed below can be
taken as events eP , identifying each projection P with the associated subspace MP .
Given an event and a state vector one can consider the probabilities

pPψ = ||Pψ ||2

corresponding to the norm of the projected vector, to be interpreted as the probability

that the event eP is verified if the state is ψ. The event is certain if ψ ∈ MP and impos-
sible if ψ ∈ M⊥
P.
Considered an observable in the standard sense of a bounded self-adjoint operator
B ∈ B(H), where B(H) denotes the Banach space of bounded linear operators on H,
assuming B takes values in R and denoting by M ∈ B(R) a generic set in the Borel σ-
algebra on the real line, one has the following expression for the probability formula

ψ (M ) = ⟨ψ |E (M )ψ⟩
µB B

= ||E B (M )ψ ||2,

@ B. Vacchini - Unimi & INFN

2 States 13

which gives the probability to obtain outcomes for the measurement of B in the set M .
Here E B (·) denotes the projection-valued measure associated to the self-adjoint oper-
ator. Note that the last line has been obtained relying on idempotency of E B (M). Note
further that starting from the projection-valued measure upon integration one obtains
all the relevant information for a statistical description, such as mean values and other
moments of the probability distribution. Assuming for simplicity that B has a non
degenerate pure point spectrum, so that
%
B = b jPubj ,
j

with Pubj = |ubj ⟩⟨ubj | one dimensional projections, one has

%
E B (M ) = Pubj
{j |b j ∈M }

and in particular E B ({b j }) = Pubj. Mean values are defined according to

#
⟨B ⟩ ψ = x dµBψ (x)
% R
= ψ ({b j })
b jµB
j
%
= b j ⟨ψ |E B ({bj })ψ ⟩
j %
= ⟨ψ | bjE B ({b j })ψ ⟩
j
= ⟨ψ |B ψ⟩,
so that mean values can be most directly expressed as expectation value of the self-
adjoint operator. Note that to obtain this result we have not exploited idempotency of
the projections E B (·). Considering however the second moment one has
#
⟨B ⟩ ψ =
2 x2 dµB ψ (x)
% R
= ψ ({b j })
b2j µB
j
%
= ⟨ψ |b2j E B ({bj })ψ ⟩
j
2 34 5
% %
= ⟨ψ | b jE ({b j })
B biE ({bi }) ψ ⟩
B

j i
= 2
⟨ψ |B ψ ⟩,
so that idempotency is now a crucial property in order to express the second moment of
the probability distribution of the observable B as expectation value of the square of the
operator. On the same premises one has
#
⟨f (B)⟩ ψ = f (x)dµB
ψ (x)
R
= ⟨ψ |f (B) ψ ⟩,
and in particular
(∆B)2ψ = ⟨B 2⟩ ψ − ⟨B ⟩2ψ.
14 Table of contents

Having in mind that a state should describe a concrete preparation procedure, let us
consider how to obtain a pure state and insofar such a state corresponds to a general
preparation. A preparation corresponding to a pure state can be obtained in the very
special situation in which we measure a complete set of commuting observables A, fixing
a unique common eigenvector, so that ψ = ua.
This is not the most general state, since the set of pure states is not closed under
convex mixtures. Let us mix two such preparation procedures corresponding to out-
comes given by common eigenvectors ua and ub with frequencies N 1/N = λ and N 2/
N = (1 − λ) respectively, with N 1 + N2 = N . Relying on conditioning and using the rela-
tion
%
P (x) = P (x, y)
y
%
= P (x|y)P (y)
y
we have
N1 N
P (B ∈ M ) = P (B ∈ M |ua1) + P (B ∈ M |ua2) 2
N N
= λ⟨ua1|E B (M )ua1⟩ + (1 − λ)⟨ua2|E B (M )ua2⟩,
which cannot be written as ||E B (M )ψ||2 with ψ a state vector. A similar result can be
obtained for the mean values starting from
%
⟨X ⟩ = x P (x)
%x
= x P (x, y)
x, y
%
= x P (x|y)P (y)
x, y
% 6% 7
= x P (x|y) P (y),
y x
so that
N1 N
⟨B ⟩ = ⟨ua1|B ua1⟩ + ⟨ua2|B ua2⟩ 2 .
N N
We are thus led to introduce an operator of the form
ρ = λ|ua1⟩⟨ua1| + (1 − λ)|ua2⟩⟨ua2|
which allows us to write
P (B ∈ M ) = Tr ρE B (M )
⟨B ⟩ = Tr ρB
and in general
#
⟨f (B)⟩ = f (x)dµB
ρ (x)

with
ρ (M ) = Tr ρE (M ).
µB B

Let us recall what we mean by trace of an operator (if it exists). Given a SONC {un } in
H one has by definition
%
Tr A = ⟨un |A un ⟩,
n

@ B. Vacchini - Unimi & INFN

2 States 15

so that the trace is a linear mapping, further satisfying invariance under a cyclic trans-
formation, namely
Tr A B = Tr B A.
An operator like ρ is called a statistical operator. We have been led to such an expres-
sion starting from a simple analysis of states as representatives of a preparation proce-
dure. One is led to consider statistical operators also when analyzing measurements in
quantum mechanics, or more generally bipartite systems, namely systems described on a
Hilbert space with a tensor product structure. Also the analysis of the classical limit in
the description of a macroscopic system is best performed considering statistical opera-
tors.
Trace class, Hilbert-Schmidt and compact operators
We now want to better specify the functional space in which a general statistical
operator can be described, thus characterizing its properties. Indeed instead of mixtures
of two one-dimensional orthogonal projections as in the previous example one can obvi-
ously consider an higher number of terms as well as a mixture, even a continuous one, of
non-orthogonal states, which thus cannot be read as classical mixture of mutually exclu-
sive events.
Let us start giving a precise meaning to the trace operation. Let A ∈ B(H) be a posi-
tive operator, namely ⟨ψ |Aψ ⟩ ! 0 for all ψ ∈ H, and consider the series of positive num-
bers
%
Tr A = ⟨un |A un ⟩,
n
for a given basis {un }. If the series converges the result is independent of the choice of
basis, indeed considered another basis {vm } we have
% %
⟨un |A un ⟩ = ∥A 1/2 un ∥2
n n
% 4% . .2 5
= . ⟨vk |A un ⟩.
1/2

n k
% 4% . . 5
= .⟨un |A 1/2 vk ⟩.2
%k n
= ∥A 1/2 vk ∥2
%k
= ⟨vk |A vk ⟩.
k
The independence from the choice of basis justifies the definition. For a generic operator
A ∈ B(H) not necessarily positive, one considers the positive operator
√
|A| = A †A ,
and if Tr|A| < ∞ one says that A is a trace class operator, then also Tr A is finite and
independent of the basis. The space T (H) of trace class operators on a given Hilbert
space turns out to be a Banach space with respect to the trace norm defined for A ∈
T (H) as
∥A∥1 = Tr|A|.
The space of trace class operators can also be obtained as closure with respect to the
trace norm of the set of degenerate operators, that is operators of the form
%n
A = |ui ⟩⟨vi |,
i=1
16 Table of contents

where the sums run over a finite index set and {ui }, {vi } are linearly independent fami-
lies.
◦
The space T (H) is also a bilateral ideal of B(H), that is to say invariant under left
and right multiplication, namely ∀A ∈ B(H), ∀T ∈ T (H) one has A T , T A ∈ T (H) and in
particular

∥A T ∥1 # ∥A∥∥T ∥1.

Another important bilateral idel of B(H) is given by the space of Hilbert-Schmidt opera-
!
tors. Given A ∈ B(H) one can consider the quantity n ⟨un |A †A un ⟩, whose value as
shown above does not depend on the basis. If this quantity is finite the operator is in
the Hilbert-Schmidt class, and the expression
√
∥A∥2 = Tr A †A

actually defines a norm which makes the space of Hilbert-Schmidt operators HS(H) a
Banach Space. It is the closure with respect to the Hilbert-Schmidt norm of the set of
degenerate operators. Again it is a bilateral ideal of B(H), indeed ∀A ∈ B(H), ∀T ∈
HS(H) one has A T , T A ∈ HS(H) and in particular

∥A T ∥2 # ∥A∥∥T ∥2.

Most importantly the space of Hilbert-Schmidt operators HS(H) is also a Hilbert space
with respect to the following scalar inner product

⟨A, B ⟩ = Tr A †B,

well defined because the product of two Hilbert-Schmidt operators is trace class. In this
setting the Cauchy-Schwarz inequality becomes

|Tr A †B | # ∥A∥2∥B ∥2

with A, B ∈ HS(H), to be compared with

|Tr A †B | # ∥A∥∥B ∥1

with A ∈ B(H) and B ∈ T (H). The latter relationship can be better understood noting
that B(H) = T ′(H), namely the space of bounded operators is the dual space with
respect to the space of trace class operators, that is the map B → Tr(B · ) is an isometric
isomorphism of B(H) onto T ′(H). The space HS(H) is self-dual since it is a Hilbert
space.
Let us recall another subspace of B(H), let us call it C(H), which is the space of
compact operators, closure of the set of degenerate operators with respect to the uni-
form norm. One has in particular

B(H) ⊇ C(H) ⊇ HS(H) ⊇ T (H),

and accordingly

∥·∥ # ∥·∥2 # ∥·∥1.

The equal sign in the inclusion between different spaces is obtained when the underlying
Hilbert space is finite dimensional, so that the distinction among the different spaces dis-
appears.
◦

@ B. Vacchini - Unimi & INFN

2 States 17

Let us now come back to the duality relation among the different spaces.
Suppose first dimH = n, then one can consider the canonical basis in T (H) given by
!
the matrices {E ij } with (E ij )kl = δik δ jl , and write any T ∈ T (H) as T = ni, j =1 Tij E ij .
We now show that any continuous linear functional f defined on T (H) can be identified
with a matrix B ∈ B(H). Indeed define

f (E ij ) = B ji ,

we have ∀T ∈ T (H)
⎛ ⎞
n
% n
%
f (T ) = f⎝ TijE ij ⎠= Tij f (E ij )
i, j=1 i, j=1
n
%
= Tij B ji = Tr T B,
i, j=1

so that the linear functional f can be identified with the matrix B, and the duality rela-
tion with the trace operation.
In the case of a generic Hilbert space given a fixed B ∈ B(H) we have recalled that

Tr B T

is a well defined complex number ∀T ∈ T (H), and furthermore

|Tr B T | # ∥B ∥∥T ∥1,

so that each bounded operator can be identified with a linear continuous functional on
the space T (H) exploiting the trace operation, and also the opposite can be shown to
hold. One has therefore indeed B(H) = T ′(H). Note however that in the infinite dimen-
sional case one has T (H) ⊂ B ′(H), as can be shown along the same lines, the inclusion is
however strict.
Considering self-adjoint operators one has the important result:
Theorem Let A ∈ B(H), self-adjoint and compact, then there exists a SONC {un }
of eigenvectors of A with corresponding eigenvalues {an }, such that the spectral repre-
sentation reads
%
A = an |un ⟩⟨un |.
n

In particular the eigenvalues {an } form a l0(C) sequence, so that limn→∞an = 0, and the
eigenspaces of the non zero eigenvalues are finite dimensional.
!
If A ∈ HS(H) the eigenvalues {an } form a l2(C) sequence, so that n |an |2 = ∥A∥22 <
!
∞; if A ∈ T (H) the eigenvalues {an } form a l1(C) sequence, so that n |an | = ∥A∥1 <
∞.
We can now introduce the set of states in the quantum probabilistic setting, fixed by
the choice of Hilbert space and given by the convex set of statistical operator

S(H) = {ρ ∈ T (H)|ρ ≥ 0, Trρ = 1}.

The convexity of the space S(H) implies that any convex combination of elements of
S(H) is still in the set, i.e.
n
% n
%
{ρi } ∈ S(H), λi ! 0, λi = 1 ⇒ λiρi ∈ S(H).
i=1 i=1
18 Table of contents

Actually it is enough to consider mixtures of two elements only, that is convexity is

implied by
ρ1, ρ2 ∈ S(H), 0 # λ # 1 ⇒ λρ1 + (1 − λ)ρ2 ∈ S(H),
! 8 9
λ λn
as follows from the simple identity ni=1 λiρi = λ1 ρ1 + (1 − λ1) 1 −2λ ρ2 + ... + ρ
1 − λ1 n
.
1
Since trace class implies compact and positive implies self-adjoint we have the following
fundamental representation of a statistical operator, which provides a orthogonal decom-
position of the statistical operator
%
ρ = pnP ϕn
%n
= pn |ϕn ⟩⟨ϕn |,
n
where note that due to the choice of one-dimensional projection P ϕn the pn are generally
not distinct. Indeed
! the {ϕn } do form a SONC and the {pn } provide a probability dis-
tribution: pn ! 0, n pn = 1. Such a decomposition always exists, though it is in general
not unique; moreover ρ also admits non-orthogonal decompositions. We will discuss this
point at length later on.
Purity
Let us first give some information on the structure of the set S(H). We call pure
states the extreme points of this convex set, i.e. those that cannot be written as a non
trivial convex mixture. It turns out that pure states are given by one-dimensional pro-
jections. In fact the following three statements are equivalent:
i. ρ is pure
ii. ρ is a one-dimensional projection
iii. Tr ρ2 = 1.

i→ii) Since ρ is a statistical operator one can consider its orthogonal decomposition,
which being extreme must read ρ = p1|ϕ1⟩⟨ϕ1|, but due to the contraint on the trace one
has p1 = 1, so that ρ is a one-dimensional projection.
ii→i) Suppose as an absurd P ψ = λρ1 + (1 − λ)ρ2 with 0 < λ < 1. Then for all ϕ ∈
{ψ }⊥ one has 0 = λ⟨ϕ|ρ1 ϕ⟩ + (1 − λ)⟨ϕ|ρ2 ϕ⟩, so that the contributions at the r.h.s.
have to be zero, and therefore the support of ρ1, ρ2 reduces to {ψ } and one must have
ρ1 = ρ2 = P ψ.
ii→iii) Obvious.
iii→ii) One has the upper bound Tr ρ2 # ∥ρ∥∥ρ∥1 = ∥ρ∥ # ∥ρ∥1 = 1. Suppose now as
an absurd that Tr ρ2 = 1 but ρ admits a non trivial decomposition ρ = λρ1 + (1 − λ)ρ2,
then
1 = Tr ρ2 = λ2Tr ρ21 + (1 − λ)2Tr ρ22 + 2λ(1 − λ)Tr ρ1 ρ2
# λ2 + (1 − λ)2 + 2λ(1 − λ)|Tr ρ1 ρ2| # 1
: :
but since due to Cauchy-Schwarz |Tr ρ1 ρ2| # Tr ρ21 Tr ρ22 # 1, this is actually a collec-
tion of identities. In particular the Cauchy-Schwarz inequality is saturated, so that we
have ρ1 = cρ2, and the constant c has to be one due to the trace constraint.
It is therefore useful to introduce the quantity
P(ρ) = Tr ρ2,
called purity of the state satisfying
0 < P(ρ) # 1,

@ B. Vacchini - Unimi & INFN

2 States 19

as immediately follows from the properties of the eigenvalues of a statistical operator,

!
indeed note that given an orthogonal resolution for ρ one also has P(ρ) = n p2n.
◦
◦
We recall the following properties of the purity
a) P is a convex map: P(λρ1 + (1 − λ)ρ2) # λP(ρ1) + (1 − λ)P(ρ2) # max {P(ρ1),
P(ρ2)}, and equality holds iff ρ1 = ρ2, so that purity is strictly convex
b) P is invariant under unitary transformations: P(UρU †) = P(ρ)
c) P(ρ) = 1 if and only if ρ is a pure state.
The extreme states thus coincide with one-dimensional projections, but do they coincide
with the boundary of the convex set? The answer is no provided dimH ! 3, as we now
show. Let us first consider the peculiar and well-known structure of states in C2.
If dimH = 2, the Hilbert space is isomorphic to C2, and a generic state can be
written as
1
ρ = (1 + a · σ),
2
1
with a ∈ R3, ∥a∥ # 1, and √ {1, σ } an orthonormal basis in B(C2) = M2(C) according
2
to the Hilbert-Schmidt scalar product.
◦
This representation allows a one to one mapping between points in the sphere S2 =
{a ∈ R3, ∥a∥ # 1} and statistical operators in C2. The geometry of the sphere reflects in
this case the geometry of the quantum states. This is called Bloch sphere (Poincarè
sphere in optics). Here the boundary coincides with the set of extreme states, and each
non extreme state can be demixed in terms of two extreme states. This is not generally
true.
The set S(H) is a convex subset of the real normed space Ts(H) of self-adjoint trace
class operators. We can characterize the boundary of a convex set in an intrinsic way as
follows. We introduce its interior S̃ (H) as the subset of the elements ρ such that ∀σ ∈
S(H) there exists a λ > 1 such that
(1 − λ)σ + λρ ∈ S(H),
that is any line segment in S(H) having ρ as one of the endpoints can be prolonged
beyond ρ without leaving S(H). We define then the boundary as ∂S(H) = S(H)\S̃ (H),
that is ρ ∈ ∂S(H) iff there exists σ ∈ S(H) such that ∀λ > 1
(1 − λ)σ + λρ ∈
/ S(H).
This condition can also be formulated saying that there exists σ ∈ S(H) such that ∀ε > 0
ρ + ε(ρ − σ) ∈
/ S(H).
To connect this intrinsic definition of boundary to the specific structure we are consid-
ering, we observe that if ρ belongs to the boundary ∀η > 0 there exists a trace class
η
/ S(H) but ∥ρ − T η ∥1 < η. Indeed consider T η = ρ + 3 (ρ − σ),
operator T η such that T η ∈
2
then ∥ρ − Tη ∥1 # 3 η < η, since for any couple of statistical operators ρ, w we have ∥ρ −
w∥1 # 2. If ρ has a zero eigenvalue, it belongs to the boundary. Indeed let ϕ ∈ H be an
1
eigenvector corresponding to the zero eigenvalue, then considering T η = ρ − 2 η |ϕ⟩⟨ϕ|, we
have that Tη is not positive and ∥ρ − Tη ∥1 < η. The converse holds true only in finite
dimensions.
20 Table of contents

◦
◦
We now show that when dimH ! 3 the boundary does not coincide with the set of
pure states. Supposing dimH ! 3 consider a orthogonal triple {ϕ1, ϕ2, ϕ3} and the sta-
tistical operator

ρ = λPϕ1 + (1 − λ)P ϕ2,

which is not pure, however ρϕ3 = 0, so that it has a zero eigenvalue and therefore
belongs to the boundary. Thinking about Euclidean space, in a sphere pure states cover
all the boundary, in a box only vertices are pure, other points on the boundary can be
demixed.
◦
Entropy
For a stochastic experiment the information content related to a certain event is a
measure for the uncertainty related to its appearance before we know it, or equivalently
of the information gained upon learning its value. In information theory the information
content of a certain news/event is related to the number of binary questions necessary to
ascertain it. Suppose the sample space Ω is a set with N = 2n elements ω1, ..., ωN , which
provide the elementary events. An element of Ω can then be characterized by a sequence
of n zeros or one. A realization of the stochastic event has an information content of I
equal to n = log 2 N bits. More generally we set I = log2N , even if N is not a power of
two. Suppose the elementary events have equal probability pi = 1/N , then the informa-
tion of the realization of the event ωi has information content I(ωi) = log2 (N ) = −
log2 (pi), and following Shannon we adopt this quantity as information content of an
event taking place with probability pi for a distribution which is not necessarily uniform.
The average information gained by a stochastic experiment measuring events with prob-
ability pi is thus quantified by the Shannon entropy
%N
H({pi }) = − pi log2 (pi).
i=1

This quantity specifies the amount of resources (bits) needed to store the information
associated to a stochastic experiment.
◦
◦

In the quantum setting the corresponding quantity is given by the von Neumann
entropy, given by

S(ρ) = −kBTrρlogρ

which does not depend on the decomposition considered for ρ, and where kB denotes the
Boltzmann constant. Since for any ρ ∈ S(H) one can consider an orthogonal decomposi-
tion in terms of pure states, we can write
%
ρ = piP ϕi
i %
S(ρ) = −kB pilog pi.
i

@ B. Vacchini - Unimi & INFN

2 States 21

In particular
S(ρ) ! −log (max {pi }),
so that the entropy is larger in the presence of many small eigenvalues, while it is equal
to zero if the state is pure. In the case of a finite dimensional Hilbert space with
dimH = n we have the bounds
0 # S(ρ) # log n
1
# P(ρ) # 1,
n
saturated by the pure states and the maximally mixed statistical operator corresponding
1
to ρ = n 1. Note that P and S introduce different orders on the set of statistical opera-
tors.
◦
◦
We have the following properties for the entropy
a) S is a concave map: S(λρ1 + (1 − λ)ρ2) ! λS(ρ1) + (1 − λ)S(ρ2) ! min {S(ρ1),
S(ρ2)}, and the equality only holds if ρ1 = ρ2, that is entropy is strictly concave
b) S is invariant under unitary transformations: S(UρU †) = S(ρ)
c) S(ρ) = 0 if and only if ρ is a pure state
d) Klein’s inequality: given two statistical operators ρ, w ∈ S(H), such that ker (w) ⊆
ker (ρ) one has
S(ρ) # −kBTr ρlog w,
and the equality sign only holds if ρ = w.
◦
◦
◦
Proof of a) We consider the concave function h(x) = −x log x, for x ! 0, put equal to
zero in x = 0. We further set
ρ = λρ1 + (1 − λ)ρ2
and consider the decomposition
%
ρ = ρ jP j
j
%
= ρ j |ujk ⟩⟨ujk |,
j ,k

where the ρ j are the distinct eigenvalues of the mixture, ρj = ⟨u jk |ρu jk ⟩, and P j the pro-
jections on the relative eigenspace, with possible degeneracy characterized by the index
k. We have
%
S(ρ) = −kBTrρlogρ = −kB ρj logρ j
% %j ,k
= kB h(ρ j ) = kB h(⟨u jk |ρu jk ⟩)
%j ,k j ,k
! kB (λh(⟨u jk |ρ1ujk ⟩) + (1 − λ)h(⟨u jk |ρ2u jk ⟩)),
j ,k
22 Table of contents

and we have exploited once the concavity of h. We now exploit it once more observing
that for a concave function

h(⟨B ⟩) ! ⟨h(B)⟩,

where B ∈ B(H) is a self-adjoint operator and ⟨B ⟩ = Tr(σB), with σ a state.

◦
Indeed the mean value of an observable is a convex mixture of its eigenvalues. More
explicitly considering the spectral decomposition for B
%
B = biPbi ,
i
and using the spectral theorem together with concavity of h we have:
2 4 53 4 % 5 4% 5
%
h(⟨B ⟩) = h Tr σ biPbi =h biTr(σPbi) = h pibi
i i 4 % i 5
% %
! pih(bi) = Tr(σPbi)h(bi) = Tr σ h(bi)Pbi
i i i
= Tr(σh(B)) = ⟨h(B)⟩.

◦
Inserting this result in the previous expression we come to
%
S(λρ1 + (1 − λ)ρ2) ! kB (λ⟨u jk |h(ρ1)ujk ⟩ + (1 − λ)⟨u jk |h(ρ2)ujk ⟩)
j ,k
= kB(λTr h(ρ1) + (1 − λ)Tr h(ρ2))
= λS(ρ1) + (1 − λ)S(ρ2).

Proof of b) Follows from the fact that the entropy only depends on the distribution of
the eigenvalues, which is not changed by a unitary transformation. Equivalently one can
observe that thanks to the ciclic property of the trace

S(UρU †) = −kBTr(UρU †log(UρU †)) = −kBTr(UρlogρU †) = −kBTrρlogρ.

Proof of c) If ρ is a pure state, all eigenvalues are zero, apart from one equal to unit,
therefore
! S(ρ) = 0. Vice versa S(ρ) = 0 implies that the sum of positive terms −
kB i pilog pi is equal to zero, so that they are all zero, which again implies all pi equal
to zero, apart from one equal to unit, and again the state is pure.
Proof of d) The statement is equivalent to

Trρlogw − Trρlogρ # 0.

Suppose
%
ρ = ρj |u j ⟩⟨u j |
j
%
w = wk |vk ⟩⟨vk |
k
we then have
%
Trρlogw − Trρlogρ = ρj (⟨u j |log w|u j ⟩ − log ρj )
j
%
= ρj |⟨u j |vk ⟩|2(log wk − log ρj )
j ,k

@ B. Vacchini - Unimi & INFN

2 States 23

where the expression is well defined since ker (w) ⊆ ker (ρ), restricting the sum to the ρ j
different from zero, and exploiting the inequality log y # y − 1 for y > 0 we have
% w
Trρlogw − Trρlogρ = ρ j |⟨u j |vk ⟩|2log k
ρj
j ,k
% 4 5
wk
# ρ j |⟨u j |vk ⟩| 2 −1
ρj
j ,k
%
= |⟨u j |vk ⟩|2(wk − ρ j )
j ,k
= 1 − 1 = 0.

◦
Decompositions
We have shown that each statistical operator has at least an orthogonal decomposi-
tion in terms of pure, that is extreme, states, thanks to the spectral theorem. Actually
this decomposition is unique iff all eigenvalues are distinct. In general one has many
orthogonal decompositions, also known as canonical convex decompositions. The
freedom lying in the change of basis in the eigenspaces. The higher the degeneracy, the
higher the number of orthogonal decompositions. All refer to the same statistical oper-
ator, and possibly describe equivalent ways to obtain the same statistical operator to be
understood as mathematical representative of an equivalence class of preparation proce-
dures.
To consider an extreme case let us work in the finite dimensional Hilbert space C2,
where statistical operators are in one to one relationships with the points on the Bloch
1
sphere, and consider the maximally mixed state proportional to the identity ρ = 2 1. We
introduce a basis {|ω+⟩, |ω−⟩}, which can be understood as eigenvectors of a polariza-
tion along a direction ω, which we can take as z-axis. Exploiting this basis for any θ ∈
[0, π] and φ ∈ [0, 2π] we obtain a new basis
θ −iφ/2 θ
|ω+(θ, φ)⟩ = cos e |ω+⟩ + sin e+iφ/2|ω−⟩
2 2
θ θ
|ω−(θ, φ)⟩ = −sin e−iφ/2|ω+⟩ + cos e+iφ/2|ω−⟩,
2 2
corresponding to eigenvectors of the polarization in an arbitrary direction. One then has
the following orthogonal decompositions
1
ρ = 1
2
1
= (Pω+ + Pω−)
2
1
= (Pω+(θ,φ) + Pω−(θ,φ)),
2
but also the non-orthogonal decomposition
# # π
1 2π
ρ = dφ dθ sinθ |ω+(θ, φ)⟩⟨ω+(θ, φ)|
4π 0 0
# # π
1 2π
= dφ dθ sinθ Pω+(θ,φ).
4π 0 0
◦
◦
24 Table of contents

Indeed one generally has an uncountable number of distinct convex decomposition in

terms of pure states, not necessarily orthogonal between them. In dimension n at most
n orthogonal pure states are required, but more are allowed. To show this consider the
following construction. Given an arbitrary statistical operator ρ and an arbitrary com-
plete orthonormal system {ϕi } we consider the numbers
; ;
λi = ⟨ϕi |ρϕi ⟩ = ; ρ1/2 ϕi; 2,

which are positive and sum up to one. For the non zero λi we introduce the vectors
ρ1/2 ϕi
ψi = √ ,
λi
which are of norm one but generally not orthogonal. We then have
%
ρ = λi |ψi ⟩⟨ψi |.
i
In fact ∀φ ∈ H one has
4% 5 %. .
⟨φ| λi |ψi ⟩⟨ψi | φ⟩ = .⟨φ|ρ1/2 ϕi ⟩.2
i %
i . .
= .⟨ρ1/2 φ|ϕi ⟩.2
i
= ⟨φ|ρφ⟩,
where we have exploited the fact that the {ϕi } form a SONC. Since the identity holds
for arbitrary φ and ρ is bounded, the equality for the diagonal matrix elements implies
the equality of the operator expressions.
◦
◦
Given another SONC {ϕi′ } one has another decomposition
%
ρ = λi′ |ψi′⟩⟨ψi′|,
i

where again the vectors ψi′and therefore the associated projections need not be orthog-
onal. Of course one can also consider decompositions in terms of non pure states. Indeed
given a decomposition
ρ = λρ1 + (1 − λ)ρ2,
one can consider infinite other decompositions of the form
4 5
λ 1 λ
ρ = ρ + 1− ρ2,
µ µ µ
with 0 < λ < µ < 1 and ρ1µ = µρ1 + (1 − µ)ρ2.
The existence of infinitely many decompositions, orthogonal or non orthogonal, for
the same statistical operator show that indeed these decompositions do not have a phys-
ical meaning. They cannot be unveiled by any subsequent measurement performed on ρ,
whose statistics for any observable is fixed by ρ only. The state is described but not
characterized by the ensembles {λi , ψi } or {λi′ , ψi′}. We cannot describe it as arising by
classical ignorance, quantified by the probability distribution of the eigenvalues, of a
known set of alternative events described by extreme elements. Indeed the decomposi-
tion is highly non unique and the different projection are generally not orthogonal, so
that they cannot be identified with exclusive alternatives.

@ B. Vacchini - Unimi & INFN

2 States 25

The different decompositions might describe different possible preparation procedures

leading to the same state. Note that this preparation procedures might even be incom-
patible, in the sense that they cannot be performed together. Coming back to the polar-
ization example, an unpolarized beam can be obtained by equal mixture of beams polar-
ized in opposite directions along the same axis. The specific choice of axis is not rele-
vant, and there is no way, by means of measurement performed on the final beam to
find along which axis the polarization was performed. All these preparation procedures
do belong to the same equivalence class. Moreover these different preparations are
incompatible, in the sense that they cannot be performed together.
Effects
Now that we have identified statistical operators as the mathematical representatives
of equivalence classes of preparation procedures, we move on to characterize the mathe-
matical representatives of equivalence classes of registration procedures. We first con-
sider registrations or measurements corresponding to the simplest events, given by a
dichotomic alternative. Such apparata provide yes-no answers, e.g. to the question
whether a certain quantity belongs to a given interval. Such elementary events or basic
building blocks of an observable can be naturally identified with maps

E: S(H) → [0, 1]
ρ " E(ρ)

which preserve the convex structure of the set of states, i.e. they are affine

E(λρ1 + (1 − λ)ρ2) = λE(ρ1) + (1 − λ)E(ρ2).

We call effects such affine maps from S(H) to the interval [0, 1], and we now show that
they can be identified with the positive bounded operators in the interval between the
zero and the identity operator.
Let us consider a self-adjoint operator B ∈ B(H), B = B †. As we have seen due to
B(H) = T ′(H) the map

ρ 6→ TrρB

provides an affine functional. Noting that for ρ = P ψ we have TrρB = ⟨ψ|Bψ ⟩ one has
that TrρB ! 0 for all ρ iff B ! 0, and in the same way considering Trρ(1 − B) = 1 −
TrρB we have that TrρB # 1 for all ρ iff B # 1. Therefore a self-adjoint bounded oper-
ator 0 # E # 1 provides an affine functional on the convex space of states through the
formula

E: S(H) → [0, 1]
ρ " TrρE.

The vice versa holds, so that elementary events or observables corresponding to yes-no
measurements can be identified with positive operators between zero and one.
◦
Theorem Given an affine map E: S(H) → [0, 1] there exists a bounded self-adjoint
operator E such that the relation

E(ρ) = TrρE

holds, and 0 # E # 1.
26 Table of contents

Proof. Let us first give the idea: we want to extend the affine functional E to a linear
bounded functional on the whole linear space T (H) to exploit the duality relation
B(H) = T ′(H) and therefore identify the map with a bounded operator, which then has
to be between zero and one to adjust the range of the functional. To extend E to a
uniquely defined linear map on T (H) we proceed in three steps, adopting a standard
procedure: we first extend the map from the convex set to the cone of positive operators
T+(H), then to the set of self-adjoint operators Ts(H), and finally to the whole linear
space T (H).
We start with the affine functional E defined on the convex set S(H), and extend it
to a functional E+ defined on the positive cone T+(H) by setting E+(0) = 0 and for T ! 0
4 5
T
E+(T ) = Tr T E ,
Tr T
homogeneous with respect to multiplication by a positive scalar s since
4 5
sT
E+(sT ) = Tr sT E
Tr sT
= sE+( T )

and furthermore additive since exploiting the fact that E is affine we have for S , T ∈
T+(H)
4 5
S +T
E+(S + T ) = Tr (S + T ) E
Tr (S + T )
4 5
Tr S S Tr T T
= Tr (S + T ) E +
Tr (S + T ) Tr S Tr (S + T ) Tr T
< 4 5 4 5=
Tr S S Tr T T
= Tr (S + T ) E + E
Tr (S + T ) Tr S Tr (S + T ) Tr T
= E+(S) + E+(T ).

To extend E+ to Ts(H) we note that each self-adjoint operator can be expressed as a

sum of positive and negative parts, according to
T = T+−T−

with
1
T+ = (|T | + T )
2
1
T− = (|T | − T )
2
and we can thus set for T ∈ Ts(H)

E r(T ) = E+(T +) − E+(T −).

Again one can check that the definition makes the functional homogeneous with respect
to multiplication by a real scalar and additive within Ts(H). We show e.g. additivity.
From the relations
S + T = (S + T )+ − (S + T )−
= S+ − S − + T + − T −
and therefore

(S + T )+ + S − + T − = (S + T )− + S + + T +,

@ B. Vacchini - Unimi & INFN

2 States 27

we have

E+((S + T )+ + S − + T −) = E+((S + T )+) + E+(S −) + E+(T −)

= E+((S + T )− + S + + T +)
= E+((S + T )−) + E+(S +) + E+(T +)

so that

E+((S + T )+) − E+((S + T )−) = E+(S +) − E+(S −) + E+(T +) − E+(T −)

and finally the additivity relation

E r(S + T ) = E r(S) + E r(T ).

As a last step we extend to the whole T (H) noting that each T ∈ T (H) can be written
as linear combination of two self-adjoint operators according to T = TR + iTI with
1
TR = (T + T †)
2
1
TI = (T − T †)
2i
and we can thus set for T ∈ T (H)

E c(T ) = E r(TR) + iE r(TI ).

Again one can check that the definition makes the functional homogeneous with respect
to multiplication by a complex scalar and additive within T (H). We have for α ∈ C

E c(αT ) = E c((αRTR − αITI ) + i(αRTI + αITR))

= E r(αRTR − αITI ) + iE r(αRTI + αITR)
= αRE r(TR) − αIE r(TI ) + i(αRE r(TI ) + αIE r(TR))
= (αR + iαI )(E r(TR) + iE r(TI ))
= αE c(T ).

And further

E c(S + T ) = E c((SR + TR) + i(SI + TI ))

= E r((SR + TR)) + iE r((SI + TI ))
= E r(SR) + iE r(SI ) + E r(TR) + iE r(TI )
= E c(S) + E c(T ).
8 9
T
The thus defined linear functional on T (H) is further bounded since due to E Tr T # 1
we have
◦

|E c(T )| = |E c(TR+ − TR− + iTI+ − iTI−)|

# |E c(TR+)| + |E c(TR−)| + |E c( TI+)| + |E c(TI−)|
# Tr TR+ + Tr TR− + Tr TI+ + Tr TI−
= Tr|TR | + Tr|TI |
1 1 . .
= Tr|T + T †| + Tr.T − T †.
2 2
# 2∥T ∥1.
28 Table of contents

But according to the duality relation B(H) = T ′(H) such bounded functional, which we
denote again with the letter E for simplicity, can be identified with an operator in B(H),
let us call it E, via

E(T ) = Tr TE ∀T ∈ T (H),

and since the functional restricted to S(H) has to take values in [0, 1], as shown above
one must have 0 # E # 1. Furthermore note that for E1 = / E2 one has in general Tr ρE1 =
/
Tr ρE2, so that different operators do correspond to different effects.
We are now in the position to define the set which provides the mathematical repre-
sentatives of equivalence classes of elementary registration procedures, corresponding to
yes-no answer in the measurement

E(H) = {E ∈ B(H), 0 # E # 1},

which we call set of effects. We thus use from now on the term effect to denote such
operators, with which the originally introduced affine maps can be identified. We have
the strict inclusions, even in the finite dimensional case

P(H) ⊂ E(H) ⊂ B(H),

so that projections are examples of effects. We note that the set of effects is convex, and
E ∈ E(H) implies E ⊥ = (1 − E) ∈ E(H). Moreover the usual sum of linear operators
defines a partial operation in E(H), since the sum of two effects E1, E2 is defined pro-
vided E1 + E2 # 1. Effects which can be summed do not necessarily commute.
◦
◦
In particular projections do coincide with its extreme elements. We stress that each
projection is an extreme element of the convex set of effects, not only one dimensional
projections. Let us show this fact. Take P ∈ E(H) ∩ P(H), so that P = P 2, and suppose
as an absurd that it can be demixed in a nontrivial way

P = λE1 + (1 − λ)E2,

with E1, E2 ∈ E(H). Taking ϕ ∈ MP⊥, that is, in the orthogonal complement of the eigen-
space of P , so that Pϕ = 0, we have

0 = ⟨ϕ|Pϕ⟩ = λ⟨ϕ|E1 ϕ⟩ + (1 − λ)⟨ϕ|E2 ϕ⟩ ! λ⟨ϕ|E1 ϕ⟩ ! 0,

so that E1 ϕ = 0 ∀ϕ ∈ MP⊥. At the same time ∀ψ ∈ MP we have, from 1 − P = λ(1 −

E1) + (1 − λ)(1 − E2),

0 = ⟨ψ |(1 − P )ψ⟩ = λ(1 − ⟨ψ |E1 ψ⟩) + (1 − λ)(1 − ⟨ψ|E2 ψ ⟩) ! λ(1 − ⟨ψ |E1 ψ ⟩) ! 0,

so that E1 and P coincides on a SONC and therefore E1 = P . Projections are therefore

extreme elements in the convex set of effects. Suppose vice versa that E ∈ E(H) is not a
projection, so that E 2 =
/ E, then it cannot be extreme, as we show providing an explicit
demixture. Set E1 = E , and E2 = 2E − E 2, both are effects since for 0 # E # 1 we have
2

E 2 # E and (1 − E2) = (1 − E)2, and we have the decomposition

1
E = (E1 + E2).
2
◦

@ B. Vacchini - Unimi & INFN

2 States 29

We note that positivity of the effects is all that we need to comply with the statis-
tical interpretation of quantum mechanics, so that given arbitrary statistical operator ρ
and effect E one has the probability formula TrρE, giving the probability that a state
preparation corresponding to ρ triggers the registration corresponding to the effect E.
We have as required for the probabilistic interpretation 0 # TrρE # 1, while idempotency
of E is not necessary and generally does not apply. By means of example we can char-
acterize effects in H = C2 as follows. A generic operator on this space can be written as
1
E = (ε1 + e · σ),
2
where in order to be self-adjoint one has ε ∈ R, e ∈ R3 and 0 # E # 1 leads to the fol-
1
lowing inequality for the two eigenvalues 2 (ε ± ∥e∥)
1 1
0 # (ε − ∥e∥) # (ε + ∥e∥) # 1,
2 2
implying ∥e∥ # 1 and
∥e∥ # ε # 2 − ∥e∥.
Generalized probability measures
So far we have considered the most general preparation procedure as given by a sta-
tistical operator ρ ∈ S(H), and we have thus identified the set of operators E ∈ E(H) as
the most general yes-no elementary measurement. The key point being that the two sets
are one dual to the other. A natural question is whether we can take the reverse route.
Taking effects as the space of basic events to be observed in a measurement, with its
partial sum operation, one can ask what is the expression of a generalized probability
measure defined on this space. Indeed a state can be naturally identified with a proba-
bility measure on the space of observables, it assigns to each elementary event the prob-
ability of its occurrence once fixed the preparation. Let us first better specify what we
define as generalized probability measure. We say that
ν: E(H) → [0, 1]
E " ν(E)
is a generalized probability measure on the set of effects provided
i. 0 # ν(E) # 1
ii. ν(1) = 1
! ! !
iii. ν( ni=1 Ei) = ni=1 ν(Ei) provided ni=1 Ei # 1.
It turns out that such a generalized probability measure can be identified with a statis-
tical operator.
◦
Theorem (Busch, 2003) Let ν be a generalized probability measure on the set of
effects E(H), then there exists a statistical operator ρ ∈ S(H) such that ν(E) = ν ρ(E) =
TrρE for all E ∈ E(H).
The proof is not difficult and is similar to the one of the previous theorem. The basic
fact is again a duality relation, while B(H) = T ′(H), in the infinite dimensional case one
has the strict inclusion B ′(H) ⊃ T (H), in particular T (H) can be identified with the
linear functionals on B(H) which besides being continuous are also normal, which is yet
another regularity condition. Given this fact the idea is to show that ν can be extended
to a functional with values in [0, 1] over the whole linear space B(H) ⊃ E(H), which is
furthermore continuous and normal. It can therefore be identified with an element of
T (H), and the restrictions on the range of the functional imply that this element is a
statistical operator.
30 Table of contents

One might wonder whether restricting the generalized probability measure to act on
the set of projections, as in the more standard formulation of quantum mechanics, leads
to consider more general states than statistical operators. The answer is negative, and
corresponds to a famous theorem by Gleason, whose proof is very difficult. Let us first
precisely define what we mean by such a measure on the lattice of projections. We say
that

µ: P(H) → [0, 1]
P " µ(P )

is a generalized probability measure on the space of projections provided

i. 0 # µ(P ) # 1

ii. µ(1) = 1
! !
iii. µ( ni=1 Pi) = ni=1 µ(Pi) provided PiP j = 0 for i =
/ j, that is the projections are
orthogonal.

It turns out that if dimH ! 3 such a generalized probability measure can be identified
with a statistical operator.
◦
Theorem (Gleason, 1957) Let µ be a generalized probability measure on the set of
projections P(H) ⊂ B(H) and dimH ! 3, then there exists a statistical operator ρ ∈ S(H)
such that µ(P ) = µ ρ(P ) = TrρP for all P ∈ P(H).
◦

Dispersion-free states
To better grasp the difference between classical and quantum probability, as well as
between projections and effects, let us discuss the issue of dispersion free states. We say
that a state in a statistical theory is dispersion free provided the variance of any observ-
able calculated with this state is zero, so that each outcome is predicted with certainty.
Consider in the classical case R as measure space, then the extreme elements of the
space of probability measures on R are those with support reduced to a point, be it x0,
and they can be formally expressed through a density given by a Dirac delta function
dµ(x) = δ(x − x0)dx. One can immediately check that this measure has zero variance, so
that this state assigns to each random variable f (x) the deterministic value f (x0), the
statistical aspect of the description is washed out. For this class of pure extreme states
no observables exhibit a nontrivial statistics. On the other hand, still within the classical
description, consider a finite sample space Ω, so that as discussed above states are of the
form p = (p1, ..., pN ), and pure states are of the form e1 = (1, 0, ..., 0). In this setting one
easily identifies the basic elementary observables corresponding to projections and
effects. Projections are N -dimensional vectors q whose entries q j are either 0 or 1,
effects are given by vectors f whose entries f j satisfy 0 # f j # 1. As a result if one con-
siders generalized observables in the sense of effects, also in the classical case there are
no dispersion free states. For every p one!can find an effect f such that the probability
formula which now simply reads f·p = N j =1 f j p j gives a value strictly different from
either 0 or 1. However, restricting to observables given by projections one still has dis-
persion free states, coinciding with the pure states {e j } j =1,...,N , indeed in this case q·p
only takes the values 0 or 1.

@ B. Vacchini - Unimi & INFN

2 States 31

Let us now consider the quantum case. It immediately appears that taking effects for
the description of elementary observations, there are no dispersion free states. Indeed
1
given any state ρ ∈ S(H) one can consider the effect E = 2 ρ, and the probability formula
1
gives the result Tr Eρ = 2 Tr ρ2 < 1, strictly less than one as follows from the property of
purity. One might still ask if there exist dispersion free states in quantum mechanics
when restricting to the set of projections. To show that this is not the case suppose
there exist ρ such that
⟨A2⟩ = ⟨A⟩2
for any self-adjoint operator A ∈ B(H), or equivalently
TrρA2 = (Tr ρA)2.
Of course for an observable A with a non empty point spectrum one can certainly find a
state such that the dispersion of A is equal to zero, just by taking a pure state fixed by
an eigenstate of A. The point is however whether the statistical description becomes
irrelevant for all observables. To show that this is not the case consider A = P φ for a cer-
tain one-dimensional projection P φ. If ρ would be dispersion free then one would have
∀φ ∈ H
(Tr ρP φ)2 = Tr ρP φ2 = Tr ρP φ ,
so that Tr ρP φ is equal to either 0 or 1. Since Tr ρP φ = ⟨φ|ρφ⟩, it cannot be always equal
to 0, otherwise ρ should be the zero operator, neither it can always be equal to 1, other-
wise ρ should be the identity operator. Therefore there exists at least a ψ ∈ H such that
⟨ψ |ρψ⟩ = 1 and therefore Trρ(1 − P ψ) = 0, in particular Tr ρP χ = 0 ∀χ⊥ψ. Let us con-
sider therefore the vector Φ = cosαχ + sinαψ, we have
TrρPΦ = sin2α + sinαcosα[⟨χ|ρψ⟩ + ⟨ψ|ρχ⟩],
this function is continuous in α and takes the values 0 for α = 0, and 1 for α = π/2, it
therefore passes through intermediate points, leading to Hilbert space vectors for which
the expectation value is neither 0 nor 1.
Maximal entropy state
In the general case, especially for large systems, the information on a state does not
correspond to the fact that it has been obtained by performing a projective measure-
ment on an impinging state, possibly selecting according to the outcomes. The available
information rather has the form of knowledge of mean values of a selection of observ-
ables, which we call relevant observables. Note that variances can also be included, con-
sidering the mean values of squares of operators. Of course this is in general a restricted
set, if all mean values were known, this would suffice to fix the state uniquely. Relevant
observables are typically given by conserved quantities or the integral of densities of con-
served quantities over suitable macroscopic space regions. Also variances can be taken
into account, considering both an operator and its square. Let
M = {Ri }i
be the set of relevant observables, so that each Ri is a self-adjoint operator, and let
m = {ri }i
be the set of real numbers corresponding to the assigned mean values. Let us further
introduce the convex set of statistical operators compatible with these mean values
M(H) = {ρ ∈ S(H) |TrρRi = ri ∀Ri ∈ M }.
32 Table of contents

◦
Among the states in M(H) we want to characterize the state which takes into
account this information but includes no further bias, namely as discussed above the
state which satisfies these constraints and has maximum von Neumann entropy. We
therefore face a maximization problem with constraints, given by the assigned mean
values as well as normalization. We will see that the solution to this problem indeed is
a statistical operator, in particular a strictly positive operator. Given the expression of
the von Neumann entropy
S(ρ) = −kBTrρlogρ
we look for stationary points of the functional
%
S ′(ρ) = S(ρ) + kB(Ω + 1)⟨1⟩ ρ − kB βi ⟨Ri ⟩ ρ
< i =
%
= −kBTrρ logρ − (Ω + 1) + βiRi ,
i
where the Lagrange
! parameters are real. Since the statistical operator can always be
written as ρ = α ρα |α⟩⟨α| with {|α⟩} a SONC, we have to consider variations both
with respect to the choice of SONC, which can be implemented via unitary transforma-
tions, and with respect to the distribution of the eigenvalues. Let us first consider varia-
tions with respect to the SONC. Since such unitary transformations are induced by a
self-adjoint operator W we can write
i
δρ = [W , ρ]δθ
!
with W = W † and δθ the parameter characterizing the infinitesimal transformation. We
therefore come to
> < = ?
%
δS ′(ρ) = −kBTr δρ logρ − (Ω + 1) + βiRi + δρ
2 < i
% =3
= −kBTr δρ logρ + βiRi ,
i
where we have used the fact that δρ is traceless. Further exploiting the explicit expres-
sion of δρ and the useful relation
Tr[A, B]C = Tr A[B, C]
due to cyclic invariance of the trace operation one has
4 5 2 3
i i %
δS ′(ρ) = −kBTr [W , ρ]logρ δθ − kBTr [W , ρ] βiRi δθ
! !
2 i
6 % 73
i
= −kBTr W ρ, βiRi δθ.
!
i
Asking the variation to be zero for arbitrary self-adjoint W we can make in particular
the choice
6 % 7
i
W = ρ, βiRi ,
!
i
so that a necessary condition for the functional S ′(ρ) to be stationary is given by
6 % 7
i
ρ, βiRi = 0,
!
i

@ B. Vacchini - Unimi & INFN

2 States 33

so that in particular !
there exist a common spectral decomposition of the commuting
self-adjoint operators i βiRi and ρ. Writing to fix the notation
% %
βiRi = Rα |α⟩⟨α|,
i α
this implies that the ρ we are looking for can be written as
%
ρ = ρα |α⟩⟨α|,
α
!
for suitable weights ρα. The commutativity of i βiRi and ρ is actually also a sufficient
condition for stationarity of the functional. Indeed suppose
%
ρ = ρ γ |γ ⟩⟨γ |,
γ

for a certain SONC {|γ ⟩}, we then have

% < % =
S ′(ρ) = −kB ρ γ logρ γ − (Ω + 1) + ⟨γ | βiRi |γ ⟩
γ i
and the latter expression due to the variational
! theorem in quantum mechanics is sta-
tionary provided the |γ ⟩ are eigenvectors of i βiRi.
◦
Indeed the functional W [ϕ] = ⟨ϕ|Hϕ⟩ is stationary if ϕ is an eigenvector of H. To
show this we consider stationarity of W [ϕ] under variations of the state vector which
have to preserve the norm. This can be included through a constraint, so that we have
to verify the relation
0 = δ(W [ϕ] − λ∥ϕ∥2)
= ⟨δϕ|Hϕ⟩ + ⟨ϕ|Hδϕ⟩ − λ⟨δϕ|ϕ⟩ − λ⟨ϕ|δϕ⟩
= 2ℜ(⟨δϕ|Hϕ⟩ − λ⟨δϕ|ϕ⟩)
= 2ℜ⟨δϕ|(H − λ)ϕ⟩.
Since this must be true ∀δϕ, it follows that one has (H − λ)|ϕ⟩ = 0, so that ϕ is an
eigenvector of H. As a result the condition
6 % 7
i
ρ, βiRi = 0,
!
i
is both necessary and sufficient in order to have stationarity with respect to variation of
the SONC. We thus have
%
S ′(ρ) = −kB ρα {logρα − (Ω + 1) + Rα }
α
and varying with respect to the eigenvalues
δS ′(ρ) = −kB{logρ β − Ω + R β }δρ β
and asking this quantity to be zero for all β we obtain
ρα = eΩ−Rα
so that the eigenvalues are indeed non-negative, and furthermore strictly positive. The
solution of the stationary problem is indeed a statistical operator. We have
%
ρ = eΩ−Rα |α⟩⟨α|
α !
= eΩe− iβiRi
34 Table of contents

Normalization implies
!
e−Ω = Tr e− iβiRi

so that defining the quantity Z according to standard notation in statistical mechanics

!
Ω = −log Tr e− iβiRi

= −log Z
the state which makes the functional S ′ stationary is identified to be
!
e− iβiRi
w = .
Z
We now show that among the states in M(H), that is with the assigned mean values, it
has indeed maximum entropy. Suppose ρ ∈ M(H), exploiting the Klein inequality
S(ρ) # −kBTr ρlog w,
we have
Trρlogρ ! Trρlog
6 % w 7
= Trρ − βiRi − log Z
6 % i 7
= Tr w − βiRi − log Z
i
= Tr w log w,
since both these operators have the same mean values for the relevant observables. As a
consequence
S(w) = −kBTr w log w
! −kBTrρlogρ
= S(ρ),
for all ρ ∈ M(H). We have in particular the expression
%
S(w) = kB βiri + kBlog Z.
i
We have thus identified the state which maximizes the entropy for fixed mean values.
This state is of Gibbs form and its expression has now been obtained adopting the max-
imum information principle to identify the state. Given its structure in terms of the rele-
vant variables and the Lagrange parameters let us consider the equations which implic-
itly fix the parameters in terms of the mean values. Starting from
%
Ω = −log e−Rα
α
we have
! ∂Rα −Rα
∂Ω α ∂βi e
= ! −R ,
α e
∂βi α

but
∂Rα ∂ %
= ⟨α| β jR j |α⟩
∂βi ∂βi
j
= ⟨α|Ri |α⟩,

@ B. Vacchini - Unimi & INFN

3 Observables 35

where always because of the variational theorem one ! need not take a variation with
respect to the basis, since the |α⟩ are eigenvectors of j β jR j . Finally we have

∂Ω ∂log Z
= −
∂βi ∂β
! i −Rα
= α ⟨α|R
! −R
i |α⟩e

α e
α
!
− jβ jR j
Tr Ri e !
= − jβ jR j
Tr e
= ⟨Ri ⟩
= ri.

The parameters appearing in the expression of the statistical operator are therefore
implicitly fixed by the solution of the equations
!
Ω = −log Tr e− iβiRi

∂Ω
= ri ,
∂βi
which can have solutions provided the set m = {ri } provides a set of expectation values
compatible with quantum mechanical requirements.
◦
◦
As a last remark note that in order to evaluate the derivatives we have actually used
the formula
∂ ∂A(λ) A(λ)
Tr eA(λ) = Tr e ,
∂λ ∂λ
which follows due to ciclic invariance of the trace from the following general relationship
keeping track of the fact that the operators ∂A(λ)/∂λ and A(λ) do generally not com-
mute
# 1
∂ A(λ) ∂A(λ) (1−u)A(λ)
e = du euA(λ) e .
∂λ 0 ∂λ
◦
◦

3 Observables
So far we have given the mathematical representative of equivalence classes of basic
experiments with yes-no answer. In terms of these basic building blocks we can now con-
struct the general notion of observable with arbitrary outcome space.
◦
Let us start from a discrete setting. Suppose that each outcome is represented by an
effect Ej corresponding to a yes-no signature in the measurement apparatus, e.g. a click.
Given a state ρ each outcome takes place with probability TrρE j , and since only one
event takes place at a time we have
%
TrρE j # 1,
j
36 Table of contents

and if!the inequality is not saturated we can always introduce a further effect given by
1 − j E j , corresponding to no event in the measurement apparatus given that the
preparation apparatus has fired. Indeed we have seen that the linear sum of effects
which still stays below the identity is also an effect, so that they can be observed
together, being different exclusive
! outcomes of the same observable. Thus we assume
without loss of generality that j TrρE j = 1. Such an observable is therefore described
!
by a collection of operators {E j } such that 0 # TrρE j # 1 and j TrρE j = 1. Since
these relations must hold for any state ρ, they uniquely identify a collection of effects
summing up to the identity.
Positive operator-valued measure
Let us put this result in a general framework. Consider the outcome space of a given
physical quantity, be it (Ω, F ), where F is the σ-algebra of measurable outcomes in the
sample space Ω. In the case of discrete or numerable outcomes, Ω will consist of a finite
or denumerable set of real numbers. In the general case Ω = Rn for a certain n, and F is
the Borel σ-algebra of open sets B(Rn).
We define as positive operator-valued measure (POVM) or probability operator-
valued measure a map
F : F → E(H) ⊂ B(H)
such that
i. 0 # F (M ) # 1 for any M ∈ F
ii. F (∅) = 0, F (Ω) = 1
@ !
iii. F ( i Mi) = i F (Mi) in the weak topology for any sequence of disjoint sets in
F , i.e. Mi ∩ M j = ∅ for i =
/ j.
◦
These properties express the fact that a POVM takes values in the set of positive
operators, between zero and unit so as to extract probabilities, and is σ-additive as a
measure should be. The weak topology allows for a natural connection with classical
scalar measures.
Indeed, given an arbitrary ρ ∈ S(H) and an arbitrary POVM F one immediately
obtains a classical probability measure according to the formula
µFρ (M ) = TrρF (M ),
as granted by linearity and continuity of the trace operation.
We take POVMs as the most general description of observables, and the classical
measure µFρ then provides the probability distribution of the observables associated to F
given a preparation ρ, that is the distribution of its possible outcomes once the state is
ρ. Otherwise stated, the map F is a POVM provided the map F ∋ M 6→ TrρF (M) is a
probability measure for any ρ ∈ S(H). To substantiate the fact that positive operator-
valued measures do indeed provide the general notion of observable, let us introduce the
convex set Prob(Ω) of all classical probability measures on the outcome space (Ω, F ).
We can show that each map associating to any element ρ of the convex set of quantum
states S(H) an element of the convex set of classical states Prob(Ω), namely an observ-
able compatible with the statistical interpretation of quantum mechanics, can be identi-
fied with a POVM. We formulate this statement as a theorem.
Theorem Given a POVM F on the outcome space (Ω, F ) the map
ΦF : S(H) → Prob(Ω)
ρ " TrρF (·)

@ B. Vacchini - Unimi & INFN

3 Observables 37

provides an affine map from the convex set of quantum states to the convex set of clas-
sical probability distributions over (Ω, F ). Vice versa, any such affine map can be real-
ized as Φ = ΦF for some POVM F .

Proof. Given a positive operator-valued measure F one immediately checks that for
any ρ the measure µFρ (M ) = TrρF (M ) belongs to Prob(Ω), and thanks to linearity of
the trace operation the map is affine in its dependence on ρ. To show that the converse
holds suppose we are given an affine map

Φ: S(H) → Prob(Ω)
ρ " Φ[ρ]

where each Φ[ρ] is a probability distribution over Ω. That is to say 0 # Φ[ρ](M ) # 1,

Φ[ρ](∅)
@ = !0, Φ[ρ](Ω) = 1 together with the proper additive behavior
Φ[ρ]( i Mi)= i Φ[ρ](Mi) for disjoint sets. For fixed M ∈ F we can consider the map

ΦM : S(H) → [0, 1]
ρ " ΦM [ρ] ≡ Φ[ρ](M )

which since Φ is affine is also affine in its dependence on ρ and takes values in the
interval [0, 1].
◦
As shown before such a map can be represented as an effect, let us call it F (M ) since
it depends on M , by expressing it as ΦM [ρ] = TrρF (M ). It remains to show that the
dependence on M is such as to make F (·) a positive operator-valued measure. Indeed
being an effect we already know 0 # F (M ) # 1 ∀M ∈ F. Moreover since Φ[ρ] ∈ Prob(Ω)
we immediately have F (∅) = 0 and F (Ω) = 1. Let us now consider a sequence of disjoint
sets in F , i.e. Mi ∩ M j = ∅ for i =
/ j. We have, since Φ[ρ] is σ-additive
4A 5 @
TrρF Mi = Φ iMi[ρ]
i 4A 5
= Φ[ρ] Mi
% i
= Φ[ρ](Mi)
%i
= ΦMi[ρ]
i %
= Trρ F (Mi)
i
@ !
and therefore σ-additivity in the weak operator topology F ( i Mi ) = i F (Mi). $

To connect with the standard notion of observable as a self-adjoint operator let us

consider the very special case in which F (M ) = F 2(M ) ∀M ∈ F , so that the measure
actually takes values in the set of projections. We therefore define as projection-valued
measure (PVM) a map

E: F → P(H) ⊂ B(H)

such that
i. E 2(M ) = E(M ), 0 # E(M ) # 1 for any M ∈ F
ii. E(∅) = 0, E(Ω) = 1
38 Table of contents

@ !
iii. E( i Mi) = i E(Mi) in the weak topology for any sequence of disjoint sets in
F , i.e. Mi ∩ M j = ∅ for i =
/ j.
Observables which do correspond to positive operator-valued measures which are in par-
ticular projection-valued measures are called sharp observables.
◦
To stress the fact that the basic difference between POVM and PVM lies in idempo-
tency and commutativity of the operators in the range, let us consider the following
statement. Let A be an observable with outcome space (Ω, F ), then the following state-
ments are equivalent
i. A is a PVM
ii. A(M)A(N ) = A(M ∩ N ) for all M , N ∈ F
iii. A(M)A(M C ) = 0 for all M ∈ F
where the second statement is known as multiplicativity and implies that the range of a
PVM is given by commuting projections. To show the equivalence of the statements we
show that each one implies the one that follows.
i.⇒ii. Since M ∩ N ⊆ M and M ⊆M ∪ N from the properties of PVM we have the
ordering A(M ∩ N ) # A(M ) # A(M ∪ N ), which implies A(M ∩ N )A(M ) = A(M ∩ N ) as
well as A(M )A(M ∪ N ) = A(M ). For a PVM we further have

A(M ∪ N ) + A(M ∩ N ) = A(M ) + A(N )

so that multiplying by A(M ) we have

A(M) + A(M ∩ N ) = A2(M ) + A(M )A(N )

and therefore thanks to idempotency

A(M )A(N ) = A(M ∩ N )

= A(N ∩ M )
= A(N )A(M )

which shows multiplicativity and commutativity.

ii.⇒iii. Immediately follows by taking N = M C .
iii.⇒i. Follows from exploiting once again the properties of being an observable

0 = A(M )A(M C )
= A(M )[1 − A(M )]
= A(M ) − A2(M )

so that we have idempotency and therefore A is a PVM.

◦
The crucial difference is idempotency, which is not necessary in order to obtain a
statistical description of the measurement outcomes in terms of a classical probability
measure µE ρ . It has however important consequences from the mathematical point of
view. For the case of projection-valued measures we know that there is a one to one cor-
respondence with self-adjoint operators. Indeed given a projection-valued measure E A
one can construct a self-adjoint operator A according to
#
A = xdE A(x),
R

@ B. Vacchini - Unimi & INFN

3 Observables 39

where we have assumed that A takes values in R. An important point is that all
moments of the probability distribution obtained from E A once fixed a state ρ can be
expressed directly in terms of this operator, which is therefore of great significance.
Indeed fixed an arbitrary state ρ we have, denoting by Meanρ(E A) the first moment of
the classical probability distribution obtained from the projection-valued measure E A
#
A
Meanρ(E ) =
A xdµEρ (x)
R
= Tr Aρ,
A
ρ (M ) = TrρE (M ). Thanks to the identity
where µE A

#
f (A) = f (x) dE A(x),
R

so that in particular
#
A2 = x2 dE A(x),
R

we can further express the variance as

Var ρ(E A) = Tr A2 ρ − (Tr Aρ)2.

Similarly any moment or combination of moments of the statistical distribution can be

directly expressed in terms of powers of the operator A. This is no more true in the case
of a POVM. The integral
#
(1)
B = xdF (x)
R

still generally defines an operator, but the latter is not necessarily self-adjoint, and even
if this is the case the operator corresponding to the second moment
#
B (2)
= x2 dF (x)
R
B C
is generally not the square of the first B (2) =/ B (1) 2. This opens the complicated
moments problem. As a general fact the focus remains on the positive operator-valued
measure itself and cannot be shifted to an operator.
As a first example of an observable in the sense of a proper POVM, let us consider
the following. We take H = C2 and consider the following POVM with three outcomes
1
F ({k }) = (α 1 + mk·σ)
2 k
with k = 1, 2, 3 and in order for F ({k }) to be an effect we further need as shown before
αk ∈ R, mk ∈ R3 together with ∥mk ∥ # αk # 2 − ∥mk ∥.
◦
In order for the triple of operators {F ({k })}k=1,2,3 to be a POVM we further need
3
%
F ({k}) = 1
k=1
! !
implying 3k=1 αk = 2, and 3k=1 mk = 0, so that the three dimensional vectors mk are
linearly dependent, lying in a plane.
◦
40 Table of contents

2
A choice compatible with these conditions corresponds to αk = 3 for k = 1, 2, 3
8 9 8 9 8 9
2 1 1 1 1
together with m1 = 0, 3 , 0 , m2 = + √ , − 3 , 0 , m3 = − √ , − 3 , 0 so that the three
3 3
vectors are coplanar, lie in the x-y plane and form an angle of 120 degrees according to
1
(mk/∥mk ∥)·(m j /∥m j ∥) = − 2 for j = / k. The probability formula if the preparation is
1
described by ρ = 2 (1 + r · σ) then implies
P ({k }) = TrρF
6 ({k}) 7
1 1
= Tr (1 + r · σ) (αk1 + mk·σ)
2 2
1 1
= + r · mk
3 2
1
= (1 + r · mk)
3
where r = (x, y, z) is a vector with norm less or equal to one and mk = mk/∥mk ∥,
where ∥mk ∥ = 2/3. In particular we have P ({k}) # 2/3, corresponding to the fact that
the outcome is never certain, whatever the state, as it should be corresponding to a joint
measurement of spin along two distinct directions. In this case we can also completely
spell out the map ΦF which associates to a quantum state a classical probability distrib-
ution thanks to the POVM
ΦF : S(H) → Prob(Ω)
ρ " TrρF (·).
For the case at hand Ω = {1, 2, 3}, so that an element p ∈ Prob(Ω) simply corresponds to
a probability vector, and the map explicitly reads
ΦF : ρ " p = (TrρF ({1}), TrρF ({2}), TrρF ({3}))
with
2 2 √ 3 2 √ 33
1 1 3 1 1 3 1
p = (1 + y), 1+ x− y , 1− x− y .
3 3 2 2 3 2 2
◦
1
We thus have in particular ∥p∥ # √ . The image of S(H) through ΦF has to be a
2
convex subset of Prob(Ω), a proper subset to comply with the constraints inherent in
the quantum states. In the case at hand Prob(Ω) is the set of three components proba-
bility vectors.
! In three dimensional space such vectors lie in the first octant, and the
constraint 3 pk = 1 implies the restriction to a plane, so that the points lie in a tri-
k=1
angle with corners (1,0,0), (0,1,0) and (0,0,1). The image of S(H) through ΦF is the
1
intersection of this triangle with the sphere ∥p∥ # √ .
2
◦
◦

Note the occurrence of a proper POVM in the space H = C2 for the case at hand is
implied by the fact that we have three outcomes. Indeed since the range of a PVM is
given by orthogonal projections, for a PVM in this space one can only have the two out-
1
comes corresponding to two orthogonal projections P±n = 2 (1 ± n·σ), so that the pre-
vious probability formula
1
TrρF ({k}) = (1 + r · mk)
3

@ B. Vacchini - Unimi & INFN

3 Observables 41

which is bound by 2/3 is to be compared with

1
TrρP±n = (1 ± r · n)
2
which can attain the value one corresponding to a sharp statement. Note further that
we have
1
F ({k }) = (1 + mk·σ)
36 7
2 1
= (1 + mk·σ)
3 2
2
= Pmk.
3
The single effects are proportional to projections along the given direction, with a factor
less than one.
In order to see how an unsharp spin measurement might arise, let us consider the fol-
lowing slightly more refined description of the Stern Gerlach experiment. We consider
the Hilbert space H = L2(R3) ⊗ C2 where both centre of mass and internal degrees of
freedom are described. We devise a preparation procedure in two steps as follows. First
one prepares a pure state of the form
Φin = ψ ⊗ ϕ
= ψ ⊗ (c+v+ + c−v−),
where ψ ∈ L2(R3) is a suitable wave packet describing the centre of mass degrees of
freedom, while {v+, v−} denote a basis in C2. The complex weights satisfy |c+|2 + |c−|2 =
1 according to normalization. As a second step a inhomogeneous magnetic field is
applied, in order to sort the spatial direction of the atoms in the sample according to
their magnetic moment. We describe this transformation through a unitary operation as
follows
Φout = U Φin
= c+ ψ+ ⊗ v+ + c− ψ− ⊗ v−,
where ψ± denote the deflected wave packets, which are generally non-orthogonal
⟨ψ+|ψ−⟩ =/ 0. These two steps combined together then lead to our preparation procedure
described by Φout. We now suppose to observe on the overall system the following PVM
on L2(R3) ⊗ C2 with two outcomes
E± = P± ⊗ 1C2,
where P± ∈ P(L2(R3)) correspond to projections on the upper and lower half-planes
respectively. The probability to have localization in the upper or lower half-plane thus
becomes
P (±) = TrL2(R3)⊗C2 E±PΦout
= ⟨Φout|P± ⊗ 1C2Φout⟩
= ⟨Φout|E±Φout⟩
= ∥|E±Φout∥2
= |c+|2∥P± ψ+∥2 + |c−|2∥P± ψ−∥2,
where we have used orthogonality of v+ and v−, and upon setting
F± = Pv+∥P± ψ+∥2 + Pv−∥P± ψ−∥2
42 Table of contents

which identifies a POVM with two outcomes on C2 one also has

P (±) = Tr F±P ϕ.
Thus the projective measurement on the overall system do correspond to a POVM for
the subsystem corresponding to the internal degrees of freedom only. Note that F± is a
proper POVM unless ∥P± ψ+∥2 and ∥P± ψ−∥2 are exactly equal to 0 or 1. In this case,
which corresponds to a sharp localization in the upper/lower plane, the observable
becomes a PVM. Coming back to the Stern Gerlach experiment, the unsharp situation
is realized when due to the width of the wave packet, one can obtain counts in the upper
plane corresponding to atoms which according to their magnetic moment have been
deflected in the down direction. A sharp observable corresponds to a situation in which
one can perfectly discriminate among the two signals, and the pattern on the screen is
given by two sharp lines, with no deposition of atoms in between the region with highest
density. Let us note that we can also write
F± = TrL2(R3) PΦin U † E±U
= ⟨ψ |U † (P± ⊗ 1C2)Uψ ⟩.
◦
Naimark’s dilation
This is actually a general situation when one has a system interacting with some
environment. A projective measurement on the overall system corresponds in general to
a POVM giving an effective description of the statistics of a certain physical quantity of
the system only, according to the external noise induced by the interaction. That in this
setting a PVM generally becomes a POVM is immediately seen as follows. Consider a
PVM on the system ES (M ), a factorized system environment initial state ρS ⊗ ρE , an
evolution described by a unitary operator U and a measurement on the side of the
system only, we then have for the distribution of the outcomes
TrS TrE {U (ρS ⊗ ρE )U †(1S ⊗ EE (M ))} = TrSρS TrE {ρEU †(1S ⊗ EE (M ))U }
= TrSρSFS (M ),
with
FS (M ) = TrE {ρEU †(1S ⊗ EE (M ))U }.
◦
According to this expression one can check that indeed FS (M ) is an observable in
the sense of a POVM whenever ES (M ) is a PVM. Actually it can be shown, as follows
by a theorem due to Naimark, that any POVM can be seen as obtained by a PVM on a
larger Hilbert space. The considered example is therefore in a sense paradigmatic. Note
however that some, but not all, unsharp observables arise as smeared version of a sharp
observable. The unsharpness brought about by coarse-graining may or may not admit
the kind of ignorance interpretation familiar from classical physics. In general the
unsharpness is the reflection of a genuine quantum indeterminacy.
We now provide a first formulation of Naimark’s dilation theorem, stating that any
POVM can be dilated to a PVM in a larger Hilbert space. A more general proof will be
provided later on, exploiting the notion of complete positivity. We restrict ourselves for
the time being to a finite Hilbert space H = Cn, and consider a POVM with finite out-
come space Ω = {1, ..., N }. A POVM on such a space can be identified with a map asso-
ciating to each k ∈ 1, ..., N a semipositive definite n × n matrix F ({k}) whose greatest
!
eigenvalue lies below one. One further has the normalization condition N k=1 F ({k}) =
1n×n.

@ B. Vacchini - Unimi & INFN

3 Observables 43

◦
Without loss of generality we take the effects F ({k}) to be given by rank one opera-
tors, so that F ({k }) =|ϕk ⟩⟨ϕk | with {ϕk } ∈ Cn, generally not orthogonal among them-
!
selves. Due to the constraint N k=1 |ϕk ⟩⟨ϕk | = 1n×n one must have N ! n, that is the
number of outcomes have to be higher than the dimensionality of the space. If N = n
the {ϕk } have to provide an orthogonal set, so that the F ({k }) are actually a collection
of mutually orthogonal projections, and the POVM is actually already a PVM. Suppose
now N > n, which is the case of interest, and consider the N × n matrix given by
⎛ ⎞
ϕ1
M = ⎝ ··· ⎠,
ϕN
where each element belongs to Cn and can therefore be expressed in terms of n compo-
nents {ϕk1, ..., ϕkn }. This matrix realizes an isometry thanks to the normalization con-
straint M †M = 1n×n.
◦
Its columns can be seen as n orthonormal vectors in a N dimensional space. It is
therefore possible to find a further matrix M ′ of dimension N × (N − n) such that the
matrix U = (M , M ′), where the first n columns come from M and the latter N − n
columns come from M ′, is a unitary matrix in a Hilbert space of dimension N . The
matrix M ′ is not uniquely identified, since there are different ways to add N − n
columns of other orthogonal vectors to the matrix M so as to make it unitary. The rows
of U therefore provide N orthonormal vectors which lead to a PVM in CN . Their pro-
jection on the original H = Cn recovers the starting POVM.
Note that this minimal construction asks an enlargement of the original Hilbert
space depending on N − n, the difference between the dimension of the outcome space of
the POVM and the dimension of the original Hilbert space.
◦
◦
Coexistence
We now want to discuss the problem of considering different observables together,
and see the major changes taking place in this respect when moving from PVM to
POVM. We first introduce the notion of range of an observable F on the outcome space
(Ω, F ) as the set

ran(F ) = {F (M), M ∈ F } ⊆ E(H).

We further say that the effects E1, E2, ..., En are coexistent if there exists an observable
F in the sense of POVM such that

{E1, E2, ..., En } ⊆ ran(F ).

!
This holds true if nj=1 E j # 1, since in this case as in previous examples we can con-
!
struct an observable adding the effect 1 − nj =1 E j . But this is only a sufficient condi-
tion, as we shall see by means of example. In this case the effects can be realized
together within a POVM, in general the fact that they are coexistent means that they
can be read within a POVM. On the same footing we say that two or more POVM are
coexistent if both their range are contained within the range of a single POVM. Other-
wise stated one can find a third observable whose statistics contains those of the other
two.
44 Table of contents

In order to clarify this notion, let us first remain in H = C2, and consider a further
spin observable, which we can describe as a spin direction observable, realized by the fol-
lowing POVM with outcome space (Ω, F ) where Ω is the surface of the sphere of radius
one in three dimensions, and F the σ-algebra naturally induced by the Borel σ-algebra
in R3
#
sphere 1
F (M ) = dn(1 + n·σ),
4π M

where n is a unit vector fixing a given direction, and the integral is over the solid angle.
The defining properties of a POVM can be easily checked.
For any direction m we can further consider the two hemispheres with poles ±m, let
us call them M±, and consider the spin observable with two outcomes, which provides a
discretization of the previous observable
#
hemisphere 1
Fm (±) = dn(1 + n·σ)
4π M±
4 5
1 1
= 1 ± m·σ
2 2
6 7
11 1 1
= 1+ (1 ± m·σ) .
22 2 2

Note that the two outcomes can be seen as a convex combination of a projection and
half the identity, the former corresponding to a sharp measurement and the latter corre-
sponding to some extra noise, which reduces the accuracy. Once again one can check
that this is a POVM. Now for any unit vector m we have different hemispheres and
therefore different POVMs. All of them are obviously within the range of F sphere, since
F hemisphere(±) = F sphere(M±), so that they are coexistent. However one can find effects
in their range whose sum is no more an effect, ! and therefore cannot be in the range of
any observable, so that indeed the condition nj =1 E j # 1 for coexistence is indeed only
sufficient.
It is further worth to indicate the relationship between coexistence and commuta-
tivity, which is the standard notion for joint measurability of observables in the stan-
dard framework of quantum mechanics. When we speak of commutativity of two observ-
ables we mean the standard notion of commutativity of the effects in their range. Given
two observables we have that coexistence coincides with commutativity if at least one of
the two is actually a PVM. If both are POVM, commutativity implies coexistence, but
not vice versa. It is just the release of commutativity that significantly enlarges the class
of physical quantities that can be considered by means of POVM.
◦

Covariance
We now want to consider a few examples of POVM and PVM concentrating on posi-
tion and momentum, showing in particular how symmetry properties can be a very
important guiding principle in the determination of meaningful observables, once we
leave the correspondence principle focussed on quantum mechanics as a new mechanics
with respect to the classical one. As we shall see while for the case of either position or
momentum alone POVM are essentially given by a suitable coarse-graining with respect
to the usual PVM, if one wants to give statistical predictions for the measurement of
both position and momentum together the corresponding observable is given by neces-
sity in terms of a POVM.

@ B. Vacchini - Unimi & INFN

3 Observables 45

Let us first start by introducing the notion of mapping covariant under a given sym-
metry group G. As we will show, this notion is of great interest in many situations, both
for the construction of POVM and general dynamical mappings. Consider a measure
space X with a Borel σ-algebra of sets B(X ). Such a space is called a G-space if there
exists an action of G on X defined as a mapping that sends group elements g ∈ G to
transformation mappings µ g on X , realizing a group omomorphism between G and the
group of automorphisms of X

µ: G → Aut(X )
g " µg

so that group composition, inverse operation and identity are preserved

µ g µh = µ gh ∀g, h ∈ G
µe = 1X
(µ g) −1 = µ g −1

If furthermore G acts transitively on X , in the sense that any two point of X can be
mapped one into the other with µ g ′ for a suitable g ′ ∈ G, then X is called a transitive G-
space. Consider for example X = R3, then X is a transitive G-space with respect to the
group of translations. The elements of the group are 3-dimensional vectors acting in the
obvious way on the Borel sets of R3, i.e. µa(M ) = M + a for all a ∈ R3 and for all M ∈
B(R3). Consider as well a unitary representation U (g) of the same group G on a Hilbert
space H
|ψ g ⟩ = U (g)|ψ⟩ ψ∈H g ∈ G,

in terms of which one also has a representation of G on a space A(H) of operators

acting on H
A g = U †(g) A U (g) A ∈ A(H).

A mapping M defined on B(X ) and taking values in A(H) is said to be covariant with
respect to the symmetry group G provided it commutes with the action of the group in
the sense that
U †(g) M(X) U (g) = M(µ g −1(X)) ∀X ∈ B(X ) ∀g ∈ G.
◦
A symmetry transformation on the domain of the mapping is mapped into the sym-
metry transformation corresponding to the same group element on the range of the map-
ping. The action of G on X commutes with the automorphism group representation of
G on A(H) induced by the unitary representation of G on H.
Considering the case of a POVM on the measure space (Ω, F ), supposing the exis-
tence of an action µ g of the group G on the space Ω, the requirement of covariance can
be written as
U †(g) F (M ) U (g) = F (µ g −1(M )) ∀M ∈ F ∀g ∈ G,

and corresponds to commutativity of the following diagram

F
F → E(H)
µ g −1↓ ↓U g
F
F → E(H)
46 Table of contents

Position observable
According to the previous results we describe observables as POVM, and look for a
position observable. Rather than relying on the usual correspondence principle with
respect to classical mechanics, we want to give an operational definition of the position
observable describing localization, fixing its behaviour with respect to the action of the
relevant symmetry group, which in this case is the isochronous Galilei group, containing
translations, rotations and boosts that is to say velocity transformations. In order for a
measurement F x(M ) to be a position measurement we have to ask that the probability
distribution of the outcomes of F x(M ) performed on a state ρ and on the correspond-
ingly transformed one ρ g = U (g)ρU †(g) should satisfy the relation

Tr ρ gF x(µ g(M )) = Tr ρF x(M ) ∀ρ ∈ S(H) ∀M ∈ B(R3) ∀g ∈ G

where now G denotes the isochronous Galilei group. The group acts in the natural way
on the Borel sets of R3, and the covariance equations that we require for an observable
to be interpreted as position observable then become

U †(a) F x(M ) U (a) =F x (M − a) ∀a ∈ R3

U †(R) F x(M ) U (R) =F x(R−1M ) ∀R ∈ S O(3)
U †(q) F x(M ) U (q) =F x(M ) ∀q ∈ R3,
i i
a·p̂ − q·x̂
where we have used the definitions U (a) = e ! , U (q) = e ! .
◦
◦
The mapping F x to be interpreted as a position observable has to transform covari-
antly with respect to translations and rotations, and to be invariant under a velocity
transformation. These equations can also be seen as a requirement on the possible
macroscopic apparata possibly performing such a measurement. The apparatus used to
test whether the considered system is localized in the translated region M − a should be
in the equivalence class to which the translated apparatus used to test localization in the
region M belongs, and similarly for rotations. Localization measurements should instead
be unaffected by boost transformations. A solution of these covariance equations, that is
to say a POVM complying with the covariance conditions, is now a position observable.
If one looks for such a solution asking moreover that the POVM be in particular a
PVM, the solution is uniquely given by the usual spectral decomposition of the position
operator
#
E (M) = χM (x̂) =
x d3x|x⟩⟨x|
M

where χM denotes the characteristic function of the set M .

The basic idea goes as follows. If E x has to be a PVM, it identifies a triple of com-
muting self-adjoint operators. At the same time according to Stone’s theorem the uni-
tary representation U (q) of the group of boosts can be uniquely expressed in terms of a
PVM EUx according to
# i
− q·y
U (q) = e ! dEUx(y).
R3

@ B. Vacchini - Unimi & INFN

3 Observables 47

The latter defines a triple of self-adjoint operators in terms of the integrals

#
x
x̂j = ydEUj(y),
R

and the latter are the generators of boosts. The invariance under boosts of our PVM
localization observable E x then implies that the generator of boosts x̂ can be identified
with the triple of self-adjoint operators defined by E x. A similar argument leads to the
identification of a PVM momentum observable and the generators of translations,
namely the usual momentum observable. Covariance under rotations then warrants that
these triples indeed transform as vectors.
The first moment of the spectral measure therefore gives the usual triple of com-
muting position operators #
x̂j = dx j x j |xj ⟩⟨x j |,
R

whose powers coincide with the higher moments of E x

#
x̂ j =
n
dx j xnj |xj ⟩⟨xj |.
R
◦

In particular for a given state ρ mean values and variances of the classical proba-
bility distribution giving the position distribution can be expressed by means of the
operator x̂

Meanρ(E x) =Trρx̂ = ⟨x̂ ⟩ ρ

Var ρ(E x) =Trρx̂ 2 − (Trρx̂)2 = ⟨x̂ 2⟩ ρ − ⟨x̂ ⟩2ρ.

◦
The couple (U , E x), where U is the unitary representation of the symmetry group,
here the isochronous Galilei group, and E x a PVM covariant under the action of U is
called a system of imprimitivity.
More generally a solution of the covariance conditions as a POVM can be obtained
as follows. Let us consider a distribution function h(x) with zero mean such that
#
h(x) ≥ 0 d3xh(x) = 1 h(Rx) = h(x),

so that it can be identified with a rotationally invariant probability density. Since it has
zero mean the variance is given by
#
Var(h) = d3xx2h(x).

One can then indeed check that the expression

# #
Fh (M ) = (h ∗ χM )(x̂) =
x 3
dy d3xh (x − y)|x⟩⟨x|
M R3

where ∗ denotes convolution, actually is a POVM complying with covariance, and in fact
provides the general solution. Indeed one immediately sees that thanks to positivity of h
the operator Fhx(M ) is positive ∀M ∈ F . Moreover since the POVM is built in terms of
an integral with an operator density one immediately has Fhx(∅) = 0, Fhx(Ω) = 1 as well
as σ-additivity.
48 Table of contents

In the case of the PVM E x we have in the position representation for a pure state
ψ∈H
⟨y|E x(M )ψ ⟩ = (E x(M )ψ)(y)
= χM (y)ψ(y)
while
⟨y|Fhx(M )ψ ⟩ = (Fhx(M )ψ)(y)
= (h
# ∗ χM )(y)ψ(y)
= d3xh (y − x)ψ(y).
M
In particular
#
⟨ψ|E x(M )ψ ⟩ = d3y|ψ(y)|2
M
while
# #
⟨ψ |Fhx(M )ψ ⟩ = d3x d3y h (x − y)|ψ(y)|2.
M R3
As a useful exercise let us directly check that Fh indeed obeys the covariance require-
ments. We have for arbitrary M ∈ F
# #
U (a) Fh (M ) U (a) =
† x 3
dy d3xh (x − y)U †(a)|x⟩⟨x|U (a)
# M # R 3

= d3y d3xh (x − y)|x − a⟩⟨x − a|

#M #R 3

= 3
dy d3xh (x + a − y)|x⟩⟨x|
#M R#3

= dy3 d3xh (x − y)|x⟩⟨x|

M −a R3
= Fhx(M − a).
As far as rotations are concerned we exploit invariance of h under rotations
# #
U (R) Fh (M ) U (R) =
† x 3
dy d3xh (x − y)U †(R)|x⟩⟨x|U (R)
# M #R 3

= d3y d3xh (x − y)|R−1x⟩⟨R−1x|

#M #R 3

= 3
dy d3xh (Rx − y)|x⟩⟨x|
#M #R 3

= 3
dy d3xh (x − R−1 y)|x⟩⟨x|
# M R 3
#
= d3y d3xh (x − y)|x⟩⟨x|.
R −1M R3
= F (R M )
x −1

Finally we have invariance under boosts, namely

# #
U †(q) F x(M ) U (q) = d3y d3xh (x − y)U †(q)|x⟩⟨x|U (q)
#M #R3
i i
q·x − q·x
= dy3 d3xh (x − y)e ! |x⟩⟨x|e !
M R3
= F x(M ).

@ B. Vacchini - Unimi & INFN

3 Observables 49

The POVM actually is a smeared version of the usual sharp position observable, the
probability density h(x) which fixes the POVM being understood as the actual, finite
resolution of the registration apparatus. For any state ρ the first moment of the associ-
ated probability density can still be expressed as the mean value of the usual position
operator, since Mean(h) = 0
#
Meanρ(Fh ) =
x
yd TrρFhx(y)
# R3
#
= 3
dy d3x yh (x − y)Trρ|x⟩⟨x|
R3 R3
= ⟨x̂ ⟩ ρ.
= Meanρ(E x)

◦
The second moment however differs
#
Var ρ(Fh ) =
x
d3y y 2 d TrρFhx(y) − ⟨x̂ ⟩2ρ
# R3
#
= 3
dy d3x y 2 h (x − y)Trρ|x⟩⟨x| − ⟨x̂ ⟩2ρ
R3 R3
= ρ − ⟨x̂ ⟩ ρ + Var(h)
⟨x̂ 2⟩ 2

= Varρ(E x) + Var(h).

◦
◦
The variance is no more expressed only by the mean value of the operator which can
be used to evaluate the first moment and by its square. Besides the variance for a sharp
position measurement, a further contribution Var(h) appears, which is state independent
and reflects the finite resolution of the equivalence class of apparata used for the local-
ization measurement. Note that the usual result is recovered in the limit of a sharply
peaked probability density h(x) → δ 3(x), corresponding to a pure state in the set of clas-
sical probability measures. Taking e.g. a distribution of the form
4 53 1 x2
1 2 − 2 σ2
σ→0
hσ(x) = e →
→
→
→→
→
→
→→ δ 3(x)
→
→
→
2πσ 2

one has that in the limit of an infinite accuracy in the localization measurement of the
apparatus exploited the POVM reduces to the standard PVM
# # 4 53
1 1
− 2 (x−y)2
Fhσ(M ) =
x
dy3 3
dx 2
e 2σ |x⟩⟨x|
M # R3 # 2 π σ2
σ→0
→
→
→→
→→
→
→→
→→
→ d3y d3xδ 3 (x − y)|x⟩⟨x|
# M R 3

= 3
d x|x⟩⟨x|.
M

Analogous results can obviously be obtained for a momentum observable, asking for the
corresponding covariance properties, namely

U †(a) F p(N ) U (a) =F p (N ) ∀a ∈ R3

U †(R) F p(N ) U (R) =F p(R−1N ) ∀R ∈ S O(3)
U †(q) F p(N ) U (q) =F p(N − q) ∀q ∈ R3.
50 Table of contents

Position and momentum observable

A more interesting situation appears when considering apparata performing both a
measurement of the spatial location of a particle as well as of its momentum. As it is
well known no observable can be associated to such a measurement in the framework of
standard textbook quantum mechanics. Let us consider the covariance equations of such
an observable in the more general framework of POVM. A position and momentum
observable should be given by a POVM F x,p defined on B (R3 × R3) satisfying the fol-
lowing covariance equations under the action of translations, rotations and boosts
respectively

U †(a) F x,p (M × N ) U (a) =F x,p (M − a × N ) ∀a ∈ R3

U †(R) F x,p (M × N ) U (R) =F x,p (R−1M × R−1N ) ∀R ∈ S O(3)
U †(q) F x,p (M × N ) U (q) =F x,p (M × N − q) ∀q ∈ R3.

Such covariance equations, defining a position and momentum observable by means of

its operational meaning, do not admit any solution within the set of PVM, while the
general solution within the set of POVM is given by
# #
x,p 1
FS (M × N ) = 3
dx d3pW (x, p) S W †(x, p)
(2 π !)3 M N

where S is a trace class operator, positive, with trace equal to one and invariant under
rotations
S ∈ T (H) S ≥ 0 TrS = 1 U †(R) S U(R) = S

so that it is in fact a statistical operator, even though it does not have the meaning of a
state, while the unitaries
i
− ! (x·p̂−x̂·p)
W (x, p) = e
i i i
− ! x·p̂ + ! x̂·p + 2! x·p
= e e e

are the Weyl operators built in terms of the canonical position and momentum opera-
tors, namely the generators of boosts and translations respectively. One can directly
check as a useful exercise that FSx,p (M × N ) is a POVM. Indeed, since it is built by a
Lebesgue integral with an operator kernel, one immediately has σ-additivity, as well as
FSx,p (∅) = 0. Furthermore normalization can be checked evaluating the matrix elements
# #
x,p 1
⟨z |FS (R × R )z ⟩ =
3 3 ′ 3
dx d3p⟨z|W (x, p) S W †(x, p)|z ′⟩
(2 π !)3 R3 R3
# # D . i . E
1 . − ! x·p̂ + !i x̂·p − !i x̂·p + !i x·p̂. ′
= 3
dx 3
d p z.e e Se e .z
(2 π !)3 R3
#R
3
# D . i .
1 3p z − x..e+ ! (z−x)·pS e − ! (z −x)·p..z ′ −
i ′
= d 3x d
(2Eπ !) R3
3
R3

x
#
= δ (z − z )
3 ′ d3x⟨x|S |x⟩
R3
= δ 3(z − z ′).

@ B. Vacchini - Unimi & INFN

3 Observables 51

The covariance of the proposed POVM can be directly checked, exploiting the properties
of S and working with the matrix elements of the operator expression. The couple (U ,
F x,p), where U is the unitary representation of the symmetry group and F x,p a POVM
covariant under its action is called system of covariance.
The connection with position and momentum observable as well as the reason why
such a joint observable can be expressed only in the formalism of POVM, where position
observables alone are generally given by smeared versions of the usual position observ-
able, and similarly for momentum, can be understood looking at the marginal observ-
ables. Starting from the POVM one can consider a measure of position irrespective of
the momentum of the particle, thus coming to the marginal position observable

FSx(M) = FSx,p (M#× R3) #

1
= d3x d3pW (x, p) S W †(x, p)
(2 π !)3 #M # R 3
# #
1 i
− ! x·p̂
i ′ i
3z ′e+ ! (z−z )·p |z⟩⟨z |S |z ′⟩⟨z ′| e+ ! x·p̂
= d3x d 3pe d3z d
#(2 π !) # M
3
R3 R3 R3

= d3x d3z |z + x⟩⟨z |S |z ⟩⟨z + x|

#M # R3

= d3y d3x|x⟩⟨x − y |S |x − y⟩⟨x|

#M #R 3

= 3
dy d3xhS x (x − y)|x⟩⟨x|,
M R3

where the function

hS x (x) = ⟨x|S |x⟩

is a well defined probability density invariant under rotations due to the fact that the
operator S has all the properties of a statistical operator invariant under rotations, so
that ⟨x|S |x⟩ would be the position probability density of a system described by the
state S. On similar grounds the marginal momentum observable is given by

FSp(M ) =F x,p
# S (R# × N )
3

= d3q d3p|p⟩⟨p − q |S |p − q ⟩⟨p|

#N #R3

= dq3 d3phS p (p − q)|p⟩⟨p|

N R3

where again the function

hS p(p) = ⟨p|S |p⟩

is a well defined probability density, which corresponds to the momentum probability

density of a system described by the statistical operator S. As it appears the marginal
observables are given by two POVM characterized by a smearing of the standard posi-
tion and momentum observables by means of the probability densities hS x (x) and
hS p(p) respectively. The latter probability densities as we have seen originate from the
same statistical operator S, so that we also have

!2
Vari(hS x )Vari(hS p ) ≥ i = x, y, z.
4
52 Table of contents

It is exactly this finite resolution in the measurement of both position and momentum,
stemming from FSx,p that allows for a joint measurement for position and momentum in
quantum mechanics, without violating Heisenberg’s uncertainty relations.
◦
Two position and momentum observables FSx and FSp obtained from confidence func-
tions hS x (x) and hS p(p) satisfying the above uncertainty relation are called a Fourier
couple. Considering the product of the variance of the two marginals FSx and FSp in a
given state ρ and restricting for the sake of simplicity to one dimension one has

Var ρ(FSx)Var ρ(FSp) = (Varρ(E x) + Var(hS x ))(Varρ(E p) + Var(hS p))

= Var ρ(E x)Varρ(E p) + Var(hS x )Var(hS p )
+Varρ(E x)Var(hS p) + Var ρ(E p)Var(hS x ),

where due to Heisenberg’s uncertainty relation and the Fourier contraints the first two
terms at the r.h.s. are both greater than !2/4. Indeed the variances of hS x and hS p sat-
isfy as¸ shown an uncertainty principle, while E x and E p are the usual PVM observ-
ables for position and momentum. Exploiting the same constraints the last two terms at
the r.h.s. obey the following inequality
6 7
!2 Var ρ(E x) Var(hS x)
Varρ(E )Var(hS ) + Var ρ(E )Var(hS x ) !
x p p +
4 Var(hS x) Varρ(E x)
!2
! ,
2
since the function x + 1/x is bounded from below by 2. As a result we have the
inequality

Varρ(FSx)Varρ(FSp) ! !2.

◦
Thus if one consider the statistics of a measurement of position obtained from the
marginal of a given joint position momentum observable, as well as measurement of
momentum obtained from the marginal of the same joint position momentum observ-
able, the product of the variances stay above the lower bound that one has for sharp
position and momentum observables, just because of the added unsharpness necessary in
order to consider a joint measurement.
In order to consider a definite example we take S to be a pure state corresponding to
a Gaussian of width σ
4 53
1 1
4 − 4σ 2 x
2
⟨x|ψ⟩ = e ,
2 π σ2

on which the Weyl operators act as a translation in both position and momentum,
leading to
4 53
1 1 i
− 2 (x−x0)2 + ! p0 ·(x−x0)
⟨x|W (x0, p0)| ψ⟩ = 4
e 4σ = ⟨x|ψx0,p0⟩.
2 π σ2
In particular one has for i = x, y, z
4 53
1 −
1
x2
h ψx(x) = 2
e 2σ 2
2 π σ2
Vari(h ψx ) = σ 2

@ B. Vacchini - Unimi & INFN

3 Observables 53

together with
4 53 2σ 2 2
2 σ2 − p
h ψ p(p) = 2
e !2
π !2
!2
Vari(h ψ p ) =
4 σ2
so that
!2
Vari(h ψx )Vari(h ψ p ) = .
4
The POVM now reads
# #
1
F x,p (M × N ) = d3x0 d3p0 |ψx0 p0⟩⟨ψx0 p0|,
(2 π !)3 M N

and it is worthwhile to verify directly the properties of POVM. Note that this is only a
POVM and not a PVM just because the vectors {ψx0 p0}x0,p0∈R3 are not orthogonal, but
rather provide an overcomplete set. We have in fact
B ′C
σ2 1 p0+ p0
− (p0 −p0′ )2 − (x0 −x0′ )2 − i ·(x0 −x0′ )
⟨ψx0 p0|ψ x0′ p0′ ⟩ = e 2! 2 e 8σ2 ! e 2 ,
8 9
1 i
and they correspond to coherent states with complex parameter α = 2 σ x0 + ! 2σ 2 p0
F x,p(∅) = 0, as well as σ-additivity directly follow from the properties of the inte-
gral. Regarding normalization we have, taking matrix elements
# #
1
⟨z |F x,p (R × R )z ⟩ =
3 3 ′ 3
d x0 d3p0 ⟨z|ψx0p0⟩⟨ψx0p0|z ′⟩
(2 π !)3 R3 R 3
4 53 # #
1 1 1 2 ′ 2 i ′
3p e − 4σ2 [(z−x0) +(z −x0) ]+ ! p0 ·(z−z )
= 2
d 3 x 0 d 0
(2 π !)3 2 π σ 2 R3 R3
4 53 # #
1 −
1
[(z−x 2 ′
0) +(z −x0) ]
2
1 i ′
3p e+ ! p0 ·(z−z )
= 2
d3x 0 e 4σ 2
d 0
2πσ 2
R3 (2 π !) R3
3

= δ 3(z − z ′),

which corresponds to the statement that the vectors {ψx0 p0} provide an overcomplete
set
# #
1
1 = 3
d x0 d3p0|ψx0 p0⟩⟨ψx0 p0|.
(2 π !)3 R3 R3

The marginals are given by

# # 4 53
1 −
1
(x−x0)2
F (M ) =
x d3x0 d3x 2
e 2σ 2 |x⟩⟨x|
M R3 2 π σ2
and
# # 4 53 2σ2
2 σ2 − (p−p0)2
F p(N ) = d3p0 d3p 2
e !2 |p⟩⟨p|
N R3 π !2

for position and momentum respectively. It is now clear that depending on the value of
σ one can have more or less coarse-grained position and momentum observables. No
limit on σ can however be taken in order to have a sharp observable for both position
and momentum. In the limit σ → 0 one has as before F x → E x, but the marginal for
momentum would identically vanish, intuitively corresponding to a complete lack of
information on momentum, and vice versa.
54 Table of contents

4 Transformations of states and observables

In order to understand and describe the basic statistical structure of quantum
mechanics, we have considered as basic statistical experiments those describable as a
direct interaction of a single particle prepared by a preparation apparatus and affecting
a registration apparatus.

[preparation apparatus] " [registration apparatus].

In this setting the preparation procedure delivers a quantum output, namely a state ρ ∈
S(H), while the registration procedure accepts a quantum input and produces a classical
output corresponding to the probability distribution of the considered observable. In
general a measurement can be performed in different steps, however the scheme can
always be traced back to this one by including intermediate transformations either on
the side of the preparation or of the registration. Note that if many intermediate steps
can be put into evidence such inclusion can generally be performed in different ways.
Think for example of the Stern Gerlach experiment. One prepares a beam of particles,
then spatially separates the contributions with different spin components along a given
axis thanks to an inhomogeneous magnetic field, finally a measurement of position is
performed on a screen. The action of the magnetic field, which gives the proper dynam-
ical interaction, can be seen as part of the state preparation corresponding to a transfor-
mation on the space of states, thus corresponding to the Schrödinger picture

[beam preparation] " [magnetic field] " [detection screen] ,

preparation apparatus registration apparatus

or equivalently it can be seen as part of the registration, determining a transformation

on the space of observables and thus corresponding to the Heisenberg picture

[beam preparation] " [magnetic field] " [detection screen].

preparation apparatus registration apparatus

Given this situation it immediately appears that it is of interest to characterize the pos-
sible transformations of states and of observables, which lead to a description of the
intermediate steps in the characterization of a measurement procedure. We will call
operations such transformations, which accept a quantum input and produce a quantum
output. These transformations of relevant spaces of operators into themselves are obvi-
ously of great importance. Each transformation can be seen in a dual way, exchanging
the role of states and observables. They allow to describe not only intermediate steps in
a given measurement setting, they describe symmetry transformations, input/output
transformations over a finite time, so-called one shot transformations, as well as a gen-
eral time dependent dynamics. Such transformations are moreover necessary if besides a
single measurement, leading from a quantum state to a classical probability distribution,
one wants to describe repeated measurements of the same or of different observables,
leading to conditional probabilities. Indeed, in order to perform two measurements one
after the other one needs to know, besides the statistics of the first measurement, the
state transformed as a consequence of the first measurement, so as to apply to this new
state the second measurement. In such a way, one can obtain conditional probabilities.
By iterating this procedure one can also think about performing a high number of subse-
quent measurements of the same observable, so that in the limit of arbitrarily small time
in between these subsequent measurements a description of measurement continuous in
time can be obtained, thus recovering in a suitable framework the notion of trajectory
also in quantum mechanics.

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 55

We focus on transformations of the space of states S(H) ⊂ T (H). Such maps have to
respect the convex structure of the space of states, and as already seen such maps can
be uniquely extended to the whole linear space T (H), which then allows to exploit
useful duality relations. In a general transformation an ensemble of quantum states will
be possibly sent to a subensemble. Think e.g. of the situation in which in the prepara-
tion one takes out of an unpolarized beam particles those which are polarized along a
given axis, thus picking up only a fraction of the beam. Such state transformations that
we have called operations will thus send elements of S(H) in the space S̃ (H) = {ρ ∈
T (H), ρ ! 0, Tr ρ # 1} ⊂ T (H) of subcollections or subnormalized statistical operators.
Denoting by O such an operation

O: T (H) → T (H)
ρ " O[ρ]

the natural requirements are: i) linearity, ii) positivity, namely O[ρ] ! 0 for ρ ! 0, and
iii) trace non-increasing, namely Tr O[T ] # Tr T for all positive T ∈ T (H). The latter
condition reflects the fact that a state can be sent to a subcollection. It turns out how-
ever that the requirement of being positivity preserving is not sufficient, since it allows
for the following unphysical situation. Suppose we have an operation acting on T (H),
and we consider our system described on H together with another system described on a
Hilbert space Cn with n ∈ N. We extend our map on T (H ⊗ Cn) by letting it act in a
trivial way on the other degrees of freedom, namely we consider the map O ⊗ 1. It
might happen that the latter map is not positive, so that states are no more sent to
states. Such a behaviour, due to the non commutativity of the space of operators, calls
for a more stringent requirement on the transformation maps known as complete posi-
tivity. It is important to consider maps of the form O ⊗ 1 when we are interested in
considering transformations on a given quantum system, which might however be cou-
pled to some other quantum system, say an environment. Such transformations, which
possibly describe a dynamical evolution, should remain meaningful when considering a
generic state of system and environment, not necessarily separable, that is given by a
mixture of factorized states. This is warranted considering completely positive maps.
Complete positivity
A property of maps on spaces of operators which becomes of relevance when the
algebra of observables is non-commutative is complete positivity. To introduce it let us
first recall that the space of bounded operators B(H) is the dual of the space of trace
class operators T (H), i.e. B(H) ≡ T ′(H) according to the duality formula

Tr: B(H) × T (H) →

6 C
(B, ρ) " Tr B † ρ

with B ∈ B(H) and ρ ∈ T (H). Given a map Φ acting on the states

Φ: T (H) 6→ T (H)

the duality relation allows to introduce the dual or adjoint map acting on the dual space

Φ ′: B(H) 6→ B(H),

whose action is defined asking

Tr(BΦ[ρ]) = Tr(Φ ′[B]ρ)

56 Table of contents

for all B ∈ B(H) and ρ ∈ T (H). Note that from the relation
Tr(Φ[T ]) = Tr(Φ[T ]1)
= Tr(T Φ ′[1])
for all T ∈ T (H) one immediately has that if Φ is trace non increasing, then Φ ′[1] is an
effect, that is to say 0 # Φ ′[1] # 1, as can be seen taking as trace class operators one
dimensional projections. In particular trace preservation of Φ corresponds to the fact
that Φ ′ is unit preserving, namely Φ ′[1] = 1. For example consider the map
Φ(t)[ρ] = U (t)ρU †(t),
which gives the time evolution in Schrödinger picture. Its adjoint is easily seen to be
given by
Φ ′(t)[X] = U †(t)X U (t),
which gives the time evolution in Heisenberg picture. The notion of dual map allows to
introduce a notion of time evolution for the observables even if the dynamics is no more
unitary.
We now define an important property of these maps, known as complete positivity.
We say that a map Φ ′
Φ ′: B(H) 6→ B(H)

is completely positive if the following equivalent conditions hold:

I. For any n ∈ N the maps
Φ ′ ⊗ 1: B(H ⊗ Cn) 6→ B(H ⊗ Cn)
send positive operators to positive operators.
II. For any n ∈ N the inequality holds
%n
B C
⟨ψi |Φ ′ Bi†B j |ψ j ⟩ ! 0 ∀{ψi } ⊂ H, ∀{Bi } ⊂ B(H) .
i, j =1

Note that due to the relation (Φ ⊗ 1) ′ = Φ ′ ⊗ 1, which can be checked starting from the
duality relation, the requirement of complete positivity can be equivalently set on the
map Φ itself rather than its adjoint, that is asking
Φ ⊗ 1: T (H ⊗ Cn) 6→ T (H ⊗ Cn)
to be positive for all n ∈ N (or up to the dimension of H in the finite dimensional case).
This condition allows to rephrase the condition asking that the map extended in the
trivial way to a tensor product structure remains positive. If Φ describes e.g. a
Schrödinger evolution, the extension of the description of the dynamics to another
system which has itself a trivial dynamics and is not coupled to the system of interest
still leads to an evolution preserving positivity. As we shall see below the difference
between positivity and complete positivity can be detected considering entangled states.
◦ F
To see the equivalence of the two definitions recall that H ⊗ Cn ≡ ni=1 H, so that a
pure state in H ⊗ Cn can be represented as
⎛ ⎞
ψ1
|Ψ⟩ = ⎝ ··· ⎠,
ψn

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4 Transformations of states and observables 57

with ψ j ∈ H, at the same time an element of B(H ⊗ Cn) can be identified as a n × n

matrix with operator entries B(H ⊗ Cn) ≡ Mn(B(H)). A positive operator A in B(H ⊗
Cn) can be written in the form
n
% n
% †
A = Bjℓ Bkℓ ⊗ |e j ⟩⟨ek |
ℓ=1 j ,k =1
n
% n
%
= Bjℓ †Bkℓ ⊗ E jk ,
ℓ=1 j ,k =1

where {e j } denote the canonical basis of vectors in Cn, and {E jk } the canonical basis of
matrices in Mn(C). Of course other bases can be considered, e.g. {ϕ j }, and in the
matrix elements relative to this basis setting E jk = |ϕ j ⟩⟨ϕk | we still have that these
operators can be represented as the canonical basis of matrices in Mn(C). We can
therefore write for a positive operator A
⎛ ⎞
ℓ† ℓ ℓ† ℓ
n
%⎜ 1 1B B ... B1 Bn
·· ·· ·· ⎟
A = ⎝ · · · ⎠
ℓ=1 ℓ† ℓ ℓ† ℓ
Bn B1 ... Bn Bn
%n
†
= Bℓ ⊗ Bℓ
ℓ=1
B C
with B ℓ ≡ B1ℓ, ..., Bn a vector of operators. Positivity of the map Φ ′ ⊗ 1 on B(H ⊗ Cn)
ℓ

is then expressed as
n 8
% 9
† I J
⟨Ψ|(Φ ′ ⊗ 1) B ℓ ⊗ B ℓ |Ψ⟩ ! 0 ∀{ψi } ⊂ (H ⊗ Cn), ∀ Biℓ ⊂ B(H)
ℓ=1

which can equivalently be written

n
% B C
⟨ψi |Φ ′ Bi†Bj |ψ j ⟩ ! 0 ∀n ∈ N, ∀{ψi } ⊂ H, ∀{Bi } ⊂ B(H),
i, j =1

since validity of for the case in which ℓ takes on a single value is necessary but also suffi-
cient to ensure the condition for the generic case.
◦
Examples of completely positive maps are given by unitary transformations

Φ[ρ] = UρU †,

directly from the definition one has

n
% n
%
B C B C
⟨ψi |Φ ′ Bi†B j |ψ j ⟩ = ⟨ψi |U † Bi†B j U |ψ j ⟩
i, j=1 i, j =1
% n
= ⟨ψi |U † Bi†U U † B j U |ψ j ⟩
i, j =1
; ;2
; n ;
;% ;
;
= ; U Bj U |ψ j ⟩;
†
;
; j=1 ;
H
! 0.
58 Table of contents

The same holds for the trace of a matrix

tr: Mn(C) 6→ C
%n
A " Aii
i=1

and note that this map can be represented as follows

n
% †
trA = eiA ei
i=1

with ei = (0, ..., 1, ..., 0), whose unique nonzero element is in the i-th position. This case
is however trivial, since positivity and complete positivity are equivalent conditions if
the elements of either the initial or final space commute. The notion of complete posi-
tivity is therefore only relevant for quantum probability.
Also the partial trace TrE is completely positive, indeed one has, ∀{ψi } ⊂ HS ,
∀{Bi } ⊂ B(HS ⊗ HE ) and ∀n ∈ N
n
% n
% %
B C
⟨ψi |TrE Bi†B j |ψ j ⟩ = ⟨ψi |⊗⟨ϕα |Bi†B j |ϕα ⟩ ⊗ |ψj ⟩
i, j =1 i, j =1; α ;2
; n ;
% ;% ;
= ; B j |ϕα ⟩ ⊗ |ψ j ⟩;
; ;
α ; j =1 ;
HS ⊗HE
! 0

Note that according to the definition the composition of completely positive maps is
completely positive, in fact, if both Φ and Λ are completely positive, then both Φ ⊗ 1
and Λ ⊗ 1 are positive, so that such is

(Φ ⊗ 1) ◦ (Λ ⊗ 1) = (Φ ◦ Λ) ⊗ 1

and therefore Φ ◦ Λ is completely positive. The same goes for linear combinations with
positive coefficients, so that the set of completely positive maps is a positive cone.
Operations
We are now in the position to actually define an operation as a map

O: T (H) → T (H)
T " O[T ]

which is linear, completely positive and trace non increasing, i.e. Tr O[T ] # Tr T for all
T ∈ T (H), T ! 0. Note that actually, as we shall see in the examples, one can consider
situations in which the initial and final Hilbert spaces are not the same. If in particular
Tr O[T ] = Tr T for all T ∈ T (H), the quantum operation is called a channel. Note that
an operation generally sends S(H) into S̃ (H), so that a statistical operator is sent to a
subcollection, while a channel sends S(H) into S(H), so that statistical operators are
transformed in statistical operators. As we have seen if O is trace preserving, then the
dual map O ′ preserves the identity and is called unital, i.e. O ′[1] = 1. In general if O is
trace decreasing O ′[1] is a positive operator between zero and one, that is an effect. Fur-
thermore, as we have seen, the map O is completely positive iff such is O ′, so that com-
plete positivity can equally well be verified on the map or its dual. Note finally that the
space of operations or channels is closed under convex combinations.

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 59

Let us consider a few examples. For a given unitary operator U ∈ U (HS ), we consider
the map AU [ρ] = UρU †, which is in particular a state automorphism, that is to say a
bijection of the space of states S(HS ) onto itself, which preserves its convex structure.
Such a map is an operation and in particular a channel, a so-called unitary channel. Lin-
earity and trace preservation immediately appear, while regarding complete positivity,
for any Hilbert space HE , taken T ∈ T (HS ⊗ HE ), T ! 0 we have, for an arbitrary ψ ∈
HS ⊗ HE

⟨ψ |(AU ⊗ 1E )[T ]ψ ⟩ = ⟨ψ|(U ⊗ 1E )T (U ⊗ 1E ) † ψ ⟩

= ⟨ψ ′|Tψ ′⟩
! 0,
which grants complete positivity. Note that if U is anti unitary, such a map does not
define a channel any more.
Let F be a non zero positive trace class operator F ∈ T (H) and define a map on the
space of trace class operators as follows
F
OF [T ] = Tr T ,
Tr F
so that each operator is sent to an operator proportional to the state F /(Tr F ). The
dual map is defined verifying the identity

Tr T OF′ [B] = Tr OF [T ]B
for all B ∈ B(H) and T ∈ T (H). We thus have

Tr T OF′ [B] = Tr 4
OF [T ]B 5
F
= Tr Tr T B
Tr F
4 5
Tr(B F )
= Tr T 1
Tr F
and therefore
Tr(B F )
OF′ [B] = 1.
Tr F
From the last expression one immediately sees complete positivity. In particular, while
the map OF in Schrödinger picture sends all the states to the state F /(Tr F ), the dual
map OF′ sends all observables in a multiple of the identity. In particular OF′ contracts
all the space on the commutative algebra of operators proportional to the identity. In
this respect we recall the fact that the notion of complete positivity coincides with the
notion of positivity if either initial of final space are commutative.
We now consider a situation in which an operation connects operators acting on dif-
ferent Hilbert spaces. Given a fixed statistical operator ρE on the Hilbert space HE we
define the map
A ρE : T (HS ) → T (HS ⊗ HE )
T " T ⊗ ρE

often called assignment map, whose complete positivity is immediately seen considering
that extending it to another ancillary Hilbert space HA, for any positive operator X ∈
T (HS ⊗ HA) one has

(A ρE ⊗ 1A)X = X ⊗ ρE
60 Table of contents

which is still a positive operator. This operation is in particular a channel. The dual
map is defined as

A ρ′ E: B(HS ⊗ HE ) → B(HS )
B " A ρ′ E [B]

satisfying

TrS TA ρ′ E[B] = TrS ⊗EA ρE [T ]B

= TrS ⊗ET ⊗ ρE B
= TrS ⊗E (T ⊗ 1E ) (1S ⊗ ρE )B
= TrS T TrE (1S ⊗ ρE )B

so that we have

A ρ′ E[B] = TrE (1S ⊗ ρE )B,

which is in particular unital, corresponding to the fact that A ρE is a channel.

The partial trace is another example of completely positive map, which sends states
to states.
◦
It is a mapping

TrE : T (HS ⊗ HE ) → T (HS )

T " TrE [T ]

defined by the requirement

TrS (TrE [T ] B) = TrS ⊗E (T (B ⊗ 1E ))

for all B ∈ B(HS ) and all T ∈ T (HS ⊗ HE ). The dual map

(TrE ) ′: B(HS ) → B(HS ⊗ HE )

B " (TrE ) ′[B]

is then obtained from the relation

TrS ⊗E (T (TrE ) ′[B]) = TrS (TrE [T ] B)

= TrS ⊗E (T (B ⊗ 1E )),

so that

(TrE ) ′[B] = B ⊗ 1E .

From the latter expression one immediately reads complete positivity of the map, as well
as trace preservation. The dual map thus amounts to embed the original observable in a
larger algebra.
◦
◦
Operations and measurement

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4 Transformations of states and observables 61

We now dwell on the physical meaning of operations in connection with conditional

measurements. As we have already seen to an operation one uniquely associates an
effect via E = O ′[1]. Vice versa many distinct operations can lead to the same effect, so
that to an effect one can associate a whole equivalence class of operations [O]E , which
are E compatible in the sense that O ′[1] = E for all O ∈ [O]E . This many to one rela-
tionship is associated to the meaning of operation as state transformation corresponding
to the measurement of the elementary observable described by the effect E. There are in
general many different way to measure the same effect, which lead to a different state
change, and they correspond to the set of operations compatible with the given effect.
Let us take the pre-measurement ρ ∈ S(H). The probability associated to the event
described by the effect E is then given by

P (E) = µE
ρ
= TrρE
= Tr ρO ′[1]
= Tr O[ρ],

where O is any operation compatible with the given effect. As it appears, the proba-
bility for the realization of the effect is given by the trace of the subcollection O[ρ],
which provides the statistical weight associated to the event. While as far as the mea-
surement of E only is concerned, any operation in the equivalence class [O]E leads to
the same result, things change when we consider to perform subsequently the measure-
ment of another effect, let us call it F . To perform such a second measurement we need
to know not only the statistics of the first measurement, but also the transformed state,
so that the notion of operation now becomes essential. The post-measurement state con-
ditioned on the occurrence of E is given by
O[ρ]
ρ|E = ,
Tr O[ρ]
which is a suitably normalized statistical operator. We can now express the conditional
probability for the occurrence of the effect F in the second measurement given the
occurrence of E in the first measurement as

P (F |E) = µFρ|E
= Tr ρ|E F
Tr O[ρ]F
=
Tr O[ρ]
Tr O[ρ]F
= ,
P (E)
which also leads to the expression of the joint probability as

P (F , E) = P (F |E)P (E)
= Tr O[ρ]F
= Tr ρ O ′[F ].

Note that the order in which the measurement are performed is actually relevant.
Indeed denoting by OE and OF operations in the equivalence class compatible with the
effects E and F respectively, we have the relations

P (F , E) = Tr OF [OE [ρ]]
= Tr OE′ [OF′ [1]]ρ
62 Table of contents

to be compared with

P (E , F ) = Tr OE [OF [ρ]]
= Tr OF′ [OE′ [1]]ρ.

Note further how the second measurement actually depends on how the state has been
transformed through the operation O in performing the first measurement. Different ele-
ments of the equivalence class [O]E lead to different results for P (F , E). While the
transformation on the state O[ρ] corresponds to a Schrödinger picture, the transforma-
tion on the effect O ′[F ] can be seen as a Heisenberg picture. As already discussed the
statistics of an experiment in which different steps are performed can be divided in dif-
ferent ways in a preparation and a registration part. We note further that while an
effect E uniquely fixes its complementary effect E ⊥, the operation O⊥ which determines
the post-measurement state linked to occurrence of the effect E ⊥ is still not fixed even
once a representative O in the equivalence class [O]E has been fixed, since the only con-
′
straint is O ⊥ [1] + O ′[1] = 1, so that once again a whole equivalence class of operations
[O]E ⊥ leads to the same statistics.
It is important to stress that while O is a linear map, the map which gives the state
conditioned on the outcome
O[ρ]
ρ " ρ|E =
Tr O[ρ]
is no more linear.
◦
Indeed considering a convex combination of states ρ = p1 ρ1 + p2 ρ2 we have
p1O[ρ1] + p2O[ρ2]
ρ|E =
p1Tr ρ1E + p2Tr ρ2E
p1Tr ρ1E O[ρ1] p2Tr ρ2E O[ρ2]
= · + ·
p1Tr ρ1E + p2Tr ρ2E Tr ρ1E p1Tr ρ1E + p2Tr ρ2E Tr ρ2E
= p(1|E)ρ1|E + p(2|E)ρ2|E ,

where p(1|E) is the probability of the first case conditioned on the outcome E and simi-
larly for p(2|E), so that a different convex combination appears, corresponding to a lack
of linearity.
◦
As a final basic example we consider a simple operation built as follows. Consider an
operator S ∈ B(H) and define

OS [T ] = S T S †.

Linearity is obvious, while regarding complete positivity we argue as before

⟨ψ|(OS ⊗ 1E )[T ]ψ ⟩ = ⟨ψ|(S ⊗ 1E )T (S ⊗ 1E ) † ψ⟩

= ⟨ψ ′|Tψ ′⟩
! 0.

For OS to be an operation we further need it to be trace non increasing. This condition

leads to

TrOS [T ] = Tr S †S T
# Tr T ,

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4 Transformations of states and observables 63

for arbitrary positive trace class operator T . The map OS then is an operation iff S †S is
an effect, that is S †S # 1 or equivalently ∥S ∥ # 1.
◦
The adjoint map has the same form with S replaced by its adjoint, namely OS′ =
!
OS †. We immediately see that more in general the map O[T ] = k Sk T Sk† still gives an
!
operation provided the effects Ek = Sk†Sk are compatible, that is †
k Sk Sk still is an
effect lying below the identity.

Operations and state preparation

We now describe the connection between operations and state preparations. As we
already discussed, a possible way to perform the preparation of a state is to exploit
apparata for the sharp measurement of a complete set of commuting observables which
can be identified with a projection-valued measure. In this setting, let us denote by A
this set of observables and a the corresponding eigenvalues, which are non degenerate
and correspond
! to certain eigenstates ua. We thus have the spectral representation
(A)i = a aiPa, where the Pa are one-dimensional projections, Pa = |ua ⟩⟨ua |. Let us
denote by ρin our pre-measurement state. We want to consider an operation Oa
describing the statistics of the measurement of our observable, so that it has to satisfy
the constraint

Tr Oa[ρin] = P (a)
= Tr ρinPa.

It is natural to consider the Ansatz

Oa[ρin] = PaρinPa,

which turns out to be a proper operation obviously compatible with our constraint. The
effect associated to each of these operations Oa′ [1] = Pa is actually a projection. These
choice of operations lead to certain special features for our state transformation which
are known as repeatability and ideality. Repeatability means that if we repeat the mea-
surement on the state selected according to a given outcome, with probability one we
recover the same outcome, i.e.
6 7
Oa[ρin]
TrOa[ρout|a] = TrOa
Tr Oa[ρin]
= 1.

◦
Ideality means that if an outcome occurs with probability one, then the initial state
is not disturbed. More specifically, suppose TrOa[ρin] = 1 for a certain a, then PaρinPa =
ρin. Indeed one has

1 = TrOa[ρin]
= Tr ρinOa′ [1]
= Tr ρinPa,

so that

Tr(1 − Pa)ρin = 0,
64 Table of contents

but if Q is a projection, Tr Qρ = 0 implies Qρ = 0. In fact, we then have 0 = TrρQ =

√ √ √
Tr( ρ Q) †( ρ Q) = ∥ ρ Q∥22, so that since the Hilbert-Schmidt norm is equal to zero one
also has 0 = ρQ = Qρ. As a result we have (1 − Pa)ρin = ρin(1 − Pa) = 0, and therefore
ρinPa = Paρin = ρin as desired. Note that these features are related to the fact that the
associated effects are actually projections, and the spectrum is discrete. The post-mea-
surement state conditioned on the occurrence of the effect Pa is given according to the
previous formulae by
Oa[ρin]
ρout|a =
Tr Oa[ρin]
P ρ P
= a in a
Tr ρinPa
= |ua ⟩⟨ua |,

that is the expected pure state.

◦
If no selection is performed, the state which a priori comes out of the measurement
procedure is the mixture
%
ρout = P (a)ρout|a
a
% Oa[ρin]
= P (a)
Tr Oa[ρin]
%
a
= PaρinPa,
a
diagonal in the basis common to the complete set of commuting observables.
◦
Mathematical characterization of operations
We now want to provide a general characterization of completely positive maps, both
in view of their structure and of their physical interpretation. This result will be accom-
plished by means of a few basic theorems, which we consider now. We first mention a
basic result on the general structure of completely positive maps between algebras.
Theorem (Stinespring, 1955) A linear map Φ ′: B(HS ) → B(HS ) is completely posi-
tive iff there exists a Hilbert space HE , a bounded operator V : HS → HS ⊗ HE and a
normal, unit preserving ∗-homomorphism π: B(HS ) → B(HS ⊗ HE ) such that Φ ′[AS ] =
V †π(AS )V holds. π can be always realized as π(AS ) = AS ⊗ 1E , so that we have the rep-
resentation

Φ ′[AS ] = V †AS ⊗ 1EV .

◦
We recall that according to its definition the ∗-homomorphism π is a map from the
algebra B(HS ) to the larger algebra B(HS ⊗ HE )

π: B(HS ) →
6 B(HS ⊗ HE )
AS " π(AS )

that satisfies

π(ASBS ) = π(AS )π(BS )

B C
π AS† = π(AS ) †
π(1S ) = 1S ⊗ 1E .

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4 Transformations of states and observables 65

These properties imply in particular that π is a positive map, and in particular a com-
pletely positive map. For a normal map this embedding in the larger algebra can always
be written as π(AS ) = AS ⊗ 1E , so that it is equivalent to the map (TrE ) ′ considered in
the previous examples. Normality is a regularity requirement, implying that convergent
monotone non-decreasing sequences of operators are transformed into convergent
monotone non-decreasing sequences.
◦
◦
As a first application of Stinespring dilation’s theorem we provide a general proof of
the already discussed Naimark’s theorem.
Theorem (Naimark, 1943) Let F be a positive operator-valued measure on the mea-
sure space (Ω, F ), so that F : F → B(HS ). Then there exists a Hilbert space HE , a
bounded operator V : HS → HS ⊗ HE and a projection-valued measure on the same mea-
sure space E: F → B(HS ⊗ HE ) such that

F (M ) = V †E(M )V .

The proof goes as follows. First we observe that the positive operator-valued measure,
defined on the measure space (Ω, F ), induces in a natural way a map acting on the
algebra C∞(Ω) of continuous functions on Ω with the supremum norm, which becomes a
Banach ∗-algebra with the supremum norm, as follows

ΦF : C(Ω) → B(HS ) #
h " ΦF (h) = h(x) dF (x),
Ω

where the operator ΦF (h) with matrix elements

#
⟨ϕ|ΦF (h)ψ⟩ = h (x)d⟨ϕ|F (x)ψ ⟩
Ω

is indeed bounded, as follows from the inequality

#
⟨ϕ|ΦF (h)ϕ⟩ = h (x)d⟨ϕ|F (x)ϕ⟩
Ω #
# ∥h∥∞ . d⟨ϕ|F (x)ϕ⟩
Ω
= ∥h∥∞∥ϕ∥2

valid ∀ϕ ∈ HS .
◦
◦
Namely, to each function one associates its integral over the measure space with the
given positive operator-valued measure.
This map is obviously positive, and therefore, since the space C(Ω) is abelian, it is
also completely positive. We can therefore exploit Stinespring theorem to write it in the
form

ΦF (h) = V †π(h)V ,

where π is a ∗-homomorphism π: C(Ω) → B(HS ⊗ HE ) between algebras. All such homo-

morphisms are of the form
#
π(h) = h (x)dE(x)
Ω
66 Table of contents

where E is a projection-valued measure on the measure space (Ω, F ) taking values in

B(HS ⊗ HE ). The defining properties of a ∗-homomorphism can be directly verified, and
in particular one has
#
⟨Φ|π(h f )Ψ⟩ = h (x)f (x)d⟨Φ|E(x)Ψ⟩
Ω
= ⟨Φ|π(h)π(f )Ψ⟩,
just thanks to idempotency of the elements of the projection-valued measure. We have
thus associated to the initial positive operator-valued measure F taking values in B(HS )
a projection-valued measure E on the same measure space taking values in B(HS ⊗ HE ).
We now exploit Stinespring to obtain a most important structural characterization of
completely positive maps. We consider for definiteness the case of a trace preserving
completely positive map Φ with equal initial and final space, however the result works as
well in the case of different initial and final spaces, as well as considering general trace
non increasing maps, the only relevant feature being complete positivity.
Theorem (Kraus, 1971) Given a linear trace preserving map Φ: T (HS ) → T (HS ) the
following statements are equivalent:
1. Φ is completely positive according to either of the two equivalent formulations
2. Φ can be written in Kraus form
% †
Φ[σ] = AKσAK
K
where the sum is!over a set which is at most denumerable, convergence is in the
†
weak sense and K AKAK = 1
3. Φ can be expressed in the form
Φ[σ] = TrE {Uσ ⊗ ρEU †}
for a certain Hilbert space HE , a state ρE , which can be taken to be pure, and a
unitary operator U on HS ⊗ HE .
The expressions appearing in 2 and 3 are known as first and second representation the-
orem for a completely positive map. While the first representation is very useful since it
gives the generic structure of an arbitrary completely positive map in terms of operators
in the Hilbert space of interest only, the second representation tells us that any com-
pletely positive transformation can be seen as originating from the interaction with an
external system, taking an initially factorized state between system and environment.
Each completely positive dynamics can be obtained taking the partial trace with respect
to a unitary evolution in a larger Hilbert space. The non uniqueness of this construction
reflects itself in the non uniqueness of the Kraus representation.
Let us prove the sequence 1 ⇒ 3 ⇒ 2 ⇒ 1.
1 ⇒ 3 By definition Φ is completely positive iff Φ ′: B(HS ) → B(HS ) has this property.
We now use Stinespring’s theorem adapted to the case in which Φ ′ is a normal map. In
this case for a completely positive map Φ ′ on the algebra B(HS ) there exists a Hilbert
space HE and a bounded operator V : HS → HS ⊗ HE such that
Φ ′[AS ] = V †AS ⊗ 1EV ,
while identity preservation implies V †V = 1. For fixed ψ ∈ HE of norm one we can there-
fore define the operator
U (ϕ ⊗ ψ) = Vϕ ∀ϕ ∈ HS ,

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4 Transformations of states and observables 67

which turns out to be an isometry thanks to the property of V and the fact that ψ is of
norm one, indeed

⟨U (ϕ ′ ⊗ ψ)|U (ϕ ⊗ ψ)⟩ = ⟨Vϕ ′|Vϕ⟩

= ⟨ϕ ′|V †Vϕ⟩
= ⟨ϕ ′|ϕ⟩
= ⟨ϕ ′ ⊗ ψ |ϕ ⊗ ψ⟩.

One can now extend U to define a unitary operator on the whole HS ⊗ HE . With this
results at hand we have the following chain of equalities, for arbitrary BS ∈ B(HS ), ρ ∈
T (HS ) and a basis {ϕ j } in HS

TrS Φ[ρ]BS = TrSρΦ ′[BS ]

= TrSρV †(BS ⊗ 1E )V
%
= ⟨ϕ j |ρV †(BS ⊗ 1E )Vϕ j ⟩
j
%
= ⟨Vρϕ j |(BS ⊗ 1E )Vϕj ⟩
j
%
= ⟨U (ρϕj ⊗ ψ)|(BS ⊗ 1E )U (ϕ j ⊗ ψ)⟩
j
%
= ⟨ϕ j ⊗ ψ |ρ ⊗ 1EU †(BS ⊗ 1E )U (ϕ j ⊗ ψ)⟩
j
= TrS ⊗Eρ ⊗ 1EU †(BS ⊗ 1E )U 1S ⊗ P ψ
= TrS ⊗E (BS ⊗ 1E )U (ρ ⊗ P ψ)U †
= TrSBS TrE {Uρ ⊗ P ψU †},

that is our statement, with ρE = P ψ given in particular by a pure state.

! 3 ⇒ 2 Follows directly by considering an orthogonal resolution for ρE , namely ρE =
α λα |ϕα ⟩⟨ϕα | and using the same orthonormal basis to evaluate the trace

Φ[ρ] = TrE {Uρ ⊗ ρEU †}

% B: C B: C
= λ β ⟨ϕα |U |ϕ β ⟩ ρ λβ ⟨ϕα |U |ϕ β ⟩ †
αβ
%
= WKρWK†
K

with K = α, β a multi-index, and the operators WK defined on HS via

:
WK = λβ ⟨ϕα |U |ϕ β ⟩,

which further satisfy

%
WK† WK = 1.
K

2 ⇒ 1 This can be proved noting that the Kraus representation immediately implies
complete positivity according to the very definition. Given the Kraus representation one
has immediately
%
Φ ⊗ 1n[T ] = (WK ⊗ 1Cn)T (WK ⊗ 1Cn) †,
K

so that Φ ⊗ 1n send positive operators into positive operators for all n.

68 Table of contents

We have thus obtained a fundamental result according to which any operation

admits a representation in the form
%
O[ρ] = AkρAk†
k
! †
with the constraint k AkAk # 1. If the Hilbert space HS of the system has dimension
d, it is always possible to use d2 or less operators in the sum. To show an example of the
non uniqueness of the Kraus representation consider the following completely positive
trace preserving map defined on HS = Cd

A: T (HS ) → T (HS )
1
T " Tr (T ) ,
d
which sends all states to the maximally mixed state. We point to two different Kraus
representation. Consider first a SONC in HS given by {ϕi }i=1,...,d and the operators
Eij =|ϕi ⟩⟨ϕ j | which provide the canonical basis in Md(C). They satisfy
d
% d
%
Eij† Eij = |ϕ j ⟩⟨ϕi |ϕi ⟩⟨ϕ j |
i, j =1 i, j=1
= d 1,
so that one has the Kraus representation
d
1%
A[T ] = Eij T Eij† .
d
i, j =1

As an alternative, consider another basis in Md(C) obtained from the unitary operators
d−1
% 2π
−i sk
Urs = e d |k ⊖ r⟩⟨k |,
k=0

where the symbol ⊖ denotes the difference modulo d, {|k⟩}k=0,...,d−1 is a basis. These
† †
operators are unitary, i.e. Urs Urs = Urs Urs = 1, as follows from their very definition, and
orthogonal according to the Hilbert-Schmidt scalar product, as follows from the identity
d−1
% 2π
i (s−s ′)k
e d = d δs,s ′,
k=0
√
so that one can normalize the basis by dividing each operator by d.
◦
We claim that one has the Kraus representation
d−1
1 %
A[T ] = Uij T Uij† .
d2
i, j =0
√
To see this relabel for convenience the operators as Uij / d → Uα, since the latter form
an orthonormal basis we have
% B C
T = UαTr Uα†T .
α
Given any two states ϕ, ψ we have
Tr P ψP ϕ = ⟨ψ |Tr(|ψ ⟩⟨ϕ|)1ϕ⟩

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 69

but also
% B B CC
Tr P ψP ϕ = Tr P ψUαTr Uα†P ϕ
α
% B C
= Tr (P ψUα)Tr Uα†P ϕ
α 4 5
%
= ⟨ψ | |Uαψ ⟩⟨Uαϕ| ϕ⟩,
α

so that we have
%
Tr(|ψ ⟩⟨ϕ|)1 = Uα |ψ ⟩⟨ϕ|Uα†
α

and since any operator can be expressed by means of rank one operators finally
%
Tr (T )1 = UαT Uα†.
α

To clarify the freedom left in connecting two different Kraus decompositions, we finally
point to the fact that two collections of Kraus operators {A1, ..., An } and {B1, ..., Bm }
define the same operation iff
m
%
Aj = ujk Bk
k=1
!n
where the complex numbers ujk satisfy j=1 u∗jl ujk = δlk . In fact given this hypothesis,
for any trace class operator T we have
n
% n
% m
%
A jT A j† = ujk BkT u∗jlBl†
j =1 j=1 k,l=1
m
%
= BkT Bk†.
k=1

Vice versa, suppose that for any trace class operator T the identity
n
% m
%
A jT A j† = BkT Bk†
j =1 k=1
!m
holds,
!n then any two such decomposition must satisfy A j = k=1 ujk Bk, with again
j =1 jl u jk = δlk .
∗
u
◦
◦
◦
◦
Trace distance
As we have seen operations have a natural interpretation as transformations on the
space of states corresponding to the measurement of an elementary observable described
by an effect. Operations can describe the preparation of a state or mixing of a state in
subcollections, as well as conditional state measurements and state preparations. We
now want to provide a way to put into evidence the measurement character of a certain
state transformation. First we stress the fact that the measurement character is related
to irreversibility.
◦
70 Table of contents

Indeed one has the result that a completely positive trace preserving map E:
T (HS ) → T (HS ) admits an inverse iff it is a unitary transformation. This can be shown
directly, or referring to Wigner’s theorem and the fact that operations described by anti-
unitary operators do not provide a channel, being related to transposition.
◦
To a completely positive trace preserving transformation one can associate a notion
of disturbance, which compares the state after and before the transformation, namely
ρin and ρout = Φ[ρin]. This leads to quantify their difference or distance, which intuitively
is maximal when input and output states are orthogonal. Such a disturbance is not
related to the measurement character, and also a unitary channel can transform a state
in another orthogonal to the given one. Another notion which can be associated to a
completely positive trace preserving map is the noise it introduces in the state, affecting
its information content. This feature thus influences the comparison of two states before
K 1,2 L
and after the action of the transformation, namely ρ1,2 1,2
in and ρout = Φ ρin , rather than
the comparison of a given input with the corresponding output. To quantify this mea-
surement character, we now introduce a quantifier of distance or distinguishability
between quantum states.
◦
On the space of states S(H) we introduce a distance according to the following defin-
ition
1
D(ρ1, ρ2) = ∥ρ1 − ρ2∥1 ρ1, ρ2 ∈ S(H).
2
One can verify that such a quantity is a proper distance, in particular the triangular
inequality follows from the property of the trace norm ∥·∥1 used in the definition. One
further has
1
D(ρ1, ρ2) = Tr|ρ1 − ρ2|
2
1%
= |λi |,
2
i

where {λi } are the eigenvalues of the self-adjoint trace class traceless operator ρ1 − ρ2,
so that

0 # D(ρ1, ρ2) # 1.

Again, since ∥·∥1 is a norm, one has D = 0 iff ρ1 = ρ2, while D = 1 iff ρ1 and ρ2 have
orthogonal support. It can be shown that if the two states are in particular pure states,
the trace distance reads
M
D(P ϕ , Pψ) = 1 − |⟨ϕ|ψ ⟩|2 .

◦
◦
We now show that this distance on the space of states, as it immediately appears
looking at the special situation of pure states, is directly connected to the distinguisha-
bility among states. To this aim we show that given a preparation procedure that leads
with equal probability to prepare the states ρ1 and ρ2, the highest probability of success
in discriminating among the two states is given by
1
Poptimal = (1 + D(ρ1, ρ2)),
2

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 71

so that the trace distance provides the bias in favor of a correct identification of the
state. To realize that the optimal strategy actually leads to this result let us first prove
the identity
D(ρ1, ρ2) = max Tr P (ρ1 − ρ2),
P
where the maximum is taken over all possible orthogonal projections. Since ρ1 − ρ2 is
self-adjoint and traceless we have the decomposition
ρ1 − ρ2 = N+ − N−,
where N+ and N− are positive operators with orthogonal finite dimensional support and
such that Tr |N+| = TrN+ = Tr |N−| = TrN−. We thus have
1
D(ρ1, ρ2) = Tr|ρ1 − ρ2|
2
1
= Tr(N+ + N−)
2
= TrN+.
If we now take the projection P+ on the eigenspace of N+, we have
Tr P+(ρ1 − ρ2) = Tr P+(N+ − N−)
= Tr P+N+
= Tr N+
1
= Tr|ρ1 − ρ2|
2
= D(ρ1, ρ2).
Moreover for any other projection P we have
Tr P (ρ1 − ρ2) = Tr P (N+ − N−)
# Tr P N+
# Tr N+
1
= Tr|ρ1 − ρ2|
2
= D(ρ1, ρ2).
Considering now the law of total probability we express our success probability as fol-
lows, in terms of conditional probabilities. We measure a projection P and associate to
the outcome 1 the state ρ1, and to the outcome 0 the state ρ2. We then have
Psuccess = p(P = 1|ρ1)p(ρ1) + p(P = 0|ρ2)p(ρ2)
1
= [Tr Pρ1 + Tr(1 − P )ρ2]
2
1
= [1 + Tr P (ρ1 − ρ2)],
2
1
since p(ρ1) = p(ρ2) = 2 , so that indeed the best strategy is to take P → P+ and leads to
1
Poptimal = (1 + D(ρ1, ρ2)).
2
As one immediately sees from the fact that a unitary transformation does not change
the eigenvalues, one has that the trace distance among two states is invariant under the
action of a unitary transformation
D(Uρ1U †, Uρ2U †) = D(ρ1, ρ2).
72 Table of contents

We have however the important fact that completely positive trace preserving maps are
contractions with respect to the trace distance, that is for any completely positive trace
preserving map Φ we have
D(Φ(ρ1), Φ(ρ2)) # D(ρ1, ρ2).
In fact we have
1
D(Φ(ρ1), Φ(ρ2)) = Tr|Φ(ρ1 − ρ2)|
2
1
= Tr|Φ(N+ − N−)|
2
1 1
# Tr|Φ(N+)| + Tr|Φ(N−)|
2 2
1 1
= Tr Φ(N+) + TrΦ(N−)
2 2
1 1
= Tr N+ + Tr N−
2 2
1
= Tr|ρ1 − ρ2|
2
= D(ρ1, ρ2),
where we have exploited the triangular inequality, positivity and trace preservation.
Indeed here complete positivity does not play any role. We thus have the result that the
action of a completely positive, or even simply positive, trace preserving map does
indeed reduce the trace distance between states. In particular the action of a trace pre-
serving operation, that is a channel, decreases the distance between states and therefore
according to the previous discussion their distinguishability.
◦
◦
◦
Instruments
We have seen that to the measurement of an effect one can associate, in a way which
is not unique, a transformation on the space of states known as operation which
describes a possible state change associated to the observation of the elementary observ-
able. Moving from elementary observables, such as effects, to the general notion of
observable as a positive operator-valued measure, we are led to ask about the general
properties of a state transformation which reproduces the statistics of the given positive
operator-valued measure. The mathematical object describing this situation is known as
instrument and defined as follows.
Given a measure space (Ω, F ) we call instrument or operation-valued measure a map
I: F → O(T (H)),
where O(T (H)) denotes the set of operations, such that
i. I(M ) is an operation ∀M ∈ F
ii. Tr I(Ω)[T ] = Tr[T ] ∀T ∈ T (H)
@ !
iii. I( i Mi) = i I(Mi) in the weak topology for any sequence of disjoint sets in F ,
i.e. Mi ∩ M j = ∅ for i =
/ j.
The first condition tells us that I(M ) sends states to subcollections for any M ∈ F,
while the last two express the notion of normalized measure. An instrument uniquely
identifies a positive operator-valued measure as follows. Setting
F (M ) = I(M ) ′[1]

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4 Transformations of states and observables 73

one can directly check that the map

F (·): F → E(H)
M " I(M ) ′[1]

is indeed a positive operator-valued measure. Given a pre-measurement state ρ ∈ S(H)

the probability to obtain an outcome in M for the observable F is given by the formula

P (M ) = µFρ (M )
= Tr ρF (M )
= Tr ρ I(M ) ′[1]
= Tr I(M )[ρ],

otherwise stated the probability of a definite outcome M is given by the weight of the
corresponding subcollection I(M )[ρ]. The instrument also provides the new state condi-
tioned on the occurrence of an outcome in M , namely

I(M )[ρ]
ρ(M ) =
Tr I(M )[ρ]
I(M )[ρ]
=
µFρ (M )
I(M )[ρ]
= .
P (M )

Note that as we noticed discussing effects, the map associating to the pre-measurement
state a new state conditioned on the measurement outcome ρ → ρ(M ) is not linear.
Moreover it is not additive in its dependence on the outcome M . If now we consider the
special case in which M becomes the whole space Ω, we have P (Ω) = 1, so that the
transformed state becomes

ρ(Ω) = I(Ω)[ρ],

which is called the a priori state. We recall that I(Ω) is a quantum channel. This is the
state that we can a priori obtain as output of the measurement, given the instrument
and the pre-measurement state only, namely if we do not look at the outcomes and
therefore do not select the state according to the measurement outcomes. If we consider
the other extreme situation in which rather than the whole space we select as outcome
an infinitesimal region d w around the point w ∈ Ω we are lead to consider the so-called a
posteriori state

I(d w)[ρ]
ρ(w) = ,
Tr I(d w)[ρ]

which is the state we can associate to the system if we perform a measurement and
obtain an outcome w. Note that while ρ(M ) is a function defined on F , the a posteriori
state is a function defined on Ω. It is a random variable depending on the outcomes dis-
tributed according to the probability distribution P (w) = Tr I(w)[ρ]. The a priori state
is then the expectation value of the a posteriori state with this probability measure
#
I(Ω)[ρ] = ρ(w)P (dw),
Ω
74 Table of contents

and in general the subcollection corresponding to a given outcome is given by

#
I(M)[ρ] = ρ(w)P (dw).
M

Given an instrument, which provides both the statistics of a positive operator-valued

measure and the transformed state, we are able to consider subsequent measurements
leading to conditional probabilities. Indeed considering another positive operator-valued
measure G on the outcome space (Γ, G), given the pre-measurement ρ and an outcome
M ∈ F, the conditional probability to obtain in a subsequent measurement an outcome
N ∈ G for the positive operator-valued measure G is given by

P (N |M ) = µG
ρ(M )(N )
= Tr G(N )ρ(M )
I(M )[ρ]
= Tr G(N )
Tr I(M )[ρ]
Tr G(N )I(M )[ρ]
= .
P (M )
Note that P (N |M ) denotes the conditional probability of obtaining first the outcome M
and then the outcome N , the order is indeed relevant since in general
P (N |M )=P (N |M ).
The joint probability distribution is then given by

P (N , M ) = P (N |M)P (M )
= Tr G(N ) I(M )[ρ]
= Tr ρ I(M ) ′[G(N )].

If we now consider an instrument K compatible with the positive operator-valued mea-

sure G in the sense that

Tr ρG(N ) = Tr K(N )[ρ] ∀N ∈ G , ∀ρ ∈ S(H),

so that it correctly reproduces the statistics of the measurement, implying in particular

G(N ) = K(N ) ′[1]

we can write the joint probability distribution as

P (N , M ) = Tr K(N ) ′[1]I(M )[ρ]

= Tr K(N )[I(M )[ρ]]
= Tr ρ I(M ) ′[K(N ) ′[1]].

The last two expressions, corresponding to a Schrödinger and Heisenberg picture respec-
tively, actually lead to a probability distribution on the rectangles M × N ∈ Ω × Γ, from
which one can construct a well defined probability on F ⊗ G. We have thus built a new
instrument, defined on the measure space (Ω × Γ, F ⊗ G), given by the composition of
the two original instruments K ◦ I, which describes two subsequent measurements. We
have thus put multiple subsequent measurements on the same formal footing as a single
measurement, thus opening the way to speak about measurements continuous in time.
◦

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 75

Similarly to the case of operations, while an instrument as we have seen uniquely

determines a positive operator-valued measure, the inverse connection is many to one, in
that many instruments, leading to different state transformations, provide the same sta-
tistics of the outcomes and therefore the same positive operator-valued measure. To
each positive operator-valued measure F corresponds a whole equivalence class of instru-
ments [I]F which are compatible with it. As usual the meaning of the appearance of
these equivalence classes [I]F is the fact that the statistics of the same observable can be
obtained affecting the state in different ways, corresponding to different registration pro-
cedures. These different registration procedures do however affect, as we have seen, sub-
sequent measurements.
We now provide some examples of instruments, originating from a given positive
operator-valued measure or in particular projection-valued measure. Let us first consider
a conditional state preparation or random state generation. Given a positive operator-
valued measure F with a finite outcome space Ω = {1, ..., N }, and a set of statistical
operators {ρk }k=1,...,N the formula

I({k })[ρ] = Tr(F ({k })ρ)ρk

!
actually defines an instrument, given that we define as usual I(M ) = k∈M I({k}) to
comply with additivity. To the pre-measurement state we thus associate different fixed
states ρk according to the probability distribution Tr(F ({k})ρ). If all the ρk are equal
to σ we have the so-called trivial instrument, and the dual map is given by

I({k }) ′[B] = Tr(Bσ)F ({k}).

In the Schrödinger picture all states are sent to a fixed set of states, in the Heisenberg
picture all observables are sent to the positive operators F ({k}).
Von Neumann instruments
Another example of major relevance of instrument is the so-called von Neumann-
Lüders instrument, obtained as follows. Consider, similarly as before in the discussion of
operations an observable in the sense of a self-adjoint operator A with discrete spec-
trum, not necessarily non degenerate.
◦
We have according to the spectral theorem
%
A = akPk
k

and denoting by E A the uniquely associated projection-valued measure we have

E A({k}) = Pk.

An instrument compatible with this projection-valued measure can be immediately con-

structed defining

M({k})[ρ] = E A({k })ρE A({k })

= PkρPk

and setting to comply with σ-additivity

%
M(M )[ρ] = E A({k})ρE A({k}).
{k|ak ∈M }
76 Table of contents

Such an instrument is called a von Neumann instrument in the absence of degeneracy,

so that the Pk are one-dimensional projections, or more generally a von Neumann-
Lüders instrument. The conditional state reads
M(M )[ρ]
ρ(M) =
Tr
! M(M)[ρ]
P ρPk
{k|ak ∈M } k
= ! .
{k|a }
Tr Pkρ
k ∈M

The a posteriori state is given by

PkρPk
ρ({k }) =
Tr Pkρ
so that in particular in the absence of degeneracy ρ({k}) is the pure state Pk = |uk ⟩⟨uk |
independently of the purity of the pre-measurement state ρ, while for a Lüders instru-
ment ρ({k}) is pure iff ρ is pure. The a priori state is given by
%
ρ(Ω) = PkρPk ,
k
diagonal in the basis determined by the original observable. Such an instrument has two
important features, which actually single it out from all other instruments, namely
repeatability and ideality. Repeatability corresponds to the fact that the state singled
out according to a certain outcome, leads to the same outcome if the instrument is
applied once again and can be expressed as follows
M({j })[M({k })[ρ]] = δ jkM({k})[ρ].
Otherwise stated the conditional state relative to a certain outcome gives with proba-
bility one the same outcome in a subsequent measurement. Denoting as usual by F ( · )
the positive operator-valued measure uniquely associated to the instrument we have
Tr ρ(M )F (M ) = 1.
Ideality corresponds to a minimality in the perturbation of the state in the measurement
of the considered observable and corresponds to the fact that
TrM(M )[ρ] = 1 ⇒ M(M )[ρ] = ρ
that is if the outcome is certain the state is not modified. As we have shown before, this
can be demonstrated just relying on the specific properties of orthogonal projections.
Otherwise stated, the state after the action of the instrument is diagonal in the basis of
the projective observable corresponding to the instrument. If the outcome for this
observable are certain, the state is already diagonal in this basis, and it does not change
under the action of the instrument.
It is an important fact that any instrument which satisfies these two properties corre-
sponding to repeatability and ideality has to be of the von Neumann-Lüders form, which
implies in particular the pure point spectrum of the associated observable. Observables
with a continuous spectrum therefore do not admit, as is well known, an ideal, that is
non perturbing, measurement. In this sense pure and continuous spectrum indeed do
have crucially different features.
Also for the general case of a positive operator-valued measure F (·) there is a simple
way to associate to it an instrument compatible with the statistics of the observable.
Still restricting to a discrete outcome space we set
% : :
I(M )[ρ] = F ({k }) ρ F ({k }) ,
{k|ak ∈M }

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 77

which can be checked to be a proper instrument, moreover compatible with F (·) as fol-
lows from
% : :
Tr I(M)[ρ] = Tr F ({k }) ρ F ({k })
{k|ak ∈M }
%
= Tr F ({k})ρ
{k|ak ∈M }
= Tr F (M )ρ.

Since one of the important advancements in introducing instruments is the possibility to

perform on an equal footing single and repeated measurements, it is interesting to see
what happens when we consider the subsequent application of two von Neumann-Lüders
instruments. We already know that their composition can be described as single instru-
ment on an outcome space given by the cartesian product of the outcome spaces. The
question is whether the class of von Neumann-Lüders instruments is stable under such a
composition law. Let us therefore consider two von Neumann-Lüders instruments MA
and MB, related to two projective measurements corresponding to the observables
%
A = akPk
%k
B = b jQ j
j

respectively. If we apply in sequence the two instruments, for the subcollection obtained
starting from the pre-measurement state ρ and selecting an outcome in M in the first
instance and an outcome in N in the second we have
% %
MB (N )[MA(M )[ρ]] = QjPkρPkQ j
I . J
j . b j ∈N {k|ak ∈M }
= MB ◦ MA(N × M )[ρ],

where in the last line we have stressed the fact that the subsequent action of the two
instruments equals the action of a single instrument obtained composing the original
two. The overall instrument is however no more of the von Neumann-Lüders type,
unless the two observables commute, i.e. [Pk , Qj ] = 0 ∀j , k so that [A, B] = 0. Indeed the
operator QjPk is positive but not idempotent in the general case. This fact is best seen
noting that the uniquely associated observable is no more a projection-valued measure
′ K ′ L
(MB ◦ MA) ′(N × M )[1] = MA (M ) MB (N )[1]
% %
= PkQ jPk ,
I . J
j . b j ∈N {k|ak ∈M }

since PkQ jPk ∈ E(H), but PkQ jPk ∈ P(H), unless A and B commute. The joint proba-
bility for an outcome in M for the observable A and an outcome in N for the observable
B is given by

P (N , M ) = Tr MB(N )[MA(M )[ρ]]

% %
= Tr Q jPkρPkQ j ,
I . J
j . b j ∈N {k|ak ∈M }

where the last expression is known as Wigner’s formula.

◦
78 Table of contents

We now provide the generic expression of an instrument which encompasses the situ-
ation in which the outcome space is not necessarily finite dimensional or discrete. Let Ω
be a measurable space, µ a positive measure on it, and suppose that a measurable func-
tion taking operator values is given
V : Ω → B(H),
such that the constraint holds
#
dµ(x)V †(x)V (x) = 1.
Ω
Then one can define an instrument as follows
#
I(M )[ρ] = dµ(x)V (x)ρV †(x).
M
The associated positive operator-valued measure is given by
F (M ) = I(M
# ) ′[1]
= dµ(x)V †(x)V (x),
M
which turns out to be well defined according to the previous constraint. The statistics of
the measurement is given by
#
P (M ) = Tr dµ(x)V †(x)V (x)ρ,
M
the a priori state reads
#
I(Ω)[ρ] = dµ(x)V (x)ρV †(x),
Ω
while the a posteriori state is
V (x)ρV †(x)
ρ(x) = .
Tr ρV †(x)V (x)
It is clear that this example includes the previously considered cases by asking the mea-
sure to have a purely discrete support.
This formalism allows us to introduce an example of instrument leading to the joint
measurement of position and momentum considered before. Given a Gaussian wave
packet ψx0,p0 centered in x0, p0
4 53
1 1 2 i
4 − 4σ 2 (x−x0) + ! p0 ·(x−x0)
⟨x|W (x0, p0)| ψ⟩ = e = ⟨x|ψx0,p0⟩.
2 π σ2
where W (x0, p0) denote as before the Weyl operators, we can consider the instrument
# #
1
I x,p (M × N )[ρ] = 3
d x0 d3p0 |ψx0 p0⟩⟨ψx0 p0|ρ ψx0 p0⟩⟨ψx0 p0|,
(2 π !)3 M N
which turns out to be well defined and associated to the joint position and momentum
observable
# #
1
I x,p (M × N ) [1] =
′ 3
d x0 d3p0 |ψx0 p0⟩⟨ψx0p0|
(2 π !)3 M N
= F x,p (M × N ).
◦

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4 Transformations of states and observables 79

To recover the expression of an instrument associated to the general form of a joint

position and momentum observable we consider
# # √
1
I x,p (M × N )[ρ] = 3
d x0 d3p0 W (x0, p0) S W (x0, p0) † ρ W (x0,
(2 π !) 3
M N
√
p0) S W (x0, p0) ,†

which has all the desired properties and lead to the positive operator-valued measure
# #
1
F x,p (M × N ) = 3
d x0 d3p0 W (x0, p0) S W †(x0, p0).
(2 π !)3 M N
Measurement models
Up to now we have considered two levels of description of the measurement, the sta-
tistics of the outcomes and the related transformation of the states, showing that the
two descriptions are generally given by positive operator-valued measures and instru-
ments respectively. As already stressed while a positive operator-valued measure only
provides the statistics of the outcomes, an instrument associates to a given pre-measure-
ment state the subcollection obtained performing a selection according to a certain out-
come. To any instrument one uniquely associates a positive operator-valued measure
whose statistics is compatible with it, while the inverse relation is many to one. Indeed a
whole equivalence class of instruments share the same statistics, though affecting the
state in different ways. We now make a further step in the description of the measure-
ment, considering a third level of description, which we call measurement model, in
which the measurement is described as an indirect measurement on a meter or measure-
ment apparatus, coupled via a unitary interaction to the considered system. We thus
provide a dynamical description of the measurement, obtained by letting the system
interact with a quantum measurement apparatus. Denoting by M a measurement
scheme, once again to a given instrument it corresponds a whole equivalence class of
measurement schemes compatible with it, in symbols
F → [I] I → [M].
As usual this construction comes about considering a suitable dilation. As we discussed
in connection to operations, we have two basic representation theorems for operations,
and therefore in particular channels. The first points to the so-called Kraus representa-
tion in terms of a denumerable set of Kraus operators, the second refers to a dilation in
a larger Hilbert space. The second representation as we shall soon see directly leads to
associate to an instrument a measurement scheme. Regarding the first representation,
which is often of great convenience, this is not in general readily available for an instru-
ment. Indeed an instrument is an operation-valued measure, so that for any fixed out-
come one has an operation, and therefore a Kraus representation, with a discrete set of
Kraus operators. However if the outcome space is not discrete, there is no general recipe
for the structural characterization of an instrument. On the contrary, if the outcome
space is finite or denumerable, one obtains a general representation of the instrument
similar to a Kraus representation by suitably summing up the Kraus representations cor-
responding to the single outcomes.
Let us now come to a general dilation of an instrument which allows to connect it to
a measurement scheme.
Theorem (Ozawa, 1984) Let I be an instrument on the outcome space (Ω, F ), then
there exist a Hilbert space K, a projection-valued measure E on K over the same out-
come space, a statistical operator σ ∈ T (K) and a unitary operator on H ⊗ K such that
the representation holds
I(M )[ρ] = TrK {U (ρ ⊗ σ)U †(1H ⊗ E(M ))} ∀M ∈ F ∀ρ ∈ T (H).
80 Table of contents

We note that the right hand side properly defines an instrument for any choice of
Hilbert space K, unitary operator U , state σ and observable E, which can also be taken
to be a positive operator-valued measure. In this model of indirect measurement we
have obtained the states transformed according to the given instrument, and therefore
also the statistics of the associated observable, by a dynamical description of the interac-
tion between system and apparatus. Let us prove this statement for the special case in
which the Hilbert space H of the system has finite dimension n, and the outcome space
is also finite dimensional. Given the finite dimension of the outcome space for any fixed
outcome we have the Kraus representation of the corresponding operation
%
I({k})[ρ] = A jρA j†,
j ∈Ik

where the index j runs over a finite set Ik with at most n2 elements, corresponding to
the cardinality of the representation of I({k}). Summing up all these representations we
have a representation for
! the
!channel I(Ω) determined by the given instrument in terms
of a finite set, say N = k j ∈Ik of Kraus operators, so that we have
N
%
I(Ω)[ρ] = A jρA j†.
j =1

We now consider a dilation of the channel I(Ω), whose existence is warranted by Stine-
spring’s theorem. For a certain Hilbert space K, whose dimension can always be taken
to be N , and a state in K, which in particular can be taken to be pure, say P η = |η⟩⟨η |,
we therefore have the representation
I(Ω)[ρ] = TrK {U (ρ ⊗ P η)U †}.
What is still missing in this description, is the pointer basis, let us introduce it as fol-
lows. First we observe that any basis, say {ϕi }i=1,...,N , in K induces a set of Kraus oper-
ators by evaluating the trace
N
%
I(Ω)[ρ] = B jρB j†,
j =1

where the operators {B j } are defined as B j = ⟨ϕ j |U η⟩, so that their matrix elements
read
⟨Φ|Bj Ψ⟩ = ⟨Φ ⊗ ϕ j |U (Ψ ⊗ η)⟩ ∀Φ, Ψ ∈ H.
Now as we know the two Kraus representation, which in particular have the same
dimension,
!N have to be related by a square N × N matrix [u jk ], whose elements satisfy
j =1 jk jl = δkl as follows
∗
u u
N
%
Aj = Bku∗jk .
k=1
Exploiting this matrix we can therefore introduce a new orthonormal system in K as
follows
%N
ϕi′ = uij ϕ j ,
k=1
and given this basis an associated projection-valued measure as follows
E ′({j }) = |ϕ j′ ⟩⟨ϕ j′ |.

@ B. Vacchini - Unimi & INFN

4 Transformations of states and observables 81

We can now consider the following expression of indirect measurement

TrK {U (ρ ⊗ P η)U †(1H ⊗ E ′({j }))},
and build in terms of it the instrument we started with. We have in fact

TrK {U (ρ ⊗ P η)U †(1H ⊗ E ′({j }))} = ⟨ϕ j′ |U η ⟩ρ(⟨ϕ j′ |U η⟩) †

N
%
= u∗jr ⟨ϕr |U η⟩ρ ujs ⟨U η |ϕs ⟩
r,s=1
%N N
%
= u∗jrBr ρ u jsBs†
r=1 s=1
= A jρA j†,

for each of the N possible values of j. The desired representation of the instrument is
obtained by suitably collecting the indexes in groups of cardinality Ik according to
⎧ ⎫
⎨ 2 3⎬
%
I({k})[ρ] = TrK U (ρ ⊗ P η)U † 1H ⊗ E ′({j }) ,
⎩ j ∈I
⎭
k
!
recalling that j ∈Ik E ′({j }) still is a projection.

As we have seen a measurement model M can be identified with a quadruple (K, U ,

σ, G), where K can be seen as the Hilbert space of the apparatus, U is a unitary oper-
ator on H ⊗ K coupling system and apparatus, σ a statistical operator for the appa-
ratus, which can be taken to be pure, G an observable for the apparatus, which can be
taken to be projection-valued. As we have already seen, given a positive operator-valued
measure F , the latter is compatible with an instrument I provided

Tr F (M )ρ = Tr I(M )[ρ] ∀M ∈ F ∀ρ ∈ T (H).

On the same footing an instrument I is said to be compatible with a measurement
model M ≡ (K, U , σ, G) if
I(M )[ρ] = TrK {U (ρ ⊗ σ)U †(1H ⊗ G(M ))} ∀M ∈ F ∀ρ ∈ T (H).

It is therefore natural to say that a positive operator-valued measure F is compatible

with a measurement model M ≡ (K, U , σ, G) if
Tr F (M)ρ = TrHTrK {U (ρ ⊗ σ)U †(1H ⊗ G(M ))} ∀M ∈ F ∀ρ ∈ T (H),
where the last equality is also known as probability reproducibility condition, as well as
Naimark’s dilation of the initial observable.
We now consider an example of measurement model leading to a von Neumann
instrument, and therefore to a sharp observable. Consider on the Hilbert space H a
sharp observable with discrete and non degenerate spectrum E({k}) = |ϕk ⟩⟨ϕk |, where
{ϕk } is a basis in H. Take as apparatus space K one with the same dimensionality of H
and consider here a basis {ψi } which has the same cardinality and allows to construct
the projective observable for the apparatus Ẽ ({i}) = |ψi ⟩⟨ψi |. Given that we initialize
the apparatus in the state P ψ1 = |ψ1⟩⟨ψ1|, for a measurement model we still only have to
specify the unitary interaction. This is obtained by considering a unitary extension of
the isometry
V (ϕk ⊗ ψ1) = ϕk ⊗ ψk ∀k.
82 Table of contents

Suppose now the system starts in a state given by a coherent superposition of eigen-
states of the considered observable E
%
ϕ = ckϕk ,
k
!
where the complex coefficients ck satisfy k |ck |2 = 1. For such a case statistics of the
outcomes of the observable E are given by the Born probabilities
P ({k }) = |ck |2.
We now put together our measurement model showing that it is compatible with this
statistics for the outcomes and that it leads to an instrument which is of the von Neu-
mann form. To this aim we have to evaluate the trace, or the partial trace with respect
to K only, of the operator
B C
V (P ϕ ⊗ P ψ1)V † 1H ⊗ Ẽ ({k }) .
We evaluate first
V (P ϕ ⊗ P ψ1)V † = V (|ϕ⟩⟨ϕ| ⊗ |ψ1⟩⟨ψ1|)V †
= |V (ϕ ⊗ ψ1)⟩⟨V (ϕ ⊗ ψ1)|
%
= c jc∗k |ϕ j ⟩⟨ϕk | ⊗ |ψ j ⟩⟨ψk |,
j ,k

so that the instrument determined by the considered measurement model reads

I B CJ
I({k})[P ϕ] = TrK V (P ϕ ⊗ P ψ1)V † 1H ⊗ Ẽ ({k})
= |ck |2|ϕk ⟩⟨ϕk |
= Tr(P ϕE({k}))E({k})
= PkP ϕPk ,
that is indeed a von Neumann instrument, while the statistics fixed by the measurement
model leads to
I B CJ
TrHTrK V (P ϕ ⊗ P ψ1)V † 1H ⊗ Ẽ ({k}) = |ck |2
= Tr(P ϕE({k})),
indeed compatible with the projective measurement considered as starting point.

@ B. Vacchini - Unimi & INFN

5 Open quantum systems as composite systems 85

“Natürlich ist man nicht so einfältig zu denken, dass solchermaßen zu

erraten sei, wie es auf der Welt wirklich zugeht.”
(Erwin Schrödinger)

5 Open quantum systems as composite systems

A ubiquitous situation in quantum mechanics is given by composite systems, i.e. a situa-
tion in which the overall set of degrees of freedom considered, the physical object one
tries to describe or observe, can be naturally divided in two parts. In this case the
overall system is described on a Hilbert space with a tensor product structure. Note
however that given the overall Hilbert space its expression as a tensor product of smaller
Hilbert spaces is generally not unique. Think for example of the hydrogen atom. The
overall system includes the centre of mass degrees of freedom of proton and electron, so
that H = L2proton(R3) ⊗ L2electron(R3). The most convenient tensor product representation
of the overall space is however given by H = L2cm(R3) ⊗ L2relative(R3). This has the advan-
tage that given an overall state factorized between centre of mass and relative degrees of
freedom, it remains factorized, these two kinds of degrees of freedom are not coupled by
the interaction. We thus concentrate on the internal degrees of freedom and obtain the
level spectrum of the hydrogen atom. This is why in a sense we can neglect that the
overall system is composite, since for a natural choice of degrees of freedom the latter
are decoupled. On the same footing when describing an electron we should generally
consider a composite structure given by centre of mass and spin degrees of freedom, but
if the latter are not coupled by some interaction, we can neglect the bipartite structure.
In these cases the Hamiltonian does not contain interaction terms so that we can ignore,
for special initial preparation and dynamics, the composite structure. This is no more
true e.g. in the Stern-Gerlach experiment.
◦
More importantly a bipartite structure may arise because the system we want to
consider does interact with some external “environment” (often called reservoir or bath if
one wants to stress that it is characterized by a system with very many degrees of
freedom at thermal equilibrium), which modifies its dynamics. If the system we are
interested in lives in a Hilbert space HS , we say that it is closed if its dynamics can be
described by a unitary evolution, which thanks to Stone’s theorem identifies a self-
adjoint Hamiltonian HS . This dynamics is then reversible. Then the closed dynamics is
given by the Schrödinger equation fixed by HS , or the Liouville von Neumann equation
to cope with mixed states. This is however an often idealized situation, since it would
correspond to a perfect shielding of our system from the rest of the universe. In general
if interaction with an external environment cannot be neglected so that the overall
Hamiltonian reads

H = HS ⊗ 1E + 1S ⊗ HE + HI

we are naturally led to consider together with the system S also a system E, to be
described on the tensor product space HS ⊗ HE . In this case we say that the system is
open, and its dynamics will generally not be given by a Schrödinger equation. This also
implies that new dynamical effects such as dissipation and decoherence can appear, not
described by a unitary evolution. Typical examples of open systems are given by a two-
level atom coupled to a laser field or a nuclear spin coupled to a magnetic field.
86 Table of contents

Given a state on the overall space ρ ∈ S(HS ⊗ HE ), with ρ = ρ †, ρ ! 0, TrS ⊗Eρ = 1

we can obtain the marginal states for system S and environment E through the partial
trace operations TrS and TrE according to

TrSρ = ρE TrEρ = ρS ,

with ρS ∈ S(HS ) and ρE ∈ S(HE ) respectively. We have already encountered the partial
trace as an example of completely positive trace preserving map among two different ini-
tial and final spaces. The partial trace TrE is the unique linear mapping

TrE : T (HS ⊗ HE ) → T (HS )

T " TrET

satisfying

TrS ⊗E [T (AS ⊗ 1E )] = TrS (TrET )AS ∀AS ∈ B(HS ), ∀T ∈ T (HS ⊗ HE ).

This constraint implies in particular for factorized operators T = TS ⊗ TE ∈ T (HS ⊗ HE )

TrS {[TrE (TS ⊗ TE )]AS } = TrS ⊗E [TS ⊗ TE (AS ⊗ 1E )]

= TrS ⊗E [TSAS ⊗ TE ]
= TrSTSAS TrETE ,

where the latter equality follows from the standard definition of trace over a given
Hilbert space. Since this relation is valid for any AS it follows that for factorized states
the partial trace reads

TrE (TS ⊗ TE ) = (TrETE ) TS .

Exploiting this result and the relation

%%
T = |ψ j ⟩⟨ψm |⊗|ϕk ⟩⟨ϕn |⟨ψ j ⊗ ϕk |Tψm ⊗ ϕn ⟩,
j ,k m,n

which expresses in a unique way the operator T in a operator basis obtained from the
SONC {ψi } and {ϕn } in HS and HE respectively, together with the identity

TrE |ψ j ⟩⟨ψm |⊗|ϕk ⟩⟨ϕn | = |ψ j ⟩⟨ψm |δk,n

as obtained above one has

%
TrET = |ψ j ⟩⟨ψm |⟨ψ j ⊗ ϕk |Tψm ⊗ ϕk ⟩
j ,k ,m
%
= ⟨ϕk |Tϕk ⟩,
k

which provides a direct way to evaluate the partial trace and defines it uniquely since we
have expressed its action on a basis of operators. The partial trace is thus obtained by
summing over the diagonal matrix element of the operator with respect to a basis in
HE .

@ B. Vacchini - Unimi & INFN

5 Open quantum systems as composite systems 87

Coming back to the action of the partial trace on a statistical operator ρ defined on
the total space, one immediately checks that ρS = TrEρ still is a statistical operator, and
as shown it is fixed by the fact that it provides the correct statistics for any observable
on the system only. Indeed for an observable of the form AS ⊗ 1E we have

⟨A⟩ρ = TrS ⊗EρAS ⊗ 1E

= TrSρSAS ,

so that ρS describes the statistics of any measurement on S only and in this sense can
be taken as state of the reduced system S. Note that ρS is mixed even if ρ is pure, a
pure ρ leads to a pure ρS iff it is the tensor product of two pure states. On the contrary
ρS can be pure even for mixed ρ (think of the trivial case ρ = |ψ ⟩⟨ψ |S ⊗ ρE ).
◦

Reduced dynamics
Given the fact that our system is in interaction with an external environment, the
question is whether we can eliminate the degrees of freedom of the environment to
obtain effective closed equations of motion, let us say master equations, for the system
only. This turns out to be feasible under the following working hypotheses:
i) factorized initial state

ρ(t0) = ρS (t0) ⊗ ρE ,

an hypothesis which essentially cannot be released to obtain a physically well-defined

closed effective dynamics;
ii) overall unitary dynamics

ρ(t) = U(t, t0)ρ(t0) = U (t, t0)ρ(t0)U †(t, t0),

hypothesis which could be released allowing e.g. for a semigroup dynamics for the whole
system.
For fixed ρE we then have the commutative diagram, now assuming t0 = 0

U (t)
ρS (0) ⊗ ρE " ρ(t)
#
A T rE T rE
! Φ(t) !
ρS (0) " ρS (t)

where A ρE is the so-called assignment map already encountered as an example of

completely positive trace preserving map

A ρE : T (HS ) →
6 T (HS ⊗ HE )
ρS " ρS ⊗ ρE

which furthermore and most importantly satisfies the compatibility condition

TrE ◦ A ρE = 1T (HS ).
88 Table of contents

As a result for fixed ρE the assignment

ρS (0) 6→ ρS (t)
= TrE ◦ U(t) ◦ A ρEρS (0)
= TrE {U (t)ρS (0) ⊗ ρE U †(t)}
= Φ(t)ρS (0)

is a linear map which preserves hermiticity and trace, further sending positive operators
to positive operators, therefore sending states to states: let us call it quantum dynamical
map. It is in particular a completely positive map being a composition of completely
positive maps. The properties of this map can be better checked considering a orthog-
onal decomposition for ρE
%
ρE = λ β |ϕ β ⟩⟨ϕ β |
β
!
so that λβ ! 0, β λ β = 1 and {ϕ β } SONC in HE , and using the same basis in HE to
evaluate the partial trace we have
%
Φ(t)σ = WK (t)σWK† (t)
K

with K a multi-index, which immediately appears to be a Kraus representation of the

completely positive trace preserving map. The operators WK (t) acting on HS are
defined via
:
WK (t) = λ β ⟨ϕα |U (t)|ϕ β ⟩,

and further satisfy

%
WK† (t)WK (t) = 1.
K

The latter equality grants trace preservation.

Bipartite states and entanglement
Considering states on a bipartite Hilbert space leads to a new feature, called entan-
glement (originally Verschränkung). Let us introduce its definition. We start with pure
states on the total system, where the characterization can be tamed.
Schmidt decomposition
Let Ψ ∈ HS ⊗ HE be a pure state describing the overall system, then one has the fol-
lowing so-called “Schmidt decomposition”
D
% √
|Ψ⟩ = λi |χSi ⟩ ⊗ |χE
i ⟩
i=1
IJ I J
with D = min {dimHS , dimHE }. Here χSi and χE i are
√orthonormal bases in HS and
HE respectively, called Schmidt bases.
!DThe coefficients λi are non-negative numbers
called Schmidt coefficients, such that i=1 λi = 1. The number of non-zero coefficients,
the Schmidt number, is uniquely determined.
◦
◦

@ B. Vacchini - Unimi & INFN

5 Open quantum systems as composite systems 89

◦
◦

Proof. For simplicity we work assuming dimHS = dimHE = D. We can always write
D
%
|Ψ⟩ = αij |ϕSi ⟩ ⊗ |ψ E
j ⟩
i, j =1
I S
J I E
J
with ϕi and ψi orthonormal basis in HS and HE respectively.
Consider the square matrix

(A)ij = αij .

The singular value decomposition theorem states that any square matrix can be brought
in the form

A = U ΛV ,

where Λ is diagonal with non-negative entries and U , V are unitary. We thus have
%% √
|Ψ⟩ = uik δkl λk vkj |ϕSi ⟩ ⊗ |ψ E
j ⟩
i, j k,l %
% %
√
= λk uik |ϕSi ⟩ ⊗ vkj |ψ E
j ⟩
k
NNNNNiNNNNNNNNNNNOPQQQQQQQQQQQQQQQQ j
NNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQ
|χS
k⟩ |χE
k⟩

◦
If dimHS = dimHE , the decomposition still holds considering block matrices, and
only at most D coefficients can be different from zero.
◦
◦
Pure entangled states
If the Schmidt number is equal to 1 the pure state is called separable or factorized.
This happens iff the state is of the form

|Ψ⟩ = |ϕS ⟩ ⊗ |ψ E ⟩,

that is to say a product state. Note that a pure state in a bipartite system is separable
iff its marginals are pure. For such a product state one has

⟨AS ⊗ AE ⟩Ψ = ⟨AS ⟩ ϕS ⟨AE ⟩ ψE

so that the statistical outcomes of product observables are uncorrelated.

If the Schmidt number is bigger than 1 the pure state is called entangled.
If the Schmidt number is equal to its maximum value D and all Schmidt coefficients
have the same modulus, that is the state is of the form
D
1 % S
|Ψ⟩ = √ |χi ⟩ ⊗ |χE
i ⟩,
D i=1
the state is said to be maximally entangled.
90 Table of contents

In a given Hilbert space one can consider a basis of maximally entangled states, also-
called Bell basis. Consider HS = HE = C2, and a basis in this space {ψ+, ψ−}, with ψ±
e.g. eigenvectors of σz . Out of it one can consider the standard basis of factorized states
on C2 ⊗ C2, given by {ψ+ ⊗ ψ+, ψ− ⊗ ψ+, ψ+ ⊗ ψ−, ψ− ⊗ ψ−}, which are joint eigenvec-
tors of σz ⊗ 1 and 1 ⊗ σz . Taking linear combinations of these vectors we can consider in
particular a basis of common eigenvectors of the commuting observables σz ⊗ σz and
σx ⊗ σx. Indeed exploiting the relation

[A ⊗ B, C ⊗ D] = [A, C] ⊗ B D + C A ⊗ [B, D],

and the usual relation for the Pauli matrices

σiσ j = δij + iεijkσk

we have

[σx ⊗ σx , σz ⊗ σz] = [σx , σz ] ⊗ σxσz + σzσx ⊗ [σx , σz ]

= 2σ y ⊗ σ y − 2σy ⊗ σ y
= 0.

◦
◦
The common eigenvectors of these operators provide a Bell basis {ψe+, ψo+, ψe−, ψo−}
according to the definition
1
ψe± = √ (ψ+ ⊗ ψ+ ± ψ− ⊗ ψ−)
2
± 1
ψo = √ (ψ+ ⊗ ψ− ± ψ− ⊗ ψ+),
2
where the subscripts mean even or odd parity.
◦
The subscripts e/o correspond to the eigenvalues ±1 for σz ⊗ σz , the superscripts ±
to the eigenvalues for σx ⊗ σx. One can check that a state diagonal in this Bell basis in
matrix form has a typical X expression, that is to say
− −
ρ = p+ +
e P ψ + + pe P ψ − + po P ψ + + po P ψ −,
e e o o

where the positive weights sum up to one in the computational basis reads
⎛ ⎞
− −
p+
e + pe 0 0 p+
e − pe
⎜ − − ⎟
1⎜ 0 p+ +
o + po po − po 0 ⎟
ρ = ⎜ ⎟.
2⎝ 0 p+
o − po
−
p+
o + po
−
0 ⎠
+ − + −
pe − pe 0 0 pe + pe

Note that for weights all equal to 1/4 one obtains the state proportional to the identity,
which can be expressed as a product of states proportional to the identity in each space.
◦
◦
Let us consider the explicit calculation for the case ρ = P ψ+. We have
e

1
Pψ+ = (P ψ+ ⊗ P ψ+ + P ψ− ⊗ P ψ− +|ψ+⟩⟨ψ−| ⊗|ψ+⟩⟨ψ−| +|ψ−⟩⟨ψ+| ⊗|ψ−⟩⟨ψ+|),
e 2

@ B. Vacchini - Unimi & INFN

5 Open quantum systems as composite systems 91

and considering the matrix representation obtained starting from the basis of eigenvec-
tors of σz , we have e.g.
4 5
1 0
P ψ+ =
0 0

and therefore
⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫
⎪
⎪ 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎪⎪
⎨⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎬
1 ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟ ⎜ 0 0 0 0 ⎟
⎟ .
P ψ+ = ⎝ 0 0 0 + + +
e 2⎪
⎪ 0 ⎠ ⎝ 0 0 0 0 ⎠ ⎝ 0 0 0 0 ⎠ ⎝ 0 0 0 0 ⎠⎪
⎪
⎩ ⎭
0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0

If one thinks of two spin one-half systems ψo+ corresponds to the χ10 triplet state, ψe±
are a linear combination of the other two triplet states, while ψo− corresponds to a χ00
singlet state.
◦
It is further instructive to consider the expression of the marginal states associated
to a total pure state. Exploiting the Schmidt decomposition we have
D
%
ρS = TrE |Ψ⟩⟨Ψ| = λi |ϕSi ⟩⟨ϕSi |
i=1
D
%
ρE = TrS |Ψ⟩⟨Ψ| = λi |ψiE ⟩⟨ψiE |.
i=1

It follows in particular that both ρS and ρE are pure states if and only if the state Ψ is
factorized. Both ρS and ρE are proportional to the identity if and only if the state Ψ is
maximally entangled.
In particular ρS and ρE have the same eigenvalues with the same multiplicity (apart
from eigenvalue zero), and therefore the same von Neumann entropy, given by
D
%
S(ρS ) = −kBTrSρS logρS = −kB λilogλi
i=1
%D
S(ρE ) = −kBTrEρE logρE = −kB λilogλi.
i=1

For a factorized pure state the von Neumann entropy of the marginals is equal to zero,
while for a maximally entangled state it takes the maximum value kB log D. Thus max-
imal correlations in the total pure state implies that the marginals describe a trivial sta-
tistics. The quantity E(|Ψ⟩) = S(ρS ) = S(ρE ) is called entanglement entropy.

Mixed entangled states

More generally one can consider mixed states on the bipartite system. In this case
the characterization is much more difficult, we can however distinguish three kinds of
states.
A state ρ ∈ T (HS ⊗ HE ) is said to be factorized if it is given by the tensor product of
two states

ρ = ρS ⊗ ρE .
92 Table of contents

In this case the statistical outcomes for a product observable is simply the product of
the outcomes

⟨AS ⊗ AE ⟩ ρ = ⟨AS ⟩ ρS ⟨AE ⟩ ρE .

A state ρ ∈ T (HS ⊗ HE ) is said to be separable or classically correlated if it can be
written as a convex combination of product states
%
ρ = pi ρiS ⊗ ρiE
i
!
where pi is a probability distribution, so that pi ! 0 and i pi = 1. Classically correlated
states correspond to a classical mixture of preparation procedures, and the statistical
outcomes for a product observable are classically correlated
%
⟨AS ⊗ AE ⟩ ρ = pi ⟨AS ⟩ ρiS ⟨AE ⟩ ρiE .
i

A state ρ ∈ T (HS ⊗ HE ) is said to be entangled otherwise.

The characterization of entangled states is a difficult issue, and requires the knowl-
edge of the notion of complete positivity.
Note that the linear combination of entangled states is not necessarily entangled, e.g.
1! 1 1
trivially as observed above 4 4i=1 PΨi = 2 12 ⊗ 2 12, which is separable. Also note that if
either ρS or ρE is a pure state, that is just one of the eigenvalues is different from zero,
then ρ is factorized. In fact if ρ is pure this immediately follows from the Schmidt
decomposition, which must have Schmidt number equal to one. Supposing ρ to be
mixed,
! we have, taking e.g. ρS to be pure and considering an orthogonal resolution of
ρ = k pk |ϕk ⟩⟨ϕk |, where pk =
/ 1 for all k since the state is mixed
%
ρS = pkTrE (|ϕk ⟩⟨ϕk |)
% k
= pkσSk
k

which implies, being ρS pure, that actually σSk does not depend on k, in particular it has
to be pure. But then according to the previous argument the ϕk have to be factorized as
ϕk = ψS ⊗ φkE and the state ρ is of the form
%
ρ = |ψS ⟩⟨ψS | ⊗ pk |φkE ⟩⟨φkE |
k
= |ψS ⟩⟨ψS | ⊗ ρE .

◦
Purification
From the Schmidt decomposition we also learn that any state can be obtained, actu-
ally in many ways, as partial trace from a pure state on a larger Hilbert space. Indeed
given ρS ∈ S(HS ) we may consider the spectral decomposition
%
ρS = λi |ϕi ⟩⟨ϕi |
i

and consider another Hilbert space HE (with dimension at least dimHS ) and a basis
{ψi } in HE . The pure state on HS ⊗ HE
%√
|Ψ⟩ = λi |ϕSi ⟩ ⊗ |ψiE ⟩
i=1

@ B. Vacchini - Unimi & INFN

5 Open quantum systems as composite systems 93

then has the required marginal. Note that for every unitary operator UE on HE the
state (1 ⊗ UE )|Ψ⟩ is another purification of ρS . The fact that a purification always
exists and is highly non unique, is reflected in the fact that the state in the ancillary
Hilbert space of a dilation can always be taken to be pure, as well as in the non unique-
ness of the dilation.
◦
Positive maps and entanglement detection
One can infer about entanglement or non-entanglement of a state by acting on it
with maps having known positivity or complete positivity properties. If Φ is a positive
map, and ρ a separable state, then (Φ ⊗ 1)ρ is positive, even if Φ is not CP. Indeed if ρ
!
is a separable state it is of the form ρ = i pi ρiS ⊗ ρiE . We then have that for positive Φ
!
the operator (Φ ⊗ 1)ρ = i pi Φ(ρi)S ⊗ ρiE is positive being the tensor product of positive
operators.
◦
As a consequence the fact that (Φ ⊗ 1)ρ # 0 for some positive but not completely
positive map Φ, so that the transformed operator has negative eigenvalues, is a sufficient
condition for entanglement of the state ρ. Indeed a theorem by the Horodecki family
states that a given state ρ is separable if and only if (Φ ⊗ 1)ρ is positive for all positive
maps Φ. If the map Φ is positive, but not completely positive, the condition (Φ ⊗ 1)ρ !
0 is a necessary condition for separability of ρ, while (Φ ⊗ 1)ρ < 0 is a sufficient condi-
tion for entanglement of ρ.
An example of a positive but not completely positive map is the transposition. Note
that in order to actually define the transposed map one has to fix a basis. Denoting by
T the transposition, we have that T is positive, but T ⊗ 1 is not. To show this we
explicitly exhibit a state which is transformed into a non positive operator. From the
above discussion, the state should not be separable, we take the projector on one ele-
ment of the previously considered Bell basis, which in the so-called computational basis
we write as
1
|ψe+⟩ = √ (|11⟩ + |00⟩).
2
We have
1
|ψe+⟩⟨ψe+| = (σ+σ− ⊗ σ+σ− + σ−σ+ ⊗ σ−σ+ + σ+ ⊗ σ+ + σ− ⊗ σ−)
2
and therefore
1
(T ⊗ 1)|ψe+⟩⟨ψe+| = (T (σ+σ−) ⊗ σ+σ− + T (σ−σ+) ⊗ σ−σ+ + T (σ+) ⊗ σ+ + T (σ−) ⊗
2
σ−)
which in the computational basis reads
⎛ ⎞
1 0 0 0
⎜
1 0 0 1 0 ⎟
(T ⊗ 1)|ψe+⟩⟨ψe+| = ⎜ ⎟
2⎝ 0 1 0 0 ⎠
0 0 0 1
S T
1 1 1 1
with eigenvalues 2 , 2 , 2 , − 2 .
◦
94 Table of contents

Note that the insurgence of negative eigenvalues together with trace preservation is
equivalent to growth of the trace norm of ρ. The trace norm of a self-adjoint trace class
operator is given by the sum of the modulus of its eigenvalues
%
∥A∥1 = |ai |.
i
!
For a statistical operator one has ∥ρ∥1 = Tr ρ = ipi = 1. In our case one has
⎛ ⎞
1 0 0 1
1⎜ 0 0 0 0 ⎟
|ψe+⟩⟨ψe+| = ⎜ ⎟
2⎝ 0 0 0 0 ⎠
1 0 0 1
with eigenvalues {1, 0, 0, 0} so that
∥|ψe+⟩⟨ψe+|∥1 = 1
but
∥(T ⊗ 1)|ψe+⟩⟨ψe+|∥1 = 2.
More generally, consider the tensor product Cn ⊗ Cn, and restrict the attention to pure
states
ρ = |Ψ⟩⟨Ψ|,
where according to the Schmidt decomposition we can write
n
% (1) (2)
|Ψ⟩ = αi |χi ⟩ ⊗ |χi ⟩
i=1

with {χi } denotes a basis on either copy of the Hilbert space. Introducing the basis of
operators
E jk = |χj ⟩⟨χk |
we thus have
n
%
ρ = αiα jEij ⊗ Eij
i, j=1

and according to the action of the transposition T (Eij ) = Eji

n
%
(T ⊗ 1)ρ = αiα jT (Eij ) ⊗ Eij
i, j =1
% n
= αiα jE ji ⊗ Eij
i, j =1
% n 8 98 9
(1) (2) (1) (2)
= αiα j |χj ⟩ ⊗ |χi ⟩ ⟨χi |⊗⟨χ j |
i, j =1

and the latter operator has negative eigenvalues as soon as the number of non zero α j is
greater or equal to two, so that according to the Schmidt decomposition the state ρ is
entangled. Indeed suppose αs =/ 0, αr =
/ 0, so that they are strictly positive. The state
8 9
1 (1) (2) (1) (2)
√ |χr ⟩ ⊗ |χs ⟩ −|χs ⟩ ⊗ |χr ⟩
2

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6 Quantum maps and their representation 95

is eigenstate of the operator (T ⊗ 1)ρ with eigenvalue −αsαr. Thus the map T ⊗ 1,
known as partial transposition, detects all pure entangled states.
◦

6 Quantum maps and their representation

Parametrization of quantum maps
We have seen that for completely positive maps important representation theorems
exist, pointing either to a Kraus representation or to a dilation to a unitary transforma-
tion in a larger Hilbert space. Despite the relevance of these results, they apply only to
completely positive maps, and involve a certain arbitrariness. Indeed as we have seen
the Kraus operators as well as their cardinality are not uniquely fixed. At the same time
the dilation can be obtained in many different ways. It is therefore convenient to obtain
other unambiguous representations for maps on space of operators, including transfor-
mations which are not necessarily completely positive. This turns out to be feasible in a
general way in finite dimension. We will provide three related ways to perform such a
task. First we will see such a map as an operator in a Hilbert space and consider its
matrix representation in a suitable basis. We will then consider different bases of maps
and express the given map in terms of these bases, leading to the so-called χ-matrix rep-
resentation. At last we will obtain Choi’s theorem on the characterization of completely
positive maps, pointing to the connection between Choi’s matrix and the χ-matrix rep-
resentation.
Linear operators on Hilbert-Schmidt space
We therefore consider a Hilbert space of finite dimension H = Cn, so that the
bounded operators on this space can be identified with n × n matrices with complex
entries B(Cn) = Mn(C). The space B(Cn) is not only a Banach space, but thanks to the
constraint on the dimensions it is also a Hilbert space, with scalar product given by the
Hilbert-Schmidt expression. Each map acting on operators on Cn can thus be seen as a
linear operator in a Hilbert space, and for a fixed basis can be expressed accordingly as
a matrix. In particular, this map is a self-adjoint operator if

⟨Λ(χ), ω⟩HS = ⟨χ, Λ(ω)⟩HS ∀χ, ω ∈ B(Cn)

and it is a positive operator if

⟨ω, Λ(ω)⟩HS ! 0 ∀ω ∈ B(Cn),

where the Hilbert-Schmidt scalar product takes the form

⟨χ, ω⟩HS = Trχ †ω.

◦
◦
To obtain a linear matrix representation for Λ we first consider a basis in the Hilbert
space B(Cn), whose elements are therefore operators rather than “vectors”, we denote it
as {τα }α=1,...,n2, orthonormal with respect to the Hilbert-Schmidt scalar product

⟨τα , τ β ⟩HS = Trτα†τ β

= δαβ .
96 Table of contents

Each operator ω ∈ B(C 8 ) can

9 be identified in this basis as a vector with components
n

†
.
ω β = ⟨τ β , ω⟩HS = Tr τ β ω . We thus have, corresponding to the expression A. ψ ⟩ =
! .
. ϕn ⟩⟨ϕn |Aϕm ⟩⟨ϕm |ψ⟩
n,m
n 2
%
Λ(ω) = ⟨τα , Λ(τ β )⟩HS⟨τ β , ω⟩HSτα
α, β=1
n2
% K L 8 9
= Tr τα†Λ(τ β ) Tr τ β†ω τα
α, β=1
% n2
= Λαβ ω β τα ,
α, β=1

where we have defined the matrix of coefficients associated to the map in this basis as
K L
Λαβ = Tr τα†Λ(τ β ) .
◦
◦
◦

Given this representation of the map, the action of the map on an operator is trans-
lated to a multiplication between a matrix and a vector, and the composition of maps is
translated to the product of matrices. One has in fact
B C
(Λ(ω))γ = Tr τ γ†Λ(ω)
n 2
%
= Λ γβ ω β ,
β=1

and also
K L
(Λ ◦ Γ)αβ = Tr τα†(Λ ◦ Γ)(τ β )
K L
= Tr τα†Λ(Γ(τ β ))
%n2 U 8 9 V
= Tr τα†Λα ′ β ′Tr τ β† ′Γ(τ β ) τα ′
α ′,β ′ =1
%n2
K L
= Λα ′ β ′Γβ ′ β Tr τα†τα ′
α ′,β ′ =1
%n2
= Λαβ ′Γβ ′ β .
β ′ =1

Hermiticity and positivity of Λ as linear operator on the Hilbert space reflect themselves
in hermiticity and positivity of the matrix Λαβ . For example one has
⟨ω, Λ(ω)⟩HS = Tr(ω †Λ(ω))
n2
% K L 8 9
= Tr(ταω †)Tr τα†Λ(τ β ) Tr τ β†ω
α, β=1
% n2
= ωα∗ Λαβω β
α, β=1
! 0 ∀ω ∈ B(Cn),

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6 Quantum maps and their representation 97

so that the linear operator Λ is positive iff such is the matrix Λαβ . Note that this prop-
erty does not correspond to the notion of positivity or complete positivity of Λ as a
map.
Canonical basis and Bloch basis representation
We now point to two particularly relevant examples of basis in B(Cn). The first is
the so-called canonical basis, given by matrices with only one non zero entry equal to
one. Given an orthonormal system in Cn, say {ϕi }i=1,...n , we have the operator basis
{Eij }i, j=1,...n with Eij = |ϕi ⟩⟨ϕ j |, often simply denoted |i⟩⟨j |. In B(C2) this basis
becomes {P+, σ−, σ+, P−} = {|+⟩⟨+|, |−⟩⟨+|, |+⟩⟨−|, |−⟩⟨−|}.
The other example is the so-called Bloch basis, in which the first element of the basis
is proportional to the identity, and the others are taken to be hermitian and traceless,
namely σ0 = 1, σi = σi†, Tr σi = 0 for i = 1, ..., n2 − 1 together with Tr σiσ j = n δij . In such
a case the action of a map has a geometric interpretation as they can be read as affine
transformations. We can in fact write
1
ρ = (1 + σ · r),
n
2
where r ∈ Rn −1 and is given by r = Tr ρσ.
◦
Putting for convenience
4 5
1 r
t = ,
n n
we also have
n −1 2
1%
ρ = σj Tr ρσ j
n
j=0
2 −1
n%
= σj tj.
j=0

The action of the map now reads, with Λαβ → Λ jk

n −1 2
1 %
Λ(ρ) = Λ jk σj Tr ρσk
n
j ,k =0
2 −1
n%
= σ j t j′
j=0

with
2 −1
n%
t j′ = Λ jk tk.
k=0

Assuming further the map to be trace preserving we have the constraint t0 = t0′ = 1/n.
This implies in particular Λ0k = δ0k so that one has

t0 → t0′ = Λ00t0 = t0
2 −1
n%
′
t j → tj = Λ jk tk + Λj0t0 j=
/ 0.
k=1
98 Table of contents

Putting Λ j0 = (b) j and Tij = Λij for i, j = 1, ..., n2 − 1, the transformation associated to
the trace preserving map Λ can be described as an affine transformation in the para-
meter space
r → r′ = T r + b
ruled by the matrix
4 5
1 0
Λ = .
b T
◦
Map basis representation
Let us now take a slightly different point of view, and look at Λ, rather than as a
linear operator in Hilbert space, as a linear map from B(Cn) into itself, Λ ∈ L(B(Cn),
B(Cn)). Such a space is also a Hilbert space of dimension n2 × n2, which can be seen as
2
a direct sum ⊕nk=1 B(Cn), and in it we have a scalar product constructed from the one in
B(Cn), namely for any basis {τ γ } γ =1,...,n2
n 2
%
⟨Λ, Γ⟩ = ⟨Λ(τ γ ), Γ(τ γ )⟩HS
γ=1
%n2
= Tr(Λ(τ γ ) †Γ(τ γ )).
γ=1
◦
In this space of maps one can introduce two relevant basis of maps, orthonormal
according to this scalar product. Let us first define a basis Eαβ as follows
8 9
Eαβ (ω) = ταTr τ β†ω ,
orthonormal as follows from
n 2
%
⟨Eαβ , Eα ′β ′⟩ = ⟨Eαβ (τ γ ), Eα ′β ′(τ γ )⟩HS
γ =1
%n2 8U 8 9V† 8 99
= Tr ταTr τ β†τ γ τα ′Tr τ β† ′τ γ
γ =1
= δαα ′δ ββ ′.
This basis is strictly related to the matrix representation we have considered before. Let
us indeed expand the generic map Λ in this basis, we have
n 2
%
Λ(ω) = ⟨Eαβ , Λ⟩Eαβ (ω)
α, β=1
%n2 %n2
= ⟨Eαβ (τ γ ), Λ(τ γ )⟩HS Eαβ (ω)
γ =1 α, β=1
%n2 % n2 U8 8 99† V
= Tr ταTr τ β†τ γ Λ(τ γ ) Eαβ (ω)
γ =1 α, β=1
%n2
K L
= Tr τα†Λ(τ β ) Eαβ (ω)
α, β=1
% n2
= Λαβ Eαβ (ω),
α, β=1

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6 Quantum maps and their representation 99

so that the matrix associated to Λ in this basis of maps, namely Λαβ = ⟨Eαβ , Λ⟩, is the
same encountered before considering it as a linear operator, and the same considerations
apply. Let us now introduce another operator basis Fαβ defined as

Fαβ (ω) = τα ω τβ†,

orthonormal as follows from

n 2
%
⟨Fαβ , Fα ′β ′⟩ = ⟨Fαβ (τ γ ), Fα ′β ′(τ γ )⟩HS
γ=1
%n2 88 9† 9
= Tr τα τ γ τ β† τα ′τ γ τ β† ′
γ=1
%n2 8 9
′
= Tr τ β τ γ† τα†τα ′τ γ τ β† ′
γ=1
8 B C ′9
= Tr τ β Tr τα†τα ′ τ β† ′
= δαα ′δ ββ ′,

where we have exploited the identity

n 2
%
τ γ† ρ τγ = 1Tr ρ,
γ =1

which can be verified exploiting invariance with respect to the choice of basis and using
the canonical basis.
◦
Let us consider the expression of the matrix of coefficients expressing a generic map
Λ in this basis
n 2
%
Λ(ω) = ⟨Fαβ , Λ⟩Fαβ (ω)
α, β=1
%n2 %n2
= ⟨Fαβ (τ γ ), Λ(τ γ )⟩HS Fαβ (ω)
γ =1 α, β=1
%n2 %n2 U8 9† V
= Tr τα τ γ τ β† Λ(τ γ ) Fαβ (ω)
α, β=1 γ=1
% n2 %n2
K L
= Tr τ β τ γ† τα†Λ(τ γ ) Fαβ (ω)
α, β=1 γ=1
% n2
′
= Λαβ Fαβ (ω),
α, β=1

where we have defined as

n 2
% K L
′
Λαβ = Tr τ β τ γ† τα†Λ(τ γ )
γ =1
100 Table of contents

◦
′
the matrix representation of the map in the basis Fαβ , that is Λαβ = ⟨Fαβ , Λ⟩. The
′
matrix Λαβ , also called χ-matrix representation, provides most important information
on the properties of Λ as a map. Indeed from
⎛ ⎞
n 2 †
% †⎠
(Λ(ω)) † = ⎝ ′
Λαβ τα ω τβ
α, β=1
n 2
%
′ ∗
= Λαβ τ β ω † τα†
α, β=1
% n2
= ′ ∗
Λ βα τα ω † τ β†,
α, β=1

one has that Λ is a hermiticity preserving map, that is

(Λ(ω)) † = Λ(ω †)

′ ′ ∗ ′
iff Λαβ = Λ βα , that is the matrix Λαβ is hermitian. Most importantly one has the result
′
that Λ is completely positive iff the matrix Λαβ is non negative. To see this suppose first
′
Λαβ is non negative, then one has

′
Λαβ = (U DU †)αβ
%n2
= λ γuαγu∗βγ ,
γ =1

where U is a unitary matrix and D a diagonal matrix with entries the positive eigen-
′
values of Λαβ . We thus obtain the Kraus decomposition
⎛ ⎞⎛ ⎞
n 2 n 2 n 2
% % %
Λ(ω) = λγ⎝ uαγτα ⎠ω⎝ u∗βγτ β† ⎠
γ =1 α=1 β=1
n2
%
= λγ τ̃ γ ωτ̃ γ†,
γ =1

which is called canonical decomposition, since the Kraus operators {τ̃ γ } are orthogonal,
and is furthermore unique if the eigenvalues are non degenerate. This decomposition,
which is a Kraus representation just because λ γ ! 0, warrants directly complete posi-
tivity of the map, not only positivity. Note in particular that by obtaining a diagonal
′
representation of Λαβ we automatically obtain a canonical Kraus decomposition of the
map.
Vice versa, suppose to have a completely positive map Λ, then it admits a Kraus
decomposition of the form

n2
%
Λ(ω) = A γ ω A γ† ,
γ=1

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6 Quantum maps and their representation 101

and expressing the A γ in the basis {τα } we have

n 2
%
Aγ = c γατα ,
α=1

and therefore
⎛ ⎞ ⎛ ⎞
n 2 n 2 n2 †
% % %
Λ(ω) = ⎝ c γατα ⎠ω⎝ c γβτ β ⎠
γ=1 α=1 β=1
⎛ ⎞
%n2 % n2
= ⎝ c γαc∗γβ ⎠τα ω τ β†,
α, β=1 γ=1

which is of the form

n 2
%
Λ(ω) = ′
Λαβ τα ω τβ†,
α, β=1

with the matrix of coefficients given by

n 2
%
′
Λαβ = c γαc∗γβ
γ=1

and therefore positive since

⎛ ⎞⎛ ⎞∗
n 2 n2 n 2 n 2
% % % %
′
vα Λαβ v∗β = ⎝ c γα vα ⎠⎝ c γβ v β ⎠
α, β=1 γ =1 α=1 β=1
! 0.
′
In the canonical basis the matrices Λαβ and Λαβ are related by a simple but crucial
exchange of indexes as follows. Let us take τα → |i⟩⟨j | and τ β → |k⟩⟨l|.
◦
We have, suitably inserting completenesses and exploiting the linearity of the map
n
%
Λ(ρ) = |i⟩⟨i|Λ(ρ)|k⟩⟨k|
i,k=1 ⎛ ⎞
n
% n
%
= |i⟩⟨i|Λ⎝ |j ⟩⟨j |ρ|l⟩⟨l| ⎠|k⟩⟨k |
i,k=1 j ,l =1
%n
= ⟨i|Λ(|j ⟩⟨l|)|k ⟩ |i⟩⟨j |ρ|l⟩⟨k |
i, j ,k ,l =1
% n2
= ′
Λαβ τα ρ τβ†,
α, β=1

leading to the identification

′
Λαβ = ⟨i|Λ(|j ⟩⟨l|)|k ⟩
′
= Λij ,kl.
102 Table of contents

In a similar way, aiming to express the map in the other basis of maps we have
n
%
Λ(ρ) = |i⟩⟨j | ⟨i|Λ(ρ)|j ⟩
i, j =1
% n
= |i⟩⟨j | Tr(|j ⟩⟨i|Λ(ρ))
i, j =1
% n n
%
= |i⟩⟨j | Tr(|j ⟩⟨i|Λ( |k⟩⟨l|Tr[|l⟩⟨k|ρ]))
i, j =1 k,l =1
% n
= Tr[|j ⟩⟨i|Λ(|k⟩⟨l|)] |i⟩⟨j | Tr(|l⟩⟨k|ρ)
i, j ,k ,l =1
% n2 8 9
= Λαβ τα Tr τ β† ρ ,
α, β=1

leading to the identification

Λαβ = ⟨i|Λ(|k⟩⟨l|)|j ⟩
′
= Λik ,jl

so that in this basis the matrices are related by a suitable exchange of indexes. Note
however their basically different meaning with reference to properties of the linear oper-
′
ator or map Λ. As we shall see for a suitable choice of basis the matrix Λαβ is also
known as Choi matrix.
◦

To stress the difference between positivity as linear operator, namely

⟨ω, Λ(ω)⟩HS ! 0 ∀ω ∈ B(Cn),
and positivity as a map, that is to say
Λ(ω) ! 0 ∀ω ∈ B(Cn), ω ! 0,
we point to two simple examples. They can be obtained taking as example maps
obtaining in the previously introduced basis Fαβ and Eαβ with the choice τα = τ β =
1
√ σz . The map
2

Λ(ω) = Fzz (ω)

1
= σz ω σz†
2
is obviously completely positive, but it is not a positive linear operator, in fact
⟨σx , Λ(σx)⟩HS = −1.
On the other side the linear operator
Λ(ω) = Ezz (ω)
1 B C
= σz Tr ω σz†
2
is positive since
1 B C
⟨ω, Λ(ω)⟩HS = Tr (ω † σz )Tr ω σz†
2
! 0,

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6 Quantum maps and their representation 103

but
1
Λ(|+⟩⟨+|) = σz
2
1
= (|+⟩⟨+| − |−⟩⟨−|),
2
so that the map is not positive, and therefore in particular it cannot be completely posi-
tive.
◦
As a last remark we point to the n4 × n4 matrix which connects the two operator
bases
K L
Λαβ = Tr τα†Λ(τ β )
⎡ ⎤
%n2
= Tr⎣ τα† ′
Λ γδ τ γτ β τδ† ⎦
γ ,δ =1
n2
% K L ′
= Tr τα† τ γ τ β τδ† Λ γδ
γ ,δ =1
% n2
′
= Mαβ ,γδ Λ γδ
γ ,δ =1
K L
where we have defined Mαβ ,γδ = Tr τα† τ γ τ β τδ† .
◦
Choi-Jamiolkowski isomorphism
A further important characterization of completely positive maps is given by a the-
orem due to Choi, which connects complete positivity of a map with positivity of a cer-
′
tain matrix, which will turn out to be related to the matrix Λαβ introduced in connec-
tion corresponding to the so-called χ-matrix representation of the map, for a suitable
choice of basis. In the proof of the theorem we will use the following operator on Cn ⊗
Cn
n
1 1 %
X = E jk ⊗ E jk
n n
j ,k =1
% n
1
= |e j ⊗ e j ⟩⟨ek ⊗ ek |,
n
j ,k =1

where as before {e j } denote the canonical basis of vectors in Cn, and {E jk } the canon-
ical basis of matrices in Mn(C). This operator is actually the projection on the maxi-
mally entangled state
n
1 %
Ψ = √ |e j ⊗ e j ⟩,
n j=1

so that it is a state. In particular X is a positive operator.

◦
We have the following important result
Theorem (Choi, 1972) Consider the linear map Λ: Mn(C) → Md(C), then the fol-
lowing statements are equivalent:
i. Λ is completely positive
ii. Λ is n-positive, that is Λ ⊗ 1Cn is a positive map
104 Table of contents

iii. For any orthonormal basis {e j } in Cn the n d × n d square matrix (known as Choi
matrix of Λ)
⎛ ⎞
Λ(|e1⟩⟨e1|) ... Λ(|e1⟩⟨en |)
ΦΛ = ⎝ ·· ·· ·· ⎠
· · ·
Λ(|en ⟩⟨e1|) ... Λ(|en ⟩⟨en |)
is positive.
Let us prove the equivalence.
i.⇒ii. Follows from the definition
ii.⇒iii. Follows observing that
ΦΛ = Λ ⊗ 1Cn[X]
and as shown above X is a positive operator.
iii.⇒i. A constructive proof which shows the insurgence of the Kraus operators goes
as follows. We have the following way to express the Choi matrix
ΦΛ = Λ ⊗ 1Cn[X]
%n
= Λ[Ekq ] ⊗ Ekq
k,q=1
%n
= Λ[|ek ⟩⟨e q |] ⊗ |ek ⟩⟨e q |
k,q=1

and under the hypothesis that ΦΛ is a positive matrix we can diagonalize it, and using
non normalized eigenvectors obtain the representation
nd
%
ΦΛ = |v j ⟩⟨vj |,
j =1

with vj ∈ Cd ⊗ Cn, so that with {ej } basis in Cn and v jk vectors in Cd we can write
n
%
vj = v jk ⊗ ek
k=1
and define an operator which relates the canonical basis in Cn to the vectors v jk
V j : Cn → Cd
ek " vjk
and finally obtain
nd %
% n
ΦΛ = |v jk ⊗ ek ⟩⟨v jq ⊗ e q |
j =1 k,q=1
%nd % n
= |v jk ⟩⟨v jq | ⊗ |ek ⟩⟨e q |
j =1 k,q=1
%nd % n
= Vj |ek ⟩⟨eq |Vj† ⊗ |ek ⟩⟨e q |
j =1 k,q=1
% n %nd
= VjEkqV j† ⊗ Ekq .
k,q=1 j=1
◦

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6 Quantum maps and their representation 105

Comparing with the previous representation and noting that the Ekq are linearly
independent this implies
nd
%
VjEkqV j† = Λ[Ekq ] ∀Ekq ,
j=1
but the {Ekq } provide a basis in Mn(C), so that the representation holds for any oper-
ator. We thus have obtained a Kraus representation for the map
nd
%
Λ[T ] = VjT V j†,
j=1
which as shown implies its complete positivity.
As shown by the proof of the theorem a Kraus representation arises from the eigen-
value decomposition of the Choi matrix representation.
This result by Choi shows in particular that for a map defined on a space of opera-
tors on a Hilbert space of dimension n, complete positivity is equivalent to n-positivity.
′
In order to see the explicit connection between the Choi matrix and the matrix Λαβ
introduced in connection with the so-called χ-matrix representation of the map, we con-
sider its defining expression
n 2
% K L
′
Λαβ = Tr τ β τ γ† τα†Λ(τ γ )
γ =1
and make the choice of basis τ γ → |e q ⟩⟨em |, so that, for α → i, j and β → k, l we have
indeed
% n
′
Λij ,kl = Tr[|ek ⟩⟨el |em ⟩⟨eq |e j ⟩⟨ei |Λ(|e q ⟩⟨em |)]
q,m=1
= ⟨ei |Λ(|e j ⟩⟨el |)|ek ⟩
= ⟨ei ⊗ e j |ΦΛ|ek ⊗ el ⟩
%n
= ⟨ei ⊗ e j | Λ[Ers ] ⊗ Ers |ek ⊗ el ⟩,
r,s=1
that is the i, k matrix element of the d × d block Λ(|e j ⟩⟨el |). Therefore Choi matrix and
′
the matrix Λαβ do coincide upon taking in the Hilbert-Schmidt space the canonical basis
of operators arising from the same Hilbert basis {e j } j=1,...,n used to obtain the Choi
matrix.
Relying on this construction one can also build a correspondence, known as Choi-
Jamiolkowski isomorphism, between the elements of the vector space of linear maps Λ ∈
L(Mn(C), Md(C))
Λ: Mn(C) → Md(C)
and the linear operators ΦΛ ∈ B(Cd ⊗ Cn) on the Hilbert space Cd ⊗ Cn
ΦΛ: Cd ⊗ Cn → Cd ⊗ Cn
according to
J : L(Mn(C), Md(C)) " B(Cd ⊗ Cn)
Λ " J [Λ] ≡ Λ ⊗ 1[PΨ],
where PΨ is the projection on a maximally entangled state, which we previously denoted
as (1/n)X.
◦
106 Table of contents

◦
This correspondence is actually an isomorphism, in fact we can introduce the inverse
mapping defined as
J −1: B(Cd ⊗ Cn) " L(Mn(C), Md(C))
Φ " (J −1Φ)[Y ] ≡ n TrCn {(1Cd ⊗ Y T )Φ},
with Y generic operator in Mn(C), and the trace is on the second factor of the tensor
product, thus leading to an operator in Md(C). In this correspondence completely posi-
tive maps are sent to positive operators, and in particular completely positive trace pre-
serving maps, that is channels, are sent to states, in a suitably higher dimensional space.
Linearity of both maps is evident by construction, so that we only have to check that
indeed they are one the inverse of the other. The operator J −1 ◦ J acts as the identity
on the space of maps, indeed if we apply it to a generic map Λ, we have ∀ Y ∈ Mn(C)
(J −1 ◦ J )Λ[Y ] = n TrCn⎧{(1Cd ⊗ Y T )Λ ⊗ 1Cn[PΨ]} ⎫
⎨ %n ⎬
1
= n TrC (1Cd ⊗ Y )
n
T Λ[Ekq ] ⊗ Ekq
⎩ n ⎭
k,q=1
n
%
= Λ[Ekq ]TrCn {Y TEkq }
k,q=1
%n
= Λ[Ekq ]⟨e q |Y T |ek ⟩
k,q=1
%n
= Λ[Ekq ]⟨ek |Y |e q ⟩
k,q=1
⎡ ⎤
n
%
= Λ⎣ ⟨ek |Y |e q ⟩Ekq ⎦
k,q=1
⎡ ⎤
n
%
= Λ⎣ |ek ⟩⟨ek |Y |e q ⟩⟨e q | ⎦
k,q=1
= Λ[Y ],
where we have used the fact that the trace only acts on the second factor and linearity
of the map Λ. Since the action of (J −1 ◦ J )Λ and Λ coincide on the generic operator,
indeed J −1 ◦ J acts as the identity. We now consider the operator J ◦ J −1 and verify
that it acts as identity on the space of operators on Cd ⊗ Cn. We have, using the pre-
vious notation
(J ◦ J −1)Φ = (J −1Φ) ⊗ 1Cn[PΨ]
n
1 %
= (J −1Φ)[Ekq ] ⊗ Ekq
n
k,q=1
n
1 % IB C J
= T
n TrCn 1Cd ⊗ Ekq Φ ⊗ Ekq
n
k,q=1
n
%
= ⟨ek |Φ|e q ⟩ ⊗ Ekq
k,q=1
%n
= |ek ⟩⟨ek |Φ|eq ⟩⟨eq |
k,q=1
= Φ,

@ B. Vacchini - Unimi & INFN

7 General dynamics of open quantum systems 107

where we recall that ⟨ek |Φ|e q ⟩ ∈ Md(C), so that again indeed J ◦ J −1 acts as the iden-
tity.
◦

7 General dynamics of open quantum systems

Quantum dynamical semigroups
Moving from closed to open systems, we consider quantum dynamical maps, that is
maps giving the dynamics of the considered system, which are not necessarily given by
unitary transformations, but we still ask for positivity and trace preservation, manda-
tory for the probabilistic interpretation. We will further keep complete positivity, as
entailed by the existence of a microscopic dynamics.
The reversible dynamics of a closed system described by a group of transformations
now is generally replaced by an irreversible dynamics, which can at most satisfy a semi-
group composition law.
An explicit general characterization of quantum dynamical maps is known only for
special cases. An important and general class is obtained just assuming a semigroup
composition law for the quantum dynamical map as a function of the time argument
Φ(t + s) = Φ(t) ◦ Φ(s) ∀t, s ! 0
where each Φ(t) is a completely positive trace preserving map. This is called the (time-
homogeneous) Markovian case. A one-parameter group of unitary operators according to
Stone’s theorem is described by its generator given by a self-adjoint operator. Likewise
for a semigroup of contraction operators a generator characterized by the Hille-Yoshida
theorem exists such that
Φ(t) = etL ,
with L given by
Φ(t) − 1
L = lim .
t→0+ t
Setting ρ(t) = Φ(t)ρ(0) one has the so-called master equation for the time evolution of
the statistical operator
d
ρ(t) = Lρ(t),
dt
which for the case of Φ(t) actually given by a unitary evolution gives back the Liouville
von Neumann equation
d
ρ(t) = −i[H(t), ρ(t)],
dt
with H(t) a self-adjoint operator.
If for any t the map is completely positive then the collection of these maps is called
a quantum dynamical semigroup.
A quantum dynamical semigroup is therefore a collection of maps Φ(t) such that
Φ(t + s) = Φ(t) ◦ Φ(s) ∀t, s ! 0
Φ(t) is completely positive trace preserving ∀t ! 0
Φ(t) → 1 for t→0
Φ(t) is continuous ∀t ! 0
108 Table of contents

Physical conditions allowing for semigroup dynamics are typically given by

τE ≪ τR
i.e. environment correlation time (decay time of correlation function) much shorter than
relaxation time of the reduced system. This separation of time scales (so-called Markov
condition) together with weak coupling (so-called Born approximation), which also justi-
fies the choice of a factorized initial state, typically allows for a description of the
dynamics in terms of a quantum dynamical semigroup.
Structure of the generator

The characterization of the structure of the generators of quantum dynamical semi-

group is given by the famous Gorini Kossakowski Sudarshan Lindblad theorem. Its finite
dimensional version reads:

Theorem (Gorini, Kossakowski and Sudarshan, 1976; Lindblad, 1976) Let dim HS =
n, a linear operator L: T (HS ) 6→ T (HS ) is the generator of a quantum dynamical semi-
group, that is to say a one-parameter continuous semigroup of completely positive trace
preserving maps Φ(t) = etL iff it is of the form
2 −1
n% 6 7
† 1I † J
L[ρ] = −i[H , ρ] + γk LkρLk − LkLk , ρ
2
k=1
n%2 −1
1 IK L K LJ
= −i[H , ρ] + γk Lk , ρLk† + Lkρ, Lk†
2
k=1
with γk ! 0; H = H †, Lk ∈ B(HS ).

The result extends to infinite dimensional Hilbert spaces provided one asks for norm
continuity of Φ(t). Most importantly, it is a necessary and sufficient condition. We give
an idea of the proof, pointing to extensions of the sufficient condition to account for
more general situations.

The key ingredients are: the Hille-Yoshida theorem for the existence of a generator,
the Kraus representation for a completely positive map, the Lindblad expression of the
generator ensuring trace and hermiticity preservation.

Necessary condition
If Φ(t) is completely positive trace preserving, according to the Kraus representation
! † ! †
at any time it can be written as Φ(t)[ρ] = i Ai(t)ρAi (t) with i Ai (t)Ai(t) = 1.
Writing the Ai(t) in terms of a basis {Fi }i=0,1,...,n2 −1 of operators orthonormal with
respect to the Hilbert-Schmidt scalar product, so that ⟨Fi , Fj ⟩HS = TrHSFi†F j = δij , set-
1 ! 2−1
/ 0, one has Ai(t) = nk=0
ting F0 = √n , so that the F j are traceless for j = vik (t)Fk and
therefore
2 −1
n%
Φ(t)[ρ] = cij (t)FiρF j†
i, j =0
!n2−1
with cij (t) = k=0 vki (t)vkj
∗
(t) a positive matrix. We know that the generator exists
and is given by
Φ(t) − 1
lim [ρ] = L[ρ].
t→0+ t

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7 General dynamics of open quantum systems 109

Relying on the existence of the limit and imposing trace preservation one comes to the
desired result.
◦
More specifically we have
Φ(t) − 1
L[ρ] = lim [ρ]
t→0+ ⎧t
⎪
⎪ n2 −1 n −12
⎨ 1 c00(t) − n 1 % ck0(t) 1 % c∗k0(t) †
= lim ρ + √ Fk ρ + √ ρFk +
t→0+ ⎪ n NNNNNNNNNNN t QQQQQQQQQQQ
NOPQ n k=1 t n k=1 t
⎪
⎩ NNNNNNNNNNNNNNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
a00 ⎫
F
2
n% −1 ⎪
⎪
ckl (t) †
⎬
FkρFl
NNN t QQ
k,l=1 NNOPQ QQQ ⎪
⎪
⎭
akl

−1
n% 2
a00
= ρ + Fρ + ρF +
† akl FkρFl†
n
k,l=1

which can also be written

6 † 7 S 2 −1
T n%
F −F 1 a00
L[ρ] = −i ,ρ + 1+F +F , ρ +
† akl FkρFl†.
2i 2 n
k,l =1

We now ask for trace preservation which implies

% n2 −1
a00
1+F +F† = − akl Fl†Fk
n
k,l =1

so that defining the self-adjoint operator

F†−F
H =
2i
one obtains the following form for the generator
2 −1
n% 6 7
† 1I † J
L[ρ] = −i[H , ρ] + akl FkρFl − Fl Fk , ρ ,
2
k,l=1

where indeed H = H †, and akl is a positive matrix known as Kossakowski matrix, as fol-
lows from positivity of ckl (t) and the relation akl = limt→0+ckl (t)/t.
◦
◦
Diagonalization of the positive matrix (A)kl = akl according to

A = U ΛU †
!
leads to an explicit Lindblad form with Lk = i uik Fi. In Schrödinger picture
2 −1
n% 6 7
1I † J
L[ρ] = −i[H , ρ] + γk LkρLk† − LkLk , ρ
2
k=1
2 −1
n%
1 IK L K LJ
= −i[H , ρ] + γk Lkρ, Lk† + Lk , ρLk†
2
k=1
110 Table of contents

corresponding to the Heisenberg picture

2 −1
n% 6 7
1I † J
L ′[X] = +i[H , X] + γk Lk†X Lk − L L ,X ,
2 k k
k=1
which can be equivalently written in so-called standard form by means of the completely
positive map
2 −1
n%
Ψ[X] = γkLk†X Lk
k=1
as follows
1
L ′[X] = +i[H , X] + Ψ[X] − {Ψ[1], X }.
2
The {Lk } are typically called Lindblad operators.
◦
◦
Note that trace preservation in Schrödinger picture corresponds to identity preserva-
tion in Heisenberg picture. Note furthermore that the generator does not uniquely fix
the operators H and {Lk }. Indeed the expression of L is invariant under the following
transformations.
√ : ! √
1. γk Lk → γk′ Lk′ = j ukj γ j Lj with (U )ij = uij unitary matrix
1! B C
2. Lk → Lk′ = Lk + ak together with H → H ′ = H + b + 2i j γj a∗jL j − a jLj† , where
ak ∈ C and b ∈ R
the latter transformation shows that (in finite dimension) H and {Lk } can be taken to
be traceless.
◦
Sufficient condition I
We want to see that a generator in Lindblad form leads to a completely positive
map. The semigroup composition law follows from the exponential representation Φ(t) =
etL. Preservation of hermiticity and trace is immediately checked.
◦
To convince oneself of complete positivity of the time evolution the following repre-
sentation for infinitesimal d t might be inspiring
⎛ ⎞ ⎛
n%2 −1
i 1 i
ρ(t + d t) = ⎝ 1 − H dt − γkLk†Lkdt ⎠ρ(t)⎝ 1 − H dt −
! 2 !
⎞ k=1
2 −1
n% †
1 †
%
⎠
γkLkLkdt + γkLkρLk† dt +O((d t)2).
2
k=1 k

We show complete positivity pointing to a perturbation expansion of the solution. Let

us put L = LR + LJ , sum of a relaxing and jump part, according to
$ &
1 % †
LR[ρ] = −i[H , ρ] − γkLkLk , ρ
2
k
B † C
= −i Heff ρ − ρHeff
%
LJ [ρ] = γkLkρLk†
k

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7 General dynamics of open quantum systems 111

i!
with Heff = H − 2 k γkLk†Lk and LJ a completely positive map. We know that
d
ρ(t) = Lρ(t)
dt
and therefore due to ρ(t) = Φ(t)ρ(0) also
d
Φ(t) = LΦ(t) Φ(0) = 1.
dt
Denoting by R(t) the solution of the relaxing part
d
R(t) = LRR(t) R(0) = 1
dt
given by
†
R(t)[ρ] = eLRt[ρ] = e−iH eff tρe+iH eff t
one has the representation, known as Duhamel’s formula
# t
Φ(t) = R(t) + dτR(t − τ )LJ Φ(τ )
0
# t
= eLRt + dτ eLR(t−τ )LJ Φ(τ )
0
= R(t) + (R ⋆ LJ Φ)(t).
Indeed with this definition one has
ρ(t) = Φ(t)ρ(0)
# t %
† †
= e−iH eff tρ(0)e+iHeff t + dτ e−iHeff (t−τ ) γkLkρ(τ )Lk† e+iHeff (t−τ )
0 k
and one can directly verify that the master equation holds since the time derivative of
the r.h.s. reads
†
%
−i Heff ρ(t) + iρ(t) Heff + γkLkρ(t)Lk† .
k

This equation is of the form G = G0 + G0V G and is solved by the Dyson series
Φ(t) = R(t) + (R ⋆ LJR)(t) + (R ⋆ LJR ⋆ LJR)(t) + ...
which is a completely positive map by construction, because such are R(t) and LJ .
Apparently it is enough to ask R(t) and LJR(t) to be completely positive, but the
inverse of R(t) is also completely positive, so that this requirement is not weaker. The
solution of the master equation can thus be explicitly written as a jump expansion as
follows
ρ(t) = Φ(t)ρ(0)
∞ #
% t # t2
= R(t)ρ(0) + dtn... dt1R(t − tn)LJR(tn − tn−1)...LJR(t1)ρ(0).
n=1 0 0

Sufficient condition II
The proof still works if we allow the operators appearing in the Lindblad structure as
well as the decay rates γk to become time dependent, provided the latter always stay
positive. This situation corresponds to a time-inhomogeneous Markovian case. Starting
from
d
Φ(t) = L(t)Φ(t)
dt
112 Table of contents

with Φ(t) ≡ Φ(t, 0) and the initial condition Φ(t, t) = 1, we can still consider a relaxing
part LR(t) and a jump part LJ (t), where the latter still is a completely positive map
thanks to the positivity of the γk(t). As done before starting from the solution of the
time-local master equation
d
R(t, s) = LR(t)R(t, s)
dt
where R(t) ≡ R(t, 0), R(t, t) = 1 and t ! s ! 0, given by
4# t 5
⃗
R(t, s) = T exp dτLR(τ )
s
⃗ denotes the chronological time ordering.
where T
◦
This is a completely positive map satisfying the two-time composition law
R(t, τ ) ◦ R(τ , s) = R(t, s) ∀t ! τ ! s.
As a result we can still write a Dyson expansion for the time evolution map, whose very
structure ensures complete positivity of the time evolution. One has
ρ(t) = Φ(t)ρ(0)
= R(t)ρ(0)
∞ # t
% # t2
+ dtn... dt1R(t, tn)LJ (tn)R(tn , tn−1)...R(t2, t1)LJ (t1)R(t1)ρ(0),
n=1 0 0

and therefore a time evolution map characterized by two time indexes, satisfying the
composition law
Φ(t, τ ) ◦ Φ(τ , s) = Φ(t, s) ∀t ! τ ! s
where each of the maps Φ(t, s) is completely positive and can be written as
4# t 5
⃗
Φ(t, s) = T exp dτL(τ ) .
s
This kind of time local master equations arise e.g. in the time-convolutionless projection
operator technique.
◦
Decoherence
Making reference to master equations in Lindblad form we now provide a phenome-
nological description of the phenomenon of quantum decoherence for a massive test par-
ticle interacting through collisions with a background gas. Now the relevant Hilbert
space for the system is L2(R3), while the environment is a gas of identical quantum par-
ticles. We have in mind to describe in most simple terms the effect of the collisions on
the dynamics of the otherwise free particle. This situation is relevant if one considers an
experiment in which interference patterns of massive particles are observed, letting them
propagate through a suitably devised interferometer, e.g. with a Mach-Zender geometry.
The interaction with the background gas, which can be neglected if the experiment is
performed at very high vacuum, but is present otherwise, will bring with itself a loss of
visibility in the interference fringes. Phenomena of this kind, in which as a consequence
of interaction with a quantum environment typical quantum coherent effects like visi-
bility fringes are suppressed, go under the name of decoherence. We derive in a heuristic
phenomenological way the master equation which describes this effect. The result can be
confirmed on the basis of a microscopic description.

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7 General dynamics of open quantum systems 113

Given that for a dilute gas the collisions can be considered to be independent, a Mar-
kovian description naturally applies. The Hamiltonian term can be considered to be
given by the free kinetic term, while we assume that the basic interaction mechanism
with the environment is given by collisions in which the particle exchanges momentum
with the gas. As a result the basic microscopic interaction event leads to a change of the
system momentum from p to p + q according to
. i .
. q·x̂ − i q·x̂.
⟨p|ρ|p⟩ → ⟨p + q|ρ|p + q ⟩ = ⟨p.e ! ρe ! .p⟩,

where we have denoted with x̂ the position operators which are also the generators of
translations in momentum space. The Lindblad operators describing the microscopic
interaction can therefore be identified with unitary operators describing a momentum
transfer. Since the exchanged momentum in the single collisions is actually a random
variable, we integrate over all possible momentum transfers, weighted according to a cer-
tain probability distribution P(q) to be determined on the basis of some phenomenolog-
ical input. Based on these suggestions we therefore introduce the following Lindblad
master equation
6 7 # U i V
d i p̂ 2 i
q·x̂ − q·x̂
ρ = − , ρ + Λ dq P(q) e ! ρe ! −ρ ,
dt ! 2m

where m is the mass of the test particle, Λ is a constant with the dimension of inverse
time, quantifying the rate of collisions, that is their number per unit interval of time,
and the loss term, generally corresponding to an anticommutator, is simply proportional
to the identity, since the Lindblad operators are unitary. It is worth looking at this
master equation in the position representation. The matrix elements then read
# U i V
d ! q·(x−y)
⟨x|ρ|y⟩ = i (∆x − ∆ y)⟨x|ρ|y⟩ + Λ dq P(q) e ! − 1 ⟨x|ρ|y⟩.
dt 2m

Neglecting in the first instance the contribution due to free motion one notices that in
this basis the action of the dissipative part is simply multiplicative in this basis. This
allows to simply write down the solution of the master equation in terms of the initial
condition

⟨x|ρ(t)|y⟩ = e−Λ[1−ΦP (x−y)]t ⟨x|ρ(0)|y⟩,

where ΦP (x) is the Fourier transform of the probability distribution P(q), also known
as its characteristic function. To see the relevance of this equation for the description of
loss of interference, let us consider the case in which relevant momentum transfers are
typically small, so that that we can approximate

i 1
ΦP (x) ≃ 1 + (x − y) · ⟨q ⟩P − 2 (x − y)2⟨q 2⟩P + ...,
! 2!

where we have introduced first and second momentum of the probability distribution of
transferred momenta. Assuming as natural isotropy, so that the first moment vanishes,
we are left with
Λ
− (x−y)2⟨q2⟩P t
⟨x|ρ(t)|y⟩ ≃ e 2! 2 ⟨x|ρ(0)|y⟩,
114 Table of contents

so that while the diagonal matrix elements of the statistical operator in the position rep-
resentation are left untouched, as follows from trace preservation, the off-diagonal matrix
elements in this basis, corresponding to spatial coherence allowing for typical quantum
phenomena such as the appearance of interference fringes, are actually suppressed. The
suppression is more effective for far-off matrix elements, and exponential in time. This
dynamical behavior, arising just due to interaction with the environment, describes the
loss of visibility in a interferometric experiment with massive particles, that is an
example of quantitative description of decoherence. The effect of the free evolution only
leads to a different quantitative description of the same phenomenology.
Pure dephasing spin-boson model
As a first example of open quantum system which allows us to highlight different
aspects of an open quantum system dynamics is a variant of the spin-Boson model
describing an effect of pure decoherence on the system dynamics. The overall Hamilto-
nian is of the form

H = HS + HE + V

with
!ω0
HS = σz
%2
HE = !ωk bk†bk
%k
B C
V = σz gkbk† + gk∗bk
k

and bk,bk† denote respectively creation and annihilation operator of a bosonic mode k, so
that one has the commutation relations
K L
bk , bk† ′ = δk,k ′
[bk , b ′] = 0
K † †k L
bk , bk ′ = 0.

◦
◦
The Hamiltonian describes the interaction of a two-level system with a collection of
bosonic modes, and gk denotes the coupling strength to each mode.
For this special model one can obtain the exact expression of the reduced dynamics,
thanks to the fact that σz and therefore the populations are constants of the motion. We
denote with |0⟩ and |1⟩ the eigenvectors of the system Hamiltonian with eigenvalues
minus and plus !ω0/2 respectively. We thus have ⟨1|ρS (t)|1⟩ = ⟨1|ρS (0)|1⟩. The interac-
tion term in interaction picture reads
% B † C
V (t) = σz gkbkeiω kt + gk∗bke−iω kt ,
k

so that in particular the commutator at different times is given by a C-number

%
[V (t), V (s)] = −2i |gk |2sin[ωk(t − s)]1.
k

The time evolution operator in interaction picture Ũ (t) obeying the equation
d i
Ũ (t) = − V (t)Ũ (t)
dt !

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7 General dynamics of open quantum systems 115

is given by
4 # 5
⃗ i t
Ũ (t) = T exp − dτ V (t) ,
! 0

where T⃗ denotes the chronological time ordering. Thanks to the fact that the commu-
tator of the operators at different times is a C-number one has the simplifying result
that the overall effect of time ordering is a phase, so that one has
4 # # τ 5 4 # 5
1 t i t
Ũ (t) = exp − dτ ds [V (τ ), V (s)] exp − dτ V (t)
2 0 0 ! 0
≡ eiΦ(t) U (t),

and therefore the dynamics is determined by the unitary operator U (t), which according
to
# t % eiω kt − 1
dτ V (t) = σz gkbk† + h.c.
0 iωk
k

and upon introducing the time dependent functions

1 − eiω kt
αk(t) = 2gk
!ωk
takes the form
2 3
1 %K † L
U (t) = exp σz αk(t)bk − α∗k(t)bk .
2
k
◦
◦
◦
In particular ∀ψ ∈ HE the action of the evolution operator on factorized states takes
the form
[ 4 5
αk(t)
U (t)(|0⟩ ⊗ |ψ⟩) = |0⟩ ⊗ Dk − |ψ ⟩
2
k
≡ |0⟩ ⊗ |ψ−(t)⟩
[ 4 5
αk(t)
U (t)(|1⟩ ⊗ |ψ⟩) = |1⟩ ⊗ Dk + |ψ ⟩,
2
k
≡ |1⟩ ⊗ |ψ+(t)⟩,

where we have introduced the operators

† −α∗b
D(α) = eαb

defined for α ∈ C, which are unitary operators known as displacement operators, such
that

D(α) † = D(α)−1
= D(−α),

and composing according to

D(α)D(β) = D(α + β).

116 Table of contents

In particular denoting by |0⟩ the state which is annihilated by the bosonic operator b
one has
† −α∗b
D(α)|0⟩ = eαb |0⟩
1
− 2 |α|2 † ∗
= e eαb e−α b |0⟩
∞
%
1
− 2 |α|2 αn † n
= e (b ) |0⟩
n!
n=0
∞
%
1
− 2 |α|2 αn
= e √ |n⟩
n=0 n!
≡ |α⟩
where the normalize states {|α⟩} are known as coherent states of amplitude α obeying
b|α⟩ = α|α⟩.
They form a socalled overcomplete set since they are not orthogonal for different values
of the complex number which defines them, we have in particular
2
|⟨α|β ⟩|2 = e−|α−β | ,
but do generate the whole Hilbert space.
Given that the diagonal matrix elements of the statistical operator in the basis of
eigenstates of σz are constant, we are only interested in the expression of the off-diag-
onal matrix element, namely ⟨1|ρS (t)|0⟩. Assuming that the initial state of system and
environment is factorized, so as to grant the existence of the reduced dynamics, we have
by definition
ρS (t) = TrEU (t)ρS (0) ⊗ ρEU (t) †
and therefore in particular
<[ =
⟨1|ρS (t)|0⟩ = TrE Dk(αk(t))ρE ⟨1|ρS (0)|0⟩,
k
so that upon defining
<[ =
Γ(t) = −log TrE Dk(αk(t))ρE
k
we have that the description of the reduced system dynamics is contained in the expres-
sion of this function, which describes the behavior in time of the coherence of the
reduced statistical operator
⟨1|ρS (t)|0⟩ = e−Γ(t)⟨1|ρS (0)|0⟩.
Let us consider the possible expression of this function for a thermal state of the envi-
ronment with inverse temperature β
e−βHE
ρE =
ZE
\ e−β!ωk bk†bk
=
ZEk
\k
= ρkE ,
k
with
†
ρkE = (1 − e−β!ωk)e−β!ωk bkbk.

@ B. Vacchini - Unimi & INFN

7 General dynamics of open quantum systems 117

We thus have to evaluate

[ B C
Γ(t) = −log ⟨exp αk(t)bk† − α∗k(t)bk ⟩ ρE
% k B C
= − log ⟨exp αk(t)bk† − α∗k(t)bk ⟩ ρE
k
Relying on the fact that for a random variable X with Gaussian distribution and zero
mean one has
4 5
1 2
⟨exp (X)⟩ = exp ⟨X ⟩
2
so that
4 5
B C 1 B C
⟨exp αk(t)bk† − α∗k(t)bk ⟩ ρE = exp ⟨ αk(t)bk† − α∗k(t)bk 2⟩ ρE
2
4 5
1 I †J
= exp − |αk(t)| ⟨ bk , bk ⟩ ρE
2
2
and therefore
1% I J
Γ(t) = |αk(t)|2⟨ bk , bk† ⟩ ρE.
2
k
◦
One immediately obtains
1 − cos (ωkt)
|αk(t)|2 = 8|gk |2 ,
!2ωk2
as well as
1 †
⟨{b, b †}⟩ = Tr{b, b †}e−β!ωb b
Z
1 †
= Tr(2b †b + 1)e−β!ωb b
Z
∞
2 % −βn!ω
= 1+ ne
Z
n=0
2 1 ∂
= 1− Z
Z !ω ∂β
2 ∂
= 1− log Z
!ω ∂β
2 ∂
= 1+ log(1 − e−β!ω)
!ω
4 ∂β 5
β
= coth !ω
2
where we have defined according to standard notation
†
Z = Tr e−β!ωb b
1
= .
(1 − e−β!ω )
◦
As a result the function determining the dynamics of the coherences is determined by
the expression
% 4 5
1 − cos (ωkt) β
Γ(t) = 4 |gk |2 coth !ωk .
ωk
2 2
k
118 Table of contents

To properly evaluate this function one needs an estimate of the relevant frequencies and
of the strength of the coupling. It it therefore natural to consider instead of a sum an
integral over
! the"modes by means of the introduction of a suitable density of modes by
replacing k → dω ρ(ω) and of a function gk → g(ω) describing the relevance of the
coupling to the different modes. These informations about both the environment and its
coupling to the system can be put in a single function, defined according to

J (ω) = 4ρ(ω)|g(ω)|2,

known as spectral density, leading to

# ∞ 4 5
1 − cos (ωt) β
Γ(t) = dω J(ω) coth !ω ,
0 ω2 2
so that the spectral density is sometimes also formally written as
%
J (ω) = 4|gk |2δ(ω − ωk).
k

The function Γ(t) is also known as decoherence function, since it describes how the
system looses coherence as a result of the interaction with the environment. Note that
as a general feature the function Γ(t) is positive, so that coherences are suppressed in
time according to a factor e−Γ(t).
◦
In the usual notation let us define

Φ(t)ρS (0) = ρS (t),

so that Φ(t) denotes the quantum dynamical map describing the reduced system time
evolution, which can also be described as follows, introducing the matrix elements of the
statistical operator in the basis of eigenvectors of the σz operator
4 5 2 3
ρ11(0) ρ10(0) ρ11(0) ρ10(0)e−Γ(t)
→ .
ρ01(0) ρ00(0) ρ01(0)e−Γ(t) ρ00(0)
The associated Choi matrix then takes the form
4 5
Φ(t)[|1⟩⟨1|] Φ(t)[|1⟩⟨0|]
ΛΦ(t) =
Φ(t)[|0⟩⟨1|] Φ(t)[|0⟩⟨0|]
⎛ ⎞
1 0 0 e−Γ(t)
⎜ ⎟
⎜ 0 0 0 0 ⎟
= ⎜ ⎟,
⎝ 0 0 0 0 ⎠
e−Γ(t) 0 0 1
so that its time dependent eigenvalues are
I J
{λ1, λ2, λ3, λ4} = 0, 0, 1 − e−Γ(t), 1 + e−Γ(t) ,

with corresponding eigenvectors which are actually time independent

⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫
⎪
⎪ 0 0 1 1 ⎪
⎪
⎨⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎬
0 ⎟⎜ 1 ⎟ 1 ⎜ 0 ⎟ 1 ⎜ 0
{v1, v2, v3, v4} = ⎜⎝ 1 ⎠,⎝ 0 ⎠, √2⎝ 0 ⎠, √2⎝ 0
⎟ .
⎠⎪
⎪
⎪ ⎪
⎩ ⎭
0 0 −1 1
◦

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7 General dynamics of open quantum systems 119

The dynamics is therefore completely positive provided the function Γ(t) is indeed
positive, as in the microscopic model considered here. Relying on the diagonalization of
the Choi matrix we can obtain a Kraus representation of the map Φ(t) as follows.
According to the previously introduced notation any map can be represented in terms of
the basis of maps Fαβ [ω] = τα ω τ β as follows
4
%
′
Φ(t)ρS (0) = Λαβ (t)Fαβ (ρS (0)),
α, β=1
′
where for the choice of operator basis τα → {|1⟩⟨1|, |0⟩⟨1|, |0⟩⟨1|, |0⟩⟨0|} the matrix Λαβ
coincide with the just introduced Choi matrix, so that we can obtain a diagonal repre-
sentation by means of the unitary matrix
U = (v1, v2, v3, v4)
according to
Λ ′ = U diag(λ1, λ2, λ3, λ4)U †,
coming to
2 3 ⎛ ⎞
4 4 4 †
% % %
Φ(t)ρS (0) = λγ uαγ τα ρS (0)⎝ u βγ τ β ⎠ ,
γ=1 α=1 β=1
4 5 4 5†
B −Γ(t)
C 1 1 1 1
= 1−e √ τ11 − √ τ00 ρS (0) √ τ11 − √ τ00
42 2 5 42 2 5†
B C 1 1 1 1
+ 1+e −Γ(t)
√ τ11 + √ τ00 ρS (0) √ τ11 + √ τ00
2 2 2 2
−Γ(t)
4 5 4 5
1−e 1 0 1 0
= ρS (0)
2 0 −1 0 −1
4 5 4 5
1+e −Γ(t)
1 0 1 0
+ ρS (0)
2 0 1 0 1
1−e −Γ(t)
1+e −Γ(t)
= σzρS (0)σz + 1ρS (0)1.
2 2
◦
◦
A map admitting such a Kraus representation is also known as phase damping
B C
channel, and introducing the positive numbers summing up to one p±(t) = 1 ± e−Γ(t) /2
it can also be written
Φ(t)ρS (0) = p−(t)σzρS (0)σz + p+(t)ρS (0).
◦
◦
Note
8 that 9 the other standard choice of basis in the space of maps, namely Eαβ [ω] =
τα Tr τ β† ω , leads for the choice of operator basis τα → {|1⟩⟨1|, |0⟩⟨1|, |1⟩⟨0|, |0⟩⟨0|} to
the representation
4
%
Φ(t)ρS (0) = Λαβ (t)Eαβ (ρS (0))
α, β=1
= |1⟩⟨1|ρS (0)|1⟩⟨1| + |0⟩⟨0|ρS (0)|0⟩⟨0|
+(|1⟩⟨1|ρS (0)|0⟩⟨0| + |0⟩⟨0|ρS (0)|1⟩⟨1|)e−Γ(t).
120 Table of contents

As a last step we consider the expression of the generator of the considered dynamical
evolution. Given that the only matrix element varying in time is the off-diagonal one
d
⟨1|ρS (t)|0⟩ = −Γ̇(t)⟨1|ρS (t)|0⟩,
dt
one can easily check that the generator of the dynamics is given by
6 7
d Γ̇(t) 1 2
ρS (t) = + σzρS (t)σz − {σz , ρS (t)}
dt 2 2
Γ̇(t)
= + [σzρS (t)σz − ρS (t)]
2
Γ̇(t)
= − [σz , [σz , ρS (t)]],
4
so that according to the previous results one has that the obtained dynamics is given by
a collection of completely positive maps if Γ̇(t) ! 0. Off-diagonal matrix elements
decrease monotonically in time, while a proper Lindblad structure, that is to say a semi-
group dynamics, is recovered if and only if Γ(t) is linear in t, say Γ(t) = 2γt, with γ a
positive constant, so that one has
d
ρS (t) = γ[σzρS (t)σz − ρS (t)].
dt
◦
◦
◦

Projection operators
Given the general structure of the master equation for the reduced system operator
which grants complete positivity of the dynamics, one might wonder whether and how
such master equations do arise from a microscopic description of the underlying
dynamics. To point to this connection we consider a general technique, devised by Naka-
jima and Zwanzig (1958 and 1960 respectively), in order to obtain a representation for
the exact equations of motion for the overall statistical operator, splitting it according to
a projection operator in a so-called relevant and irrelevant part. The basic idea comes
from non equilibrium statistical mechanics: we have a complex system and try to obtain
a manageable dynamics by eliminating degrees of freedom by means of some projection
operator, thus considering the dynamics of relevant variables only, to be described in
terms of effective master equations.
A projection operator is a map which sends states to states

P: ρ 6→ Pρ

being linear, positive, trace preserving and idempotent P 2 = P. The latter requirement
corresponding to the defining property of a projection. Having in mind that the total
space has a bipartite structure H = HS ⊗ HE we consider projection operators of the
form

P = 1S ⊗ Λ

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7 General dynamics of open quantum systems 121

with Λ a completely positive, trace preserving idempotent map on HE . This choice

grants the crucial property
ρS = TrEρ = TrEPρ
so that knowledge of the dynamics of the relevant part is enough to recover the reduced
state ρS . It further implies that separable (product) states are sent to separable (pro-
duct) states, so that the space of separable (product) states is invariant under P, namely
no artificial entanglement is introduced by means of the chosen projection. Note that
indeed in the preparation procedure only classical correlations are usually introduced.
For such a projection operator one has the following representation theorem.
Theorem (Breuer, 2007) A projection operator with the above mentioned properties
can be written as
%
Pρ = TrE {Aiρ} ⊗ Bi
i

with {Ai }, {Bi } linearly independent sets of observables on HE with the properties
1. TrE {AiB j } = δij
!
2. i TrE {Bi }Ai = 1E
! T
3. i Ai ⊗ Bi ! 0
To prove the result, instead of resorting to the Kraus representation for a completely
positive map, we first observe that a linear and idempotent map Γ on a finite dimen-
sional Hilbert space can always be represented in the form
%
Γ|ϕ⟩ = |fk ⟩⟨ek |ϕ⟩,
k

with {ek } and {fk } two sets of linearly independent vectors, satisfying ⟨ek |f j ⟩ = δk,j .
◦
In our setting the Hilbert space is given by the space of Hilbert-Schmidt operators
over the Hilbert space HE which has dimension n. We thus have the representation
n 2
% B C
Λ[X] = BkTrE Ak†X ,
k=1
B C
with {Ak } and {Bk } two sets of linearly independent operators, satisfying TrE Ak†B j =
δk,j . Further asking hermiticity of Λ, we have that these operators have to be self-
adjoint, so that they can be taken as observables. Trace preservation requires the
validity of
⎧ ⎫
⎨% n2 ⎬
TrE Λ[X] = TrE BkTrE (AkX)
⎩ ⎭
k=1
⎧⎡ ⎤ ⎫
⎨ % n2 ⎬
= TrE ⎣ ⎦
TrE (Bk)Ak X
⎩ ⎭
k=1
= TrEX
for all operators X, thus leading to the requirement
n 2
%
TrE (Bk)Ak = 1E .
k=1
122 Table of contents

We now have to put conditions on these operators in order to have complete positivity
of the map Λ. To this end we study positivity of the associated Choi’s matrix. The
Choi’s matrix associated to the map is given by
⎡ ⎤
%n
(1 ⊗ Λ)[X] = (1 ⊗ Λ)⎣ E jk ⊗ Ejk ⎦
j ,k =1
n
%
= E jk ⊗ Λ[E jk ]
j ,k =1
% n n2
%
= E jk ⊗ BiTrE (AiE jk )
j ,k =1 i=1
% n %n2
= E jk ⊗ Bi ⟨ek |Ai |e j ⟩
j ,k =1 i=1
% n %n2
. .
= E jk ⊗ Bi ⟨e j.ATi .ek ⟩
j ,k =1 i=1
% n %n2
. .
= |ek ⟩⟨ek.ATi .e j ⟩⟨e j | ⊗ Bi
i=1 j ,k =1
n 2
%
= ATi ⊗ Bi ,
i=1
so that complete positivity is granted provided the obtained operator is positive
n 2
%
ATi ⊗ Bi ! 0.
i=1
◦
We have thus recovered the three constraints. Vice versa, given sets of observables
{Ai }, {Bi } linearly independent satisfying the above constraints, one can checks that the
operator
%
Pρ = TrE {Aiρ} ⊗ Bi
i
provides a well defined projection with the required properties. Note that in general dif-
ferent sets of observables can represent the same mapping, e.g. we can take
%
Ai′ = uik Ak
%k
Bi′ = vik Bk ,
k
where the real non singular matrices U and V satisfy U TV = 1.
Standard projection
The standard projection operator onto a factorized state is obtained for the choice
A = 1E B = ρE
with ρE a fixed environmental state
Pρ = TrEρ ⊗ ρE = ρS ⊗ ρE .
Note that at the r.h.s. we do not have the product of the marginals of ρ, but rather the
product of the first marginal with a fixed environmental state.

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7 General dynamics of open quantum systems 123

Correlated projection
A correlated projection operator is obtained considering an orthogonal decomposition
!
of unit in HE according to Πi = Πi†, ΠiΠ j = δij Πi, i Πi = 1E , and defining
ΠiρE Πi
Ai = Πi Bi = ρiE =
TrE {ΠiρE }
with ρiE is the collection of statistical operators obtained from a fixed environmental
state ρE upon the action of a map which implements a von Neumann instrument. In
this case we have the expression
% ΠiρE Πi
Pρ = TrE {Πiρ} ⊗ .
TrE {ΠiρE }
i
Projected equations of motion
In order to obtain equations of motion for the reduced statistical operator only we
start from the overall unitary dynamics. For the total system one has the Liouville von
Neumann equation with
H = HS + HE + αV
NNNNNNNNNNOPQQQQQQQQQQQ
H0

and therefore in interaction picture with V (t) = eiH0tV e−iH0t

d
ρ(t) = −iα[V (t), ρ(t)] ≡ L(t)ρ(t),
dt
where we have defined the Liouvillian operator L(t) as
L(t)[ · ] = −iα[V (t), ·].
To simplify the notation we omit the index I in ρ(t) and V (t), though they are indeed
operators in interaction picture.
We now use P and the complementary projection operator Q = 1 − P to write the
statistical operator in terms of a relevant and irrelevant part according to ρ = Pρ + Qρ.
The two contributions obey the equations
d
Pρ(t) = P L(t)Pρ(t) + P L(t)Qρ(t)
dt
d
Qρ(t) = Q L(t)Pρ(t) + Q L(t)Qρ(t).
dt
◦
By analogy with the case of functions, in which a differential equation of the form
d
α(t) = β(t)(α(t) + γ(t))
dt
has the solution
" t # t " t
dτβ (τ ) dσβ(σ)
α(t) = e 0 α(0) + dτ e τ β(τ )γ(τ ),
0
the second equation can be solved introducing the operator
4# t 5
⃗ Q(t, t1) = T
G ⃗ exp dt2Q L(t2) ,
t1

where T⃗ denotes chronological time ordering, solution of the homogeneous equation

d⃗ ⃗Q(t, t1)
GQ(t, t1) = Q L(t)G
dt
corresponding to the initial condition
⃗Q(t, t) = 1.
G
124 Table of contents

The solution of the equation for the irrelevant part therefore reads
# t
⃗
Qρ(t) = GQ(t, 0)Qρ(0) + ⃗Q(t, t1)Q L(t1)Pρ(t1)
dt1G
0
and substituting in the first equation one obtains
# t
d ⃗ ⃗Q(t, t1)Q L(t1)Pρ(t1).
Pρ(t) = P L(t)Pρ(t) + P L(t)GQ(t, 0)Qρ(0) + dt1P L(t)G
dt 0
Before going on we consider a couple of simplifying assumptions of general validity.
First we suppose that the inhomogeneous term vanishes, so that
Qρ(0) = 0,
which means that the initial state is an eigenoperator of the projection on the relevant
part. In the case of the standard projection this corresponds to a factorized initial state,
which grants in particular, as already discussed, the existence of the reduced dynamics
at the level of the Hilbert space of the system only. The other assumption is that
P L(t1)...L(t2n+1 )P = 0,
which corresponds to
TrEV (t1)...V (t2n+1)ρE = 0
and is typically satisfied in applications. To see the connection between the two equiva-
lent conditions let us consider the simplest case, with a single action of the Liouvillian
L(t). We have, considering a standard projection
P L(t)Pρ = TrE (L(t)Pρ) ⊗ ρE
= −iαTrE ([V (t), ρS ⊗ ρE ]) ⊗ ρE
= −iαTrE (V (t)ρE )ρS ⊗ ρE + iαρS TrE (ρEV (t)) ⊗ ρE ,
where we have denoted as usual ρS = TrEρ and TrE (V (t)ρE ) is an operator on HS . So
that the term vanishes whenever TrE (V (t)ρE ) = 0. The evolution equation for the rele-
vant part thus reads
# t
d ⃗Q(t, t1)L(t1)Pρ(t1),
Pρ(t) = dt1P L(t)G
dt 0
where in the integral we have replaced Q = 1 − P with the identity because of the pre-
vious condition. We now want to obtain perturbation expansions of this equation,
allowing for approximate solutions, recalling that TrEPρ(t) = ρS (t). This equation is
valid for different choices of P and leads to equations for different relevant states, which
all lead to the same exact equations for ρS , though rearranged in a non perturbative
way.
Introducing the integral kernel
⃗Q(t, t1)L(t1)P
KNZ(t, t1) = P L(t)G
where NZ stands for Nakajima-Zwanzig one has the so-called Nakajima-Zwanzig master
equation
# t
d
Pρ(t) = dt1KNZ(t, t1)Pρ(t1).
dt 0

Nakajima-Zwanzig master equation

One can now consider a perturbation expansion of the memory kernel
⃗Q(t, t1)L(t1)P
KNZ(t, t1) = P L(t)G

@ B. Vacchini - Unimi & INFN

7 General dynamics of open quantum systems 125

relying on the natural expansion of G ⃗Q(t, t1), where we recall that each Liouvillian L(t)
brings with itself a factor α
4# t 5
⃗ ⃗
GQ(t, t1) = T exp dt2Q L(t2)
# t t1 # t # t2
= 1+ dt2Q L(t2) + dt2 dt3Q L(t2)Q L(t3) + ...
t1 t1 t1
(2) (4)
so that we can write KNZ = KNZ + KNZ + ... with
(2)
KNZ(t, t1) = P L(t)L(t1)P
# t # t2
(4)
KNZ(t, t1) = dt2 dt3[P L(t)L(t2)L(t3)L(t1)P − P L(t)L(t2)P L(t3)L(t1)P],
t1 t1
where the order of the perturbation corresponds to the order in powers of α. The
expression up to second order obtained replacing G ⃗Q(t, t1) with the identity operator
reads
# t
d
Pρ(t) = dt1P L(t)L(t1)Pρ(t1).
dt 0 #
t
= −α2 dt1P[V (t), [V (t1), Pρ(t1)]].
0
To obtain the master equation for the reduced state ρS (t) we now have to specify the
choice of projection, following the above introduced representation so that
%
Pρ = TrE {Aiρ} ⊗ Bi.
i
Let us first consider the case of a standard projection, corresponding to A → 1E , B →
ρE , we obtain
# t
d
ρS (t) ⊗ ρE = − dt1TrE {[V (t), [V (t1), ρS (t1) ⊗ ρE ]]} ⊗ ρE ,
dt 0
so that upon taking the partial trace with respect to the environment of the Nakajima-
Zwanzig perturbation expansion we obtain the following equation for ρS (t) only
# t
d
ρS (t) = − dt1TrE {[V (t), [V (t1), ρS (t1) ⊗ ρE ]]}
dt 0
which is sometimes called generalized master equation.
◦
If we consider instead a correlated projection where Ai → Πi, Bi → ρiE = Πi/TrE Πi we
obtain
⎧ ⎡ >
% # t % ⎨
d Πi
TrE {Πiρ(t)} ⊗ = − dt1 TrE Πk⎣ V (t), V (t1),
dt TrE Πi 0 ⎩
i k
?⎤⎫
⎬
% Πj
TrE {Π jρ(t1)} ⊗ ⎦ ⊗ Πk .
TrE Π j ⎭ TrE Πk
j

From this expression defining the sub-collections

wi(t) = TrE {Πiρ(t)},
which are trace class operators on HS with trace less or equal to one we come to
# t $ > > ??&
d % Πi
w j (t) = − dt1TrE Π j V (t), V (t1), wi(t1) ⊗ .
dt 0 TrE Πi
i
126 Table of contents

According to the relationship

%
ρS (t) = w j (t)
j

we finally have
# $> > ??&
d t % Πi
ρS (t) = − dt1TrE V (t), V (t1), wi(t1) ⊗ .
dt 0 TrE Πi
i
Note however that in accordance to the fact that the initial state is not necessarily fac-
torized, but is only an eigenoperator of the correlated projection, there generally is no
closed dynamics at the level of the reduced system only. Indeed to solve the last equa-
tion one does not simply need to know ρS (0), the knowledge of !all sub-collections at
time zero wi(0) is required. The initial state ρS (0) only fixes i wi(0), and not the
single contributions.

8 Weak coupling master equation

Taking as starting point the second order expansion within the Nakajima-Zwanzig pro-
jection operator technique, for the distinguished case of the standard projection, we will
make contact to the so-called weak copuling master equation, which can also be
obtained along different approaches using similar approximations.
As we have seen, up to second order and with the standard choice for the projection
operator the Nakajima-Zwanzig projection operator technique leads to the master equa-
tion
# t
d
ρS (t) = − dt1TrE {[V (t), [V (t1), ρS (t1) ⊗ ρE ]]}.
dt 0
With the replacement ρS (t1) → ρS (t) this equation is also known as Redfield master
equation. The approximation involving the restriction of the perturbation expansion to
second order is also-called Born approximation, in analogy with the Born approximation
in scattering theory. Upon performing the change of variables t1 = t − τ in the integral
one has
# t
d
ρS (t) = − dτ TrE {[V (t), [V (t − τ ), ρS (t − τ ) ⊗ ρE ]]},
dt 0
and we can further simplify this expression extending the integral to the whole real line
and replacing ρS (t − τ ) ≈ ρS (t). The latter approximation is known as Markov approxi-
mation. The basic justification behind these approximations is a separation of time
scales between a typical time scale of the environment τE which characterizes a decay
time of its correlation functions, and the time scale τR of the dynamics of the reduced
system. If we have
τE ≪ τR ,
then the actual contribution to the integral only refers to a short time interval, thus jus-
tifying the extension of the integral to the whole real line, as well as the fact that the
small retardation in the time argument of the statistical operator of the reduced system
is neglected. Thus in the so-called Born-Markov approximation the master equation
reads
# ∞
d
ρS (t) = − dτ TrE {[V (t), [V (t − τ ), ρS (t) ⊗ ρE ]]}.
dt 0

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 127

We now proceed to further elaborate this expression in order to obtain the so-called
weak-coupling master equation.
◦
As a first step we notice that the interaction term can be generally written in the
form
%
V = Aα ⊗ Bα ,
α
where due to self-adjointness of V the operators {Aα } and {Bα }, acting on the Hilbert
space of system and environment respectively, can also be taken self-adjoint. In order to
sort out different contributions to the dynamics according to their time dependence,
which will allow to later perform a so-called secular approximation, it is convenient to
decompose the interaction term into eigenoperators of the free system Hamiltonian HS .
Supposing HS to have a pure point spectrum, and denoting by Ei its eigenvalues and
Π({Ei }) the projections on the corresponding eigenspace, so that
%
HS = EiΠ({Ei })
Ei
we can introduce the following decomposition for the system operators appearing in the
interaction term
Aα = 2
1Aα1 3 2 3
% %
= Π({Ei }) Aα Π({Ej })
E Ej
%i %
= Π({Ei })AαΠ({E j })
ω E j −Ei =ω
%
≡ Aα(ω)
ω
%
= Aα† (ω),
ω
where the real quantities ω do correspond to all possible energy differences among dis-
tinct eigenvalues of the system.
◦
Note that the Π({Ei }) are not necessarily one dimensional. In such a way we have
defined a collection of operators
%
Aα(ω) = Π({Ei })AαΠ({E j }),
E j −Ei =ω

which together with their adjoints

Aα† (ω) = Aα(−ω)
are indeed eigenoperators of the free system Hamiltonian corresponding to the eigen-
values ±ω
[HS , Aα(ω)] = −ωAα(ω)
K L
HS , Aα† (ω) = +ωAα† (ω),
where the index α can now be seen as a degeneracy index for the eigenoperators corre-
sponding to the eigenvalue ±ω. Exploiting these commutation relations we also immedi-
ately have
K L
HS , Aα† (ω)A β (ω) = 0.
128 Table of contents

These operators also have a very simple expression in interaction picture, since we have
eiHStAα(ω)e−iHSt = e−iωt Aα(ω)
which leads to the following two equivalent expressions for the interaction term in inter-
action picture, where we omit the standard lower index I
%%
V (t) = e−iωtAα(ω) ⊗ Bα(t)
%ω % α
= e+iωtAα† (ω) ⊗ Bα† (t).
ω α
◦
Let us notice that the condition
P L(t)P = 0
which we used to come to the second order Nakajima-Zwanzig master equation now
translates into
TrEBα(t)ρE = ⟨Bα(t)⟩ρE
= 0.
If we now insert the obtained expressions for the interaction term in the master equa-
tion, using the first one for the term V (t − τ ) and the second one for the term V (t), we
obtain
# ∞
d
ρS (t) = − dτ TrE {−V (t − τ )ρS (t) ⊗ ρEV (t) + V (t)V (t − τ )ρS (t) ⊗ ρE + h.c.}
dt # ∞0 % I ′
= dτ TrE e−iω(t−τ )A β (ω) ⊗ B β (t − τ )(ρS (t) ⊗ ρE )e+iω tAα† (ω ′) ⊗
0 ω,ω ′
α, β
′
Bα† (t) − e+iω tAα† (ω ′) ⊗ Bα† (t)e−iω(t−τ )A β (ω) ⊗ B β (t − τ )ρS (t) ⊗ ρE +
J
h.c.
% # ∞
+i(ω ′ −ω)t
K LI
= e dτ eiωτ TrE Bα† (t)B β (t − τ )ρE A β (ω)ρS (t)Aα† (ω ′) −
ω,ω ′ 0
α, β
J
Aα† (ω ′)A β (ω)ρS (t) + h.c.,
◦
where we have used the cyclic property of the trace and the fact that operators on
system and environment only simply commute. In view of the obtained expression it is
now natural to define the following C-number function of the energy differences of the
system ω
# ∞
K L
Γαβ (ω) = dτ eiωτ TrE Bα† (t)B β (t − τ )ρE
#0 ∞
= dτ eiωτ ⟨Bα† (t)B β (t − τ )⟩ρE.
0
One can immediately check that provided
[HE , ρE ] = 0,
so that the state of the environment is a function of its free Hamiltonian, the functions
Γαβ (ω) are actually independent of the time t and only depend on the time difference in
the argument of the two interaction picture operators. We can thus write
# ∞
Γαβ (ω) = dτ eiωτ ⟨Bα† (τ )Bβ (0)⟩ ρE.
0

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8 Weak coupling master equation 129

As anticipated by introducing the eigenoperators of HS we have put into evidence an

′
explicit time dependence through the oscillating functions e+i(ω −ω)t. We now perform a
so-called secular approximation, consisting in keeping only terms with ω = ω ′, which
provide contributions which cumulate in time, while the terms with ω = / ω ′ do not give a
significant contribution to the time evolution if they oscillate very rapidly. This approxi-
mation thus requires that
τS ≪ τR ,
where τS is a typical time scale of the free system evolution, which can be estimated as
τS ≈ 1/|ω − ω ′|, while τR is the typical time scale of the dynamics of the system coupled
to the environment, roughly of the order of the inverse of the coupling constants
appearing in the master equation.
◦
Under this hypothesis the secular approximation applies. We now consider the split-
ting
1
Γαβ (ω) = γαβ (ω) + i Sαβ (ω)
2
where

γαβ (ω) = Γαβ (ω) + Γ∗βα(ω)

and
1
Sαβ (ω) = (Γ (ω) − Γ∗βα(ω))
2i αβ
∗ ∗
are hermitian matrices for any ω, that is γαβ (ω) = γβα (ω) and Sαβ (ω) = S βα (ω). Rear-
ranging terms according to
d %% I J
ρS (t) = Γαβ (ω) A β (ω)ρS (t)Aα† (ω) − Aα† (ω)A β (ω)ρS (t) + h.c.
dt
ω α, β
% % †
% % 41
∗
= (Γαβ (ω) + Γβα(ω))A β (ω)ρS (t)Aα(ω) − γαβ (ω) +
2
ω α, β ω α, β
5
i Sαβ (ω) Aα† (ω)A β (ω)ρS (t)
% %41 5
− γαβ (ω) − i Sαβ (ω) ρS (t)Aα† (ω)A β (ω),
2
ω α, β

we finally have
> ?
d % %
ρS (t) = −i Sαβ (ω)Aα† (ω)A β (ω), ρS (t) +
dt
ω α, β
%% 6 7
† 1I † J
γαβ (ω) A β (ω)ρS (t)Aα(ω) − Aα(ω)A β (ω), ρS (t) .
2
ω α, β

In order to show that this is indeed a sum of contributions in Lindblad form we have to
prove that
! the Kossakowski matrix γαβ (ω) is a positive matrix for any ω, that is the
function α, β wαγαβ (ω)w β are positive for all {w} ∈ Cn. To this aim it is now crucial to
∗

observe that the ω-dependent matrix γαβ (ω) can be written as

# ∞
γαβ (ω) = dτ eiωτ ⟨Bα† (τ )Bβ (0)⟩ρE ,
−∞
130 Table of contents

which actually warrants its positivity. To see this let us first introduce the notion of
positive definite function. A function f (t) on the real line is said to be positive definite
if for arbitrary t1, ..., tn and ∀n ∈ N
n
%
vi∗ f (ti − t j )v j ! 0,
i, j =1

where that is to say the n × n matrices aij = f (ti − t j ) are positive for any n.
{v } ∈ Cn,
◦
One can check that the Fourier transform of a positive function is positive definite.
Vice versa according to Bochner’s theorem the Fourier ! transform of a positive definite
function is positive. In our case we have, setting B = β B βwβ
n
% n
%
vi∗⟨B †(ti − t j )B(0)⟩ ρEv j = vi∗TrE {B †(ti − t j )B(0)ρE }vj
i, j=1 i, j=1
% n
I J
= vi∗TrE eiH EtiB †e−iHEtieiHEtjBe−iH EtjρE vj
i, j=1
⎧ 2 n 3†⎛ n ⎞ ⎫
⎨ % % iH t ⎬
= TrE eiH EtiBe−iHEtivi ⎝ e E jBe−iHEtjvj ⎠ρE
⎩ ⎭
i=1 j=1
= TrE †
{X Xρ E },

so that f (τ ) = ⟨B †(τ )B(0)⟩ρE is positive definite, and therefore thanks to Bochner’s the-
orem its Fourier transform is positive. Given the actual expression of B we have there-
fore actually shown that the matrices γαβ (ω) are indeed positive.
◦
Defining the Hamiltonian contribution
%%
HLS = Sαβ (ω)Aα† (ω)A β (ω),
ω α, β

where LS stands for Lamb shift, and the dissipator

%% 6 7
† 1I † J
D[σ] = γαβ (ω) A β (ω)σAα(ω) − Aα(ω)A β (ω), σ
2
ω α, β

we finally have the master equation

d
ρS (t) = −i[HLS, ρS (t)] + D[ρS (t)],
dt
which thanks to positivity of the matrix γαβ (ω) is indeed of Lindblad form and therefore
provides a proper generator of quantum dynamical semigroup.
◦
◦
We further note that thanks to the fact that the operators A β (ω) are eigenoperators
of the system free Hamiltonian the dissipator D is covariant with respect to the unitary
transformation generated by HS . As a result in going from the present equation in inter-
action picture to the master equation in Schrödinger picture one only has to add a com-
mutator term with HS .
General properties of the master equation
We now introduce other relevant properties exhibited by the master equation under
suitable further hypothesis.

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 131

Detailed balance and equilibrium state

If the environment is in a canonical equilibrium state, so that its two-point correla-
tion function obeys the so-called β-KMS condition

⟨Bα† (t)B β (0)⟩th = ⟨Bβ (0)Bα† (t + iβ!)⟩th,

where KMS stands for Kubo, Martin and Schwinger, which is easily verified if the envi-
ronment is a thermal reservoir described by the canonical equilibrium distribution, then
the following thermal state for the system
e−βHS
ρth =
TrS e−βHS
is a stationary solution of the master equation. Indeed using the β-KMS condition one
has
# ∞
γαβ (−ω) = dτ e−iωτ ⟨Bα† (τ )Bβ (0)⟩th
−∞
= e−βωγ βα(ω),

which is often called detailed balance condition.

◦
Exploiting further

ρthAα(ω) = e+βωAα(ω)ρth
ρthAα† (ω) = e−βωAα† (ω)ρth,

◦
together with [ρth, HLS] = 0 as follows from the already shown property [HS , HLS] = 0
one immediately obtains, using again Aα† (ω) = Aα(−ω)
%% 6 7
† 1I † J
D[ρth] = γαδ (ω) Aδ(ω)ρthAα(ω) − Aα(ω)Aδ(ω), ρth
2
ω α,δ
% % 6
1 †
= γαδ (ω) e−βωAδ(ω)Aα† (ω)ρth − A (ω)Aδ(ω)ρth −
2 α
ω α,δ 7
1 †
A (ω)Aδ(ω)ρth
2 α
% % 6 1
= γαδ(−ω)e βωAδ†(ω)Aα(ω)ρth − γαδ(ω)Aα† (ω)Aδ(ω)ρth −
2
ω α,δ 7
1 †
γ (ω)Aα(ω)Aδ(ω)ρth
2 αδ
%%
= [γδα(−ω)e βω − γαδ(ω)]Aα† (ω)Aδ(ω)ρth
ω α,δ
= 0.

◦
Approach to equilibrium
A semigroup is said to be relaxing, if the stationary state is unique and all initial
states converge to it. A sufficient condition to ensure this fact is that γαβ (ω) > 0, and
I J
that any operator which commutes with the Hamiltonian and all {Aα(ω)} and Aα† (ω)
I J′
is proportional to the identity operator, that is HS , HLS, Aα(ω), Aα† (ω) α,ω = c 1.
132 Table of contents

Classical master equation

If the pure point spectrum of HS is non degenerate, so that all the Π({Ei }) are one-
dimensional projections, say
%
HS = En |ϕn ⟩⟨ϕn |
En

with the {En } distinct eigenvalues, then the equations for populations and coherences
get decoupled. That is to say the evolution equations for the diagonal matrix elements
of the statistical operator are closed and do not involve off-diagonal matrix elements. To
show this one starts from the identity
%
⟨ϕm |Aα(ω)ϕn ⟩ = ⟨ϕm |ϕk ⟩⟨ϕk |Aαϕl ⟩⟨ϕl |ϕn ⟩
k,l
El −Ek =ω
= δEn −Em ,ω ⟨ϕm |Aαϕn ⟩

so that

⟨ϕn |[HLS, ρS (t)]ϕn ⟩ = 0,

and as far as the dissipative part is concerned, recalling that the Aα are self-adjoint, one
has
% % 6
⟨ϕn |D[ρS (t)]ϕn ⟩ = γαβ (ω) ⟨ϕn |A β (ω)ρS (t)Aα† (ω)ϕn ⟩ −
ω α, β 7
1 I J
⟨ϕn | Aα† (ω)A β (ω), ρS (t) ϕn ⟩
2
⎡
% % %
= γαβ (ω)⎣ ⟨ϕn |A β (ω)ϕr ⟩⟨ϕr |ρS (t)ϕs ⟩⟨ϕs |Aα† (ω)ϕn ⟩ −
NNNNNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQQQQ NNNNNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQQQQ
ω α, β r,s δEr −En ,ω δEs −En ,ω
1
⟨ϕn |Aα† (ω)ϕr ⟩⟨ϕr |A β (ω)ϕs ⟩⟨ϕs |ρS (t)ϕn ⟩ ⎤ −
2
1
⟨ϕn |ρS (t)ϕr ⟩⟨ϕr |Aα† (ω)ϕs ⟩⟨ϕs |A β (ω)ϕn ⟩ ⎦
2
⎧
% ⎨2 %
= γαβ (Er −
r
⎩ α, β 3 2
%
En)⟨ϕn |A βϕr ⟩(⟨ϕn |Aαϕr ⟩)∗ ⟨ϕr |ρS (t)ϕr ⟩ − γαβ (En −
⎫ α, β
3 ⎬
Er)⟨ϕr |A βϕn ⟩(⟨ϕr |Aαϕn ⟩)∗ ⟨ϕn |ρS (t)ϕn ⟩ ,
⎭

where we have used δEr −En,ω δEs −En,ω = δr,s δEr −En,ω thanks to non degeneracy of the
spectrum.
Upon defining the transition rates
%
W (n|r) = γ µν (Er − En)⟨ϕn |Aνϕr ⟩⟨ϕr |Aµ† ϕn ⟩,
µ,ν

which are actually positive, the evolution equation for the diagonal matrix elements in
the energy eigenbasis

⟨ϕn |ρS (t)ϕn ⟩ = Pn(t)

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 133

therefore reads
d %
Pn(t) = [W (n|r)Pr(t) − W (r|n)Pn(t)].
dt
r
◦
This equation is also known as Pauli master equation. As it appears the equation for
the populations has the form of the master equation for a classical Markovian process,
with transition rates W (n|r). If the state of the environment is in canonical form these
rates obey the so-called detailed balance condition
W (n|r)e−βEr = W (r|n)e−βEn
as follows from γαβ (−ω) = e−βωγ βα(ω) and therefore
%
W (n|r) = γ µν (En − Er)⟨ϕr |Aνϕn ⟩⟨ϕn |A µ† ϕr ⟩e−β(En −Er)
µ,ν
= W (r|n)e−β(En −Er).
The detailed balance condition ensures the canonical stationary solution for the classical
process Pn(t), so that the equilibrium populations are distributed according to the
Boltzmann law.
Quantum optical master equation
Starting from the general expression of the weak coupling master equation, we now
consider a definite model of environment, later also specifying a particular system.
We suppose the environment to be the quantized electromagnetic field. Taking the
field to be confined in a box of volume V and assuming periodic boundary conditions,
apart from an infinite C-number contribution related to normal ordering the free electro-
magnetic Hamiltonian can be written
% %
HE = !ωkbλ† (k)bλ(k),
k λ=1,2
where the sum is over the two polarization states for the photon λ and the wave vector
k, while the frequencies are fixed by the dispersion relation ωk = c|k|. The creation and
annihilation operators are assumed to satisfy the canonical commutation relations, so
that we have a Bosonic environment
K L
bλ(k), bλ† ′(k ′) = δλ,λ ′δk,k′
[b (k), bλ ′(k ′)] = 0
K †λ L
bλ(k), bλ† ′(k ′) = 0.
We denote by eλ(k) the unit polarization vectors, in the plane orthogonal to the wave
vector, chosen so as to form Cartesian axes
k · eλ(k) = 0
eλ(k) · eλ ′(k) = δλ,λ ′.
Since the components of the polarization vectors can be seen as their direction cosine
along the coordinate axes one has the relation
4 54 5
k k
(e1(k))i(e1(k)) j + (e2(k))i(e2(k)) j + = δij
|k| i |k| j
or equivalently
% kik j
(eλ(k))i(eλ(k)) j = δij − ,
|k|2
λ=1,2
134 Table of contents

which identifies the linear operator

kik j
Pij = δij −
|k|2
as projection of a vector on its transverse component with respect to the direction k.
◦
We consider the system to be matter, e.g. an atom, coupled to the field in dipole
approximation, so that the interaction term reads
%
V = − Di ⊗ Ei
i=1,2,3
= −D · E,

and the previous sum over α now runs over a Cartesian index. The triple of operators
Di are the components of the dipole operator for the system, while Ei are the three
Cartesian components of the quantized electric field. One can show that the dipole
interaction term V = −D · E arises from the standard interaction term V = −e(p · A)/m
for a massive particle coupled to the electromagnetic potential by performing a suitable
time dependent gauge transformation and working in the dipole approximation. In inter-
action picture the coupling term to be inserted in the master equation can be written
%
V (t) = − e−iωtD(ω) · E(t)
% ω
= − e+iωtD †(ω) · E(t),
ω

where the sum over α is replaced by the sum over a Cartesian index and the electric
field can be written
]
% % !ωk K L
E(t) = i eλ(k) bλ(k)e−iωkt − bλ† (k)e+iωkt .
2ε0V
k λ=1,2
◦
The condition

P L(t)P = 0

now therefore reads

TrEE(t)ρE = ⟨E(t)⟩ ρE
= 0,

which holds if the electromagnetic field is in a thermal state or in the vacuum.

According to the expression of the coupling the matrix of ω dependent coefficients in the
master equation now reads
#
1 ∞
Γαβ (ω) → Γij (ω) = 2 dτ eiωτ ⟨Ei(t)E j (t − τ )⟩ρE
! 0
and therefore
] ] # ∞
1% % % !ωk !ωk ′
Γij (ω) = 2 (eλ(k))i(eλ ′(k )) j′ dτ eiωτ
! 2ε 0 V 2ε 0 V 0
k,k ′ λ=1,2 λ ′ =1,2
I
× ⟨bλ(k)bλ† ′(k ′)⟩ ρE e−iωkteiωk ′(t−τ ) + ⟨bλ† (k)bλ ′(k ′)⟩ ρE e+iωkte−iωk ′(t−τ ) −
J
⟨bλ(k)bλ ′(k ′)⟩ ρEe−iωkte−iωk ′(t−τ ) − ⟨bλ† (k)bλ† ′(k ′)⟩ ρE e+iωkte+iωk ′(t−τ ) .

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 135

We now assume the electromagnetic field to be in a thermal state at inverse temperature

β, identifying β ≍ ∞ with the vacuum state. We can thus write
e−βHE
ρE =
Tr e−βHE
[E †
= (1 − e−β!ωk)e−β!ωkbλ(k)bλ(k),
λ,k
and exploit ⟨E(t)⟩ ρE = 0 as well as
⟨bλ(k)bλ† ′(k ′)⟩ ρE = δλ,λ ′δk,k′(1 + N β (ωk))
⟨bλ† (k)bλ ′(k ′)⟩ ρE = δλ,λ ′δk,k′N β (ωk)
⟨bλ(k)bλ ′(k ′)⟩ ρE = 0
⟨bλ† (k)bλ† ′(k ′)⟩ ρE = 0,
where N β (ωk) is the Planck distribution
1
Nβ (ωk) = .
e β!ωk − 1
The expression of the matrix Γij (ω) thus simplifies to
< # ∞
1 % !ωk %
Γij (ω) = 2 (eλ(k))i(eλ(k)) j (1 + N β (ωk)) dτ e−i(ωk −ω)τ +
! 2ε0V 0
k # λ=1,2 =
∞
N β (ωk) dτ e+i(ωk +ω)τ .
0
We now consider the continuum limit, so as to consider the electromagnetic field in free
space rather than in a finite volume. Recalling the relevant dispersion relation and using
the formula
#
1% d3 k
→
V (2π)3
k # ∞ #
1
= dωk ωk dΩ,
2
(2π)3c3 0
further noting that the sum over polarizations only involves the polarization vectors,
recalling the previous identity for this sum and using the following result for the integral
over the solid angle
# 4 5
kk 8
dΩ δij − i 2j = πδij
|k| 3
we come to
# ∞ < # ∞
31 1 8 1
Γij (ω) = δij dωk ωk π (1 + N β (ωk)) dτ e−i(ωk −ω)τ +
0 ! 2ε0 3 (2π)3c3 0
# ∞ =
N β (ωk) dτ e +i(ωk +ω)τ
.
0
We now evaluate the integral over τ exploiting the so-called Sokhotski Plemelj formula
# ∞ # ∞
f (x) f (x)
lim dx = PV dx ± iπf (y),
ε→0 −∞ x − y ∓ iε −∞ x −y
valid for a function f (z) holomorphic and bounded in a strip 0 < Im z < η. We thus have
# ∞ # ∞ # ∞
f (ω)
lim dωf (ω) dτ e −i(ω −Ω−iε)τ
= −i lim dω
ε→0 0 0 ε→0 0
# ∞ Ω − iε
ω −
f (ω)
= πf (Ω) − i PV dω
0 ω −Ω
136 Table of contents

and therefore
# ∞
1 1 8 1
Γij (ω) = δij dωk ωk3 π {(1 + Nβ (ωk))πδ(ωk − ω) + N β (ωk)πδ(ωk +
0 ! 2ε0 3 (2π)3c3
ω)} < =
# ∞
1 8 1
1 1 + N β (ωk) N β (ωk)
+iδij PV dωk ωk3
π +
0 ! 2ε0 3 (2π)3c3 ω − ωk ω + ωk
# <
3 1 + N β (ωk )
∞
1 1 4ω 3 1 2
= δij (1 + N β (ω)) + iδij PV dωk ωk +
4πε0 2=3!c3 4πε0 3π!c3 0 ω − ωk
Nβ (ωk)
.
ω + ωk
◦
According to the previous notation we rewrite this expression as
4 5
1
Γij (ω) = δij γ(ω) + i S(ω)
2
identifying the contributions
1 4ω 3
γ(ω) = (1 + N β (ω))
4πε0 3!c3
and
# ∞ < =
1 2 3 1 + N β (ωk ) N β (ωk)
S(ω) = PV dωk ωk + .
4πε0 3π!c3 0 ω − ωk ω + ωk
◦
In the latter expression one distinguishes a Lamb shift contribution independent on
the photon number, and a Stark shift contribution proportional to N β (ωk). Further
noting that
1 + N β (−ωk) = −N β (ωk)
◦
◦
and inserting all the obtained contributions in the expression for the weak coupling
master equation we finally have
6 7
d i %
ρS (t) = − !S(ω)D (ω) · D(ω), ρS (t) + D[ρS (t)]
†
dt !
ω
with
% 6 7
1
D[ρS (t)] = πJ(ω)(1 + N β (ω)) D(ω)ρS (t)D (ω) − {D (ω) · D(ω), ρS (t)}
† †
2
ω>0 6 7
% 1
+ πJ(ω)N β (ω) D (ω)ρS (t)D(ω) − {D(ω) · D (ω), ρS (t)} ,
† †
2
ω>0
where a sum over Cartesian indexes is either implicit or expressed through the dot scalar
product and we have introduced a spectral density J(ω) defined through the expression
1 1 4ω 3
J (ω) =
π 4πε0 3!c3
for the present case, but which might also arise from a phenomenological characteriza-
tion of the environment.
◦

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 137

The upper and lower terms at the r.h.s. are connected to the transfer of a quantum
from the system to the bath or from the bath to the system respectively. To see this
note that if the vector ϕn is an eigenvector of the system Hamiltonian HS corresponding
to the eigenvalue En
HSϕn = Enϕn ,
then the vector Ai(ω)ϕn still is an eigenvector corresponding to the lower eigenvalue
En − ω, as follows from the commutation relations between HS and Ai(ω)
HSAi(ω)ϕn = Ai(ω)(HS − ω1)ϕn
= (En − ω)Ai(ω)ϕn.
Thus the action of the Lindblad operators appearing in the upper line at the r.h.s.
removes an excitation from the system. Note that the rate corresponding to this event
apart from an overall factor is proportional to (1 + N β (ω)), so that this contribution is
non vanishing even if the electromagnetic field is in the vacuum. The term in the lower
line at the r.h.s. on the contrary describes the inverse process and vanishes in the limit
of the vacuum field, which cannot transfer excitations to the system. Note that given a
typical matrix element for the dipole operator of the system the reference rate appearing
in the master equation reads as
1 4ω 3 2
γ0 = |d| .
4πε0 3!c3
To assess the validity of the weak coupling master equation this rate γ0 ≈ 1/τR has to be
compared with a typical frequency of the electromagnetic field ω0 ≈ 1/τE .
Bloch equations
We now consider as specific example of matter interacting with the electromagnetic
field a two-level atom stimulated by a laser described as a classical field. This system is
often called driven damped two-level system, and the equations describing it are known
as optical Bloch equations. They are of the same form as the equations introduced by
Bloch for the dynamics of a spin coupled to a driving magnetic field in the presence of
an environment.
For the case at hand the eigenoperators of the free Hamiltonian
1
HS = ω0σz
2
where ω0 is the frequency of the two-level system, are easily seen to correspond to the
σ+ and σ− operators, so that we have
[HS , σ±] = ±ω0σ±.
Assuming no permanent dipole moment, so that ⟨±|V | ±⟩ = 0, where +, − denote the
two level of the atoms singled out by the relevant transition, and introducing the dipole
moment
d = ⟨+|e r̂ | −⟩,
where r̂ now denotes the position operator for the centre of mass of the two-level system
we can write
V = −σ−d · E + h.c.
The system operators appearing in the master equation are now therefore given by
D(ω0) = dσ−
D †(ω0) = d∗σ+.
138 Table of contents

We thus end up for the dissipative part with

6 7
1
D[ρS (t)] = γ0(1 + Nβ (ω0)) σ− ρS (t)σ+ − {σ+σ−, ρS (t)}
2
6 7
1
+γ0N β (ω0) σ+ ρS (t)σ− − {σ−σ+, ρS (t)} .
2
We now couple the atom to a classical monochromatic electromagnetic field in the elec-
tric dipole and rotating wave approximation. The driving electric field can be written as
]
ω
E = iε [Ae−iωt − A∗e+iωt]
2ε0V
with A the classical amplitude and ε the polarization. With no permanent dipole
moment in the rotating wave approximation one has
2 ] 3∗ 2 ] 3
ω ω
V = id·ε A∗ e−iωtσ+ + i d · ε A∗ eiωtσ−
2ε0V 2ε0V
NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
Ω
−2

and the quantity Ω, called Rabi frequency, can be taken real for a suitable choice of
phase in the definition of the states |±⟩. It is essentially the matrix element of −r̂ · E.
Including the classical driving field the Hamiltonian term is of the form
1 Ω
H(t) = ω0σz − (e−iωtσ+ + eiωtσ−)
2 2
◦
where Ω is called Rabi frequency. Leaving out the index S of the statistical operator
for the system we thus have the master equation
d
ρ = −i[H(t), ρ] + Dρ
dt
with
6 7
1
Dρ = γ0(1 + N β (ω0)) σ− ρσ+ − {σ+σ−, ρ}
NNNNNNNNNNNNNNNNNNNNNNOPQQQQQQQQQQQQQQQQQQQQQQ 2
γ↓
6 7
1
+γ0Nβ (ω0) σ+ ρσ− − {σ−σ+, ρ}
NNNNNNNNNNNOPQQQQQQQQQQQ 2
γ↑

where γ0 is called spontaneous emission rate, N β (ω0) as defined above is the mean
number of photons in the electromagnetic field at inverse temperature β at the resonant
frequency ω0.
The first term describes induced and spontaneous emission, the second induced
absorption. In this case we have only two Lindblad operators
:
γ1L1 = γ0(1 + N β (ω0)) σ−
:
γ2L2 = γ0N β (ω0) σ+.
This master equation can describe dynamics of populations and coherences, spectrum of
emitted radiation, statistics of photon detection. The atom is driven by the laser and
can decay in the electromagnetic field.
The time dependence in the Hamiltonian can be removed by a unitary transforma-
tion of the form
ω ω
i σ t −i σ t
ρI (t) = e 2 z ρe 2 z
= U (ωt)ρU (ωt) †,

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8 Weak coupling master equation 139

where we have put

ω
+i 2 σzt
U (ωt) = e ,
which provides a representation of the group of the rotation around the z axis.
◦
If ω = ω0 that is to say the system is on resonance, this would correspond to a I-pic-
ture with respect to the free Hamiltonian. In general the quantity ∆ = ω0 − ω is called
detuning. Here it has to be interpreted as moving to a time dependent rotating frame
around the σz direction. Note that the Schrödinger picture is recovered via
ω ω
−i 2 σzt +i 2 σzt
ρ = e ρI e
2 3
(ρI )11 (ρI )10e−iωt
= .
(ρI )01eiωt (ρI )00
One is left with
6 7
d 1 i σ td
ω ω
−i σ t
ρI = i ωσz , ρI + e 2 z ρe 2 z
dt 2 dt
= LIρI
The expression of LI can be obtained by explicit calculation, but also noting the covari-
ance of the dissipative part D. Due to
U (ωt)σ±U (ωt) † = e±iωtσ±
one can check that the dissipative part D is covariant under U (ωt), therefore
U (ωt)DρU (ωt) † = DρI .
The same trivially holds for the free evolution term. It only remains to evaluate
Ω Ω
i U (ωt)[e−iωtσ+ + eiωtσ−, ρ]U (ωt) † = i [σ+ + σ−, ρI ]
2 2
Ω
= i [σx , ρI ].
2
We now want to study the equation
d
ρI = LIρI
dt 6 7
∆ Ω
= −i σz − σx , ρI + DρI .
2 2
To this aim we consider the matrix representation of the map in the Bloch basis, that is
%
LI [ρ] = ΛαβEαβ (ρ)
α, β
% 8 9
= ΛαβσαTr ρσ β† ,
α, β
where the matrix of coefficients is given by
B C
Λαβ = Tr σα† LI [σ β ] .
Recalling that any statistical operator in C2 can be represented as
1
ρ(t) = (1 + ⟨σ(t)⟩ · σ)
22 3
1 1 + ⟨σz (t)⟩ ⟨σx(t)⟩ − i⟨σy (t)⟩
= ,
2 ⟨σx(t)⟩ + i⟨σy (t)⟩ 1 − ⟨σz (t)⟩
140 Table of contents

so that
Tr ρ(t)σ = ⟨σ(t)⟩,
the dynamics is equivalently determined by the evolution equations for ⟨σ(t)⟩ given by
d %
⟨σα(t)⟩ = Λαβ ⟨σβ (t)⟩.
dt
β

Setting γ = γ0(1 + 2N β (ω0)) = γ ↑ + γ ↓ we have

LI 1 = −γ0σz
γ
LIσx = − σx + ∆σy
2
γ
LIσy = −∆σx − σy − Ωσz
2
LIσz = Ωσy − γσz
and therefore
γ
⟨σ̇x(t)⟩ = − ⟨σx(t)⟩ − ∆⟨σ y(t)⟩
2
γ
⟨σ̇ y(t)⟩ = ∆⟨σx(t)⟩ − ⟨σ y(t)⟩ + Ω⟨σz (t)⟩
2
⟨σ̇z(t)⟩ = −Ω⟨σ y(t)⟩ − γ ⟨σz (t)⟩ − γ0
4 5
γ↑ − γ↓
= −Ω⟨σ y(t)⟩ − (γ↑ + γ↓) ⟨σ+σ−(t)⟩ − ⟨σ−σ+(t)⟩ − .
γ↑ + γ↓
It is not a trivial fact that we have closed equations of motion for the mean values or
first moment. This is due to the fact that we are dealing with the generators of the
su(2) Lie algebra.
To better follow the behavior of populations and coherences we can write the equa-
tions in the form
d1 Ω γ γ
(1 + ⟨σz(t)⟩) = +i (⟨σ+(t)⟩ − ⟨σ−(t)⟩) − ⟨σz(t)⟩ − 0
dt 2 8 2 9 2 2
d γ Ω
⟨σ (t)⟩ = +i∆ − ⟨σ+(t)⟩ + i ⟨σz (t)⟩
dt + 2 2
◦
These equations are known as optical Bloch equations. Here we note that coherences
(that is to say off-diagonal matrix elements of the statistical operator) and populations
(that is to say diagonal matrix elements of the statistical operator) are coupled only
through the driving field. For the dissipative part we recover the decoupling between
populations and coherences discussed when deriving the general form of the master
equation.
◦
Coherent driving
In the absence of environment, i.e. γ = 0 and on resonance, i.e. ∆ = 0 we have the
Rabi oscillations described by
⟨σx(t)⟩ = ⟨σx(0)⟩
⟨σ y(t)⟩ = ⟨σ y(0)⟩cos(Ωt) − ⟨σz(0)⟩sin(Ωt)
⟨σz (t)⟩ = ⟨σ y(0)⟩sin(Ωt) + ⟨σz (0)⟩cos(Ωt)
caused by interaction with a laser with intensity proportional to Ω2. In this case pure
states remain pure. √
For ∆ > 0 one still has oscillations with higher frequency Ω2 + ∆2 .

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 141

◦
Otherwise stated the solution reads
4 5
1 0
⟨σ(t)⟩ = ⟨σ(0)⟩
0 R(Ωt)
and formally describes a precession around the axis of the driving field.
We notice that if
1
ρI (t) = (1 + ⟨σ(t)⟩ · σ)
2
with ⟨σ(t)⟩ solution of the equations given above, by the relation between ρI and ρ we
have
1
ρ(t) = (1 + ⟨σ ′(t)⟩ · σ)
2
with
4 5
R(ωt) 0
⟨σ ′(t)⟩ = ⟨σ(t)⟩
0 1
= (⟨σx(0)⟩cos(ωt) − ⟨σ y(0)⟩sin(ωt), ⟨σx(0)⟩sin(Ωt) + ⟨σ y(0)⟩cos(Ωt), ⟨σz(0)⟩).
General solution
To handle the equations for ⟨σ̇(t)⟩ it is convenient to write them in matrix form as
follows
⟨σ̇ (t)⟩ = b − A⟨σ(t)⟩
◦
with
⎛ ⎞
γ
⎜ ∆ 0 ⎟
⎜ 2 ⎟
A = ⎜ γ
⎝ −∆ −Ω ⎟
⎠
2
0 Ω γ
⎛ ⎞
0
b = ⎝ 0 ⎠
−γ0
and detA > 0 for γ > 0. The matrix A is normal, i.e. [A, A †] = 0, since A A † = A †A, so
that it admits a spectral decomposition.
◦
The solution reads
# t
⟨σ(t)⟩ = e −At ⟨σ(0)⟩ + ds e−A(t−s)b.
0
√
For γ = 0 the non zero eigenvalues of A are purely imaginary and given by ±i Ω2 + ∆2 ,
corresponding to a unitary dynamics. For detA =/ 0 the solution can be written
1 − e−At
⟨σ(t)⟩ = e−At ⟨σ(0)⟩ + b.
A
The stationary state corresponds to ⟨σ ⟩eq
⟨σ ⟩eq = A−1b ⎛ ⎞
−4Ω∆
γ 1 ⎜ ⎟
= − 0 2 ⎝ 2Ωγ ⎠.
γ (γ + 2Ω2 + 4∆2)
γ 2 + 4∆2
142 Table of contents

Resonant case
From now on we assume ∆ = 0, i.e. we are on resonance. We thus have in particular
γ γ
⟨σz ⟩eq = − 2 0 2
γ + 2Ω
γ Ω
⟨σ+⟩eq = −i 2 0 2 .
γ + 2Ω
In the absence of driving the stationary state has only one non zero component
γ
⟨σz ⟩eq = − 0
γ
1
= −
1 + 2N β (ω0)
8 9
ω
= −Tanh β 0
2
◦
so that the stationary population in the upper level is given by
1
ρeq
11 = (1 + ⟨σz ⟩eq)
2
N β (ω0)
=
1 + 2N β (ω0)
ω0
−β
e 2
= 8 9.
βω
2 Ch 2 0
The stationary state is therefore of the canonical form, as expected thanks to the
detailed balance condition obeyed by the coefficients in front of the different channels in
the master equation
γ ↑ = γ↓e−βω 0.
We thus have
e−βH0
ρeq =
Tr e−βHσ0z
−βω0 2
e
= σz
−βω0
Tr e 2
⎛ ⎞
ω0
−β
1 e 2 0
= 8 9⎝ ω0
⎠.
βω0 +β
2 Ch 2 0 e 2

In the considered case of zero detuning the eigenvalues of A become

< =
γ 3
λi = , γ ± iµ
2 4
with
] 8 92
γ
µ = Ω2 − ,
4
while for ∆ =/ 0 one should solve an algebraic equation of the third order. Note that the
real part of the eigenvalues is strictly positive. As we shall see the dynamics is relaxing
in the sense that the stationary state is reached with elapsing time for any initial condi-
tion. To see this consider the quantity ⟨σ(t)⟩ − ⟨σ ⟩eq which obeys the homogeneous
equation
d
(⟨σ(t)⟩ − ⟨σ ⟩eq) = −A(⟨σ(t)⟩ − ⟨σ ⟩eq)
dt

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 143

predicting convergence to the stationary state for t → ∞ due to Reλi > 0. One has there-
fore ⟨σ ⟩eq = ⟨σ(∞)⟩. This is just a special case of the general result stated previously,
according to which the dynamics is relaxing if the commutant of Hamiltonian and Lind-
blad operators reduces to multiple of the identities, indeed in this case one has {σz , σ+,
σ−} ′ = c 1.
NMR equations
Before considering the solution of the equations for the case of zero detuning, let us
point to the connection between the optical Bloch equation and the standard Bloch
equations introduced by Bloch in 1946 to describe the behavior of nuclear spins. The
equations for ⟨σ(t)⟩ can be recast in the following form
⟨σx, y (t)⟩
⟨σ̇x, y (t)⟩ = −[⟨σ(t)⟩ × H]x, y −
T2
⟨σz(t)⟩
⟨σ̇z (t)⟩ = −[⟨σ(t)⟩ × H]z − − γ0
T1
⟨σ (t)⟩ − ⟨σz (∞)⟩
= −[⟨σ(t)⟩ × H]z − z
T1
upon the identification 1/T1 = γ, and 1/T2 = γ/2, together with H = (Ω, 0, −∆).
◦
These equations are known as (magnetic) Bloch equations. In the nuclear magnetic
resonance (NMR) terminology T1 is called longitudinal relaxation time (spin-lattice
relaxation time), while T2 is the transverse relaxation time (spin-spin relaxation time).
Here 2T1 = T2, while in general complete positivity requires 2T1 ! T2, which is the typical
experimental case.
◦
Recalling that for a spin 1/2 the magnetic moment is proportional to σ one can
write

⟨M (t)⟩ = −M (t) × H − R[M (t) − M (∞)]

where the relaxation matrix R reads

⎛ ⎞
1
⎜ T 0 0 ⎟
⎜ 2 ⎟
⎜ 1 ⎟
R = ⎜
⎜ 0 T2 0 ⎟⎟.
⎜ ⎟
⎝ 1 ⎠
0 0
T1
Let us now consider the general solution of the equations for ∆ = 0. To this aim and for
later reference it is useful to change basis and consider instead of ⟨σ(t)⟩ − ⟨σ⟩eq = (σx −
⟨σx ⟩eq, σx − ⟨σx ⟩eq, σz − ⟨σz ⟩eq) the vector with components Σ = (σ+ − ⟨σ+⟩eq, σ− −
⟨σ−⟩eq, σz − ⟨σz ⟩eq). In the corresponding equations the matrix A has to be replaced by
another matrix G of the form
⎛ ⎞
γ Ω
⎜ +i∆ − 2 0 i
2 ⎟
⎜ ⎟
G = ⎜ γ Ω ⎟,
⎝ 0 −i∆ − −i ⎠
2 2
iΩ −iΩ −γ
so that

⟨Σ̇(t)⟩ = −G⟨Σ(t)⟩.
144 Table of contents

The solution of our system of equations, recalling that we take zero detuning ∆ = 0,
reads
3
6 7
− 4 γt γ
⟨Σz (t)⟩ = e cos (µt) − sin(µt) ⟨Σz (0)⟩
4µ
Ω − 3 γt
+i e 4 sin(µt)(⟨Σ+(0)⟩ − ⟨Σ−(0)⟩)
µ
and
1 − 12 γt
⟨Σ+(t)⟩ = e (⟨Σ+(0)⟩ + ⟨Σ−(0)⟩)
2 6 7
1 − 34 γt γ
+ e cos (µt) + sin(µt) (⟨Σ+(0)⟩ − ⟨Σ−(0)⟩)
2 4µ
Ω − 3 γt
+i e 4 sin(µt)⟨Σz (0)⟩,
2µ
together with ⟨Σ−(t)⟩ = ⟨Σ+(t)⟩∗
We now consider the solution of these equations for the case of the atom initially in
the ground state, so that ρ(0) = |−⟩⟨−|, and for a bath at T = 0 i.e. β % ∞, so that γ =
γ0. The initial conditions for the homogeneous equations become

⟨Σ+(0)⟩ = ⟨σ+(0)⟩ − ⟨σ+⟩eq

= −⟨σ+⟩eq
γ Ω
= i 2 0 2
γ0 + 2Ω
and

⟨Σz (0)⟩ = ⟨σz (0)⟩ − ⟨σz ⟩eq

= −1 − ⟨σz ⟩eq
γ2
= −1 + 2 0 2
γ0 + 2Ω
Ω2
= −2 2 .
γ0 + 2Ω2
For the population in the upper level, if it is initially not occupied one then has
1
Pexcited(t) = (1 + ⟨σz (t)⟩)
2
1
= (1 + ⟨σz ⟩eq + ⟨Σz (t)⟩)
2
Ω2
= 2
γ0 + 2Ω2 <6 7 =
1 − 34 γ0t γ0 −2Ω2 Ω 2iγ0 Ω
+ e cos (µt) − sin(µt) 2 + i sin(µt) 2
2 4µ γ0 + 2Ω2 7= µ γ0 + 2Ω2
< 6
Ω2 3
− γ t 3γ
= 2 1 − e 4 0 cos (µt) + 0 sin(µt)
γ0 + 2Ω 2 4µ
γ γ0
For Ω > 40 one has exponentially damped oscillations, for Ω < 4
the system is over-
damped. The stationary value of the upper population is
Ω2
Peq = ,
γ02 + 2Ω2

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 145

that is approximately 1/2 for the case of strong driving. In a similar way one obtains for
the coherences
⟨σ+(t)⟩ = ⟨σ+⟩eq + ⟨Σ+(t)⟩
< 6 4 5 7=
γ0 Ω 3
− 4 γ0t γ0 Ω2
= −i 2 1−e cos (µt) + − sin(µt) .
γ0 + 2Ω2 4µ γ0 µ
γ0
In the case of strong driving, i.e. Ω ≫ 4
,
one has
U V
1 3
− 4 γ0t
Pexcited(t) ≃ 1−e cos(Ωt)
2
i − γt
3
⟨σ+(t)⟩ ≃ − e 4 0 sin(Ωt)
2
so that for long times Pexcited(t) ≃ 1/2. As an application of these equations we study
the fluorescence spectrum of the emitted radiation in the stationary state. According to
the Wiener-Khintchine theorem the spectrum of the emitted radiation is proportional to
the Fourier transform of the autocorrelation function of the field generated by the atom.
The latter can be shown to be proportional to the σ− operator, so that we essentially
need to know the correlation function ⟨σ+(τ )σ−(0)⟩eq. The expression of multi-time cor-
relation functions of this form is generally not warranted by knowledge of the time
evolved statistical operator for the system ρS (t), nor by knowledge of the time evolution
map Φ(t). Under certain conditions they can however be shown to obey the same evolu-
tion equations as the mean values. This results goes under the name quantum regression
theorem. Let us start providing a statement of the Wiener-Khintchine theorem.
Wiener-Khintchine theorem
Let us consider a stochastic process X(t), taking values in C. We say that such a
process is wide-sense or second-order stationary if the one and two dimensional distribu-
tions only depend on the time difference, so that, denoting by E the expectation value,
we have
E[X(t)] = E[X(0)] ∀t ∈ R
E[X(t + τ )X (t)] = E[X(τ )X (0)]
∗ ∗ ∀t, τ ∈ R
≡ GX (τ ),
where we define GX (τ ) as autocorrelation function of the process. More generally a
strictly stationary stochastic process is such that all finite dimensional distributions only
depend on the difference of the time arguments. Suppose that GX (τ ) is absolutely inte-
grable, so that it admits Fourier transform, which we suppose to be also absolutely inte-
grable, so that the inverse Fourier transform is well defined. We consider the truncated
Fourier transform of the process, which we define as
# +T
2
X̂T (ω) = dt e−iωtX(t),
T
−2

so that we do not necessarily ask X(t) to admit a Fourier transform. We further define
as. truncated
. power spectral density ST (ω) the expectation value of the random variable
1.
T
X̂T (ω).2
1 K.. . L
ST (ω) = E X̂T (ω).2 .
T
◦
The Wiener-Khintchine theorem states that the limit for large T of this quantity
SX (ω) = lim ST (ω),
T →∞
146 Table of contents

which we call the spectral density of the process, exists and is given by the Fourier
transform of the autocorrelation function of the process
# +∞
SX (ω) = dt e−iωtGX (t).
−∞
The power spectral density of the process is therefore given by the Fourier transform of
the autocorrelation function, and vice versa the autocorrelation function can be obtained
as the inverse Fourier transform of the power spectrum
# +∞
GX (t) = dω e+iωtSX (ω).
−∞
The functions GX (t) and SX (ω) therefore constitute a Fourier couple. The result can be
described stating that the autocorrelation function of a wide-sense stationary random
process has a spectral decomposition given by the power spectrum of the process. It is
important to stress that for such a stationary stochastic process the autocorrelation
function is positive definite, in fact
⎡2 3⎛ n ⎞∗ ⎤
%n n
% %
viGX (ti − t j )v∗j = E⎣ viX(ti) ⎝ v jX(t j ) ⎠ ⎦
i, j =1 i=1 j =1
! 0,
for arbitrary t1, ..., tn, {vi } ⊂ C and ∀n ∈ N. Thanks to Bochner’s theorem this means
that indeed its Fourier transform is a positive function, and therefore a well-defined
spectral density. Moreover note that the value at τ = 0 of the autocorrelation function,
which is by construction positive, provides the integrated spectral density
# +∞
GX (0) = dωSX (ω).
−∞
It remains to be proven the existence of the limit. We have
># T # T ?
1 K.. .2 L 1 +2 +2
E X̂T (ω). = E dt dτ e−iω(t−τ )X(t)X ∗(τ )
T T T
−2
T
−2
# +T # +T
1 2 2
= dt dτ e−iω(t−τ )E[X(t)X ∗(τ )]
T − T
−2
T
2
# +T # +T
1 2 2
= dt dτ e−iω(t−τ )GX (t − τ ),
T − T
−
T
2 2

where we have used the stationarity property of the process. We exploit the result
# +T # +T # T
2 2
dt dτ f (t − τ ) = dτ (T − |τ |)f (τ ),
T T
−2 −2 −T

to obtain the expression

# T 4 5
1 K.. .2 L |τ |
E X̂T (ω). = dτ 1 − GX (τ )e−iωτ
T −T T
# +∞
= dτ GX ,T (τ )e−iωτ ,
−∞
where we have defined the function
⎧8 9
⎨ |τ |
1− GX (τ ) |τ | < T
GX ,T (τ ) = T .
⎩0 |τ | ! T

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 147

We now use Lebesgue dominated convergence theorem with the identification

fn → GX ,T (τ )e−iωτ ,

converging to the function

f → GX (τ )e−iωτ ,

and the sequence is dominated by the function

g → |GX (τ )|,

which is by hypothesis absolutely integrable. Since |GX ,T (τ )e−iωτ | # |GX (τ )|, we can
exchange limit and integral, thus indeed obtaining
1 K. . L
SX (ω) = lim E .X̂T (ω).2
T
T →∞ #
+∞
= lim dτ GX ,T (τ )e−iωτ
T →∞ −∞
# +∞
= dτ lim GX ,T (τ )e−iωτ
T →∞
#−∞
+∞
= dτ GX (τ )e−iωτ .
−∞
◦

Multi-time correlation functions

Suppose we have a quantum reduced dynamics described by a collection of com-
pletely positive maps of the form
4# t 5
⃗ exp
Φ(t, s) = T dτL(τ ) ,
s

where L(τ ) is a generator in Lindblad form, so that they satisfy the equation
d
Φ(t, s) = L(t)Φ(t, s)
dt
with initial condition Φ(t, t) = 1, and obey the composition law

Φ(t, τ ) ◦ Φ(τ , s) = Φ(t, s) ∀t ! τ ! s.

We now consider two generic system operators AS , BS ∈ B(HS ), and consider the two-
time correlation function

⟨AS (t + τ )BS (t)⟩ ≡ TrS ⊗E eiH (t+τ )AS e−iH (t+τ )eiHtBS e−iHtρSE(0),

which we define as the expectation value with respect to the initial state of the overall
system of the product of the two system operators, in the Heisenberg picture with
respect to the full Hamiltonian. Exploiting the ciclic property of the trace and the fact
that the operators act in the system Hilbert space only we have

⟨AS (t + τ )BS (t)⟩ = TrS ⊗E eiH (t+τ )AS e−iH (t+τ )eiHtBS e−iHtρSE(0)
= TrS AS TrE {e−iHτ [BS (e−iHtρSE(0)e+iHt)]e+iHτ }
= TrS AS TrE {e−iHτ [BS ρSE(t)]e+iHτ }
= TrS AS TrEXt(τ ).
148 Table of contents

Here the operator Xt(τ ) ∈ T (HS ⊗ HE ), if the initial condition is factorized, thus war-
ranting the existence of a dynamical map for the reduced system, reads

Xt(τ ) = e−iHτ [BS (e−iHtρS (0) ⊗ ρE e+iHt)]e+iHτ .

It therefore obeys the Liouville-von Neumann equation

d
Xt(τ ) = −i[H , Xt(τ )],
dτ
with initial condition Xt(0) = BS (e−iHtρS (0) ⊗ ρE e+iHt). Under the same physical
approximations under which the reduced system state ρS (t) = TrEρSE(t) obeys a time
dependent Lindblad dynamics as given by the map Φ(t, 0), we can assume that the
dynamics of the operator TrEXt(τ ) in its τ dependence is given by the same map. This
implies in particular that we approximate ρSE(t) ≃ ρS (t) ⊗ ρE , in keeping with the Born-
Markov approximation. We thus take

TrEXt(τ ) = Φ(t + τ , t){TrEXt(0)}

= Φ(t + τ , t){BS TrE e−iHtρS (0) ⊗ ρE e+iHt }
= Φ(t + τ , t){BS Φ(t, 0){ρS (0)}}
= Φ(t + τ , t){BSρS (t)},

where as stressed by the curly brackets any superoperator acts on all operators to its
right.
◦
We thus obtain the important relation

⟨AS (t + τ )BS (t)⟩ = TrS AS Φ(t + τ , t){BS Φ(t, 0){ρS (0)}},

which has the very important feature that it provides the expression of a multi-time cor-
relation function in terms of the dynamical map Φ(t, s) which provides the time evolu-
tion of the mean values.
◦
◦
On a similar footing one can consider the correlation function ⟨BS (t)AS (t + τ )⟩ and
obtain the result

⟨BS (t)AS (t + τ )⟩ = TrSAS Φ(t + τ , t){Φ(t, 0){ρS (0)}BS },

where the curly brackets delimit the action of the superoperators. We have indeed

⟨BS (t)AS (t + τ )⟩ = TrS ⊗E eiHtBS e−iHteiH (t+τ )AS e−iH (t+τ ) ρSE(0)
= TrS AS TrE {e−iHτ [(e−iHtρSE(0)e+iHt)BS ]e+iHτ },

and considering similarly as before the quantity

Y (τ , t) = e−iHτ [(e−iHtρS (0) ⊗ ρE e+iHt)BS ]e+iHτ ,

in the same approximations

⟨BS (t)AS (t + τ )⟩ = TrS AS TrEY (τ , t)

= TrS AS Φ(t + τ , t){TrEY (0, t)}
= TrS AS Φ(t + τ , t){Φ(t, 0){ρS (0)}BS }
= TrS AS Φ(t + τ , t){ρS (t)BS }.

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 149

This procedure can be extended to consider arbitrary multi-time correlation functions of

the form

⟨A1(s1)...Am(sm)Bn(tn)...B1(t1)⟩ ≡ TrS ⊗E {Bn(tn)...B1(t1)ρSEA1(s1)...Am(sm)},

with time ordering tn > ... > t1 ! 0, sm > ... > s1 ! 0, where all operators at the r.h.s. are
system operators considered in Heisenberg picture with respect to the full Hamiltonian.
For a suitable choice of times the expression can be used to describe the repeated action
of a completely positive map, and therefore an instrument. As already stressed the cru-
cial fact here is that one can express multi-time correlation functions by means of the
same operator which provides the mean values.
In some cases these relationships can be used to obtain the expression of multi-time
correlation functions as follows. Suppose that for the considered system a set of system
operators {Ai } ⊂ B(H) can be considered such that their mean values obey closed homo-
geneous evolution equations, as it was the case for the previously considered triple Σ, so
that
d %
⟨A j (t)⟩ = G jk ⟨Ak(t)⟩,
dt
k 4 5
%
= TrS G jk Ak ρS (t),
k

with a suitable matrix of coefficients G jk . We then have for any initial system state
d d
⟨A j (t)⟩ = TrSA jρS (t)
dt dt
d
= TrS A j Φ(t){ρS (0)}
dt
= TrSA jLΦ(t){ρS (0)}
= TrS (L ′{A j })Φ(t){ρS (0)}
= TrS (L ′{A j })ρS (t),

so that we obtain the operator relation

%
L ′{A j } = G jkAk.
k

If we combine this result with the previous regression relation for the multi-time correla-
tion function we have for its time dependence, considering a time homogeneous evolution
equation
d d
⟨A j (t + τ )C(t)⟩ = TrS A j Φ(t + τ , t){CΦ(t, 0){ρS (0)}}
dτ dτ
= TrS A jLΦ(t + τ , t){CΦ(t, 0){ρS (0)}}
= TrS L ′{A j }Φ(t + τ , t){CΦ(t, 0){ρS (0)}}
%
= TrS G jk AkΦ(t + τ , t){CΦ(t, 0){ρS (0)}}
% k
= G jk ⟨Ak(t + τ )C(t)⟩.
k

As a result the higher order correlation functions have the same time dependence, so
that they can be obtained once the time evolution for the mean values is known. This
powerful result is known as quantum regression theorem.
◦
150 Table of contents

Resonance fluorescence
Having a proper handle on two-time correlation functions for our model of a two-
level atom interacting in dipole approximation with the electromagnetic field, we can
consider relevant physical expression such as the fluorescence spectrum and statistics of
the emitted photons, which depend on multi-time correlation function of second and
fourth order respectively. We will focus on the spectrum of the emitted radiation in the
stationary state ρeq introduced before. The expression of the positive frequency compo-
nent of the retarded electromagnetic field radiated by the atomic dipole is given by
8 9
ω2 r
E (+)(t, x) = 20 [(n × d) × n]σ− t − ,
cr c
where x = n r so that r = |x|.
◦
The spectral density radiated per unit solid angle by the dipole is then given by the
expression
#
dI c r 2 +∞ dτ iωτ (−)
(ω) = e ⟨E (t, x)E (+)(t + τ , x)⟩eq
dΩ 4π −∞ 2π
= I0(x)S(ω),

where we have introduced the quantities

ω04
I0(x) = |(n × d) × n|2
4π 2#c3
1 +∞
S(ω) = dτ e−iωτ ⟨σ+(τ )σ−(0)⟩eq.
2π −∞
To evaluate the stationary correlation function ⟨σ+(τ )σ−(0)⟩eq we use the quantum
regression theorem applied to the present model. The expression of the correlation func-
tion in our case is explicitly given by

⟨σ+(τ )σ−(0)⟩eq = TrS {σ+eτL {σ− ρeq}},

where L is the superoperator in Schrödinger picture. To exploit the quantum regression

theorem we consider the previously introduced operators

Σ = (σ+ − ⟨σ+⟩eq, σ− − ⟨σ−⟩eq, σz − ⟨σz ⟩eq)

= υ − ⟨υ ⟩eq

which obey the homogeneous equations

⟨Σ̇(t)⟩ = −G⟨Σ(t)⟩.

The quantum regression theorem then tell us that multi-time correlation functions of the
form ⟨Σ(t + τ )C(t)⟩ obey the same equation of motions as the mean values. In our case,
considering the stationary state for the system we have

⟨Σ(τ )C(0)⟩eq = ⟨(υ(τ ) − ⟨υ ⟩eq)C(0)⟩eq

= ⟨υ(τ )C(0)⟩eq − ⟨υ ⟩eq⟨C(0)⟩eq
≡ (υ(τ ), C(0)).

The correlation function defined in the last line obeys

d
(υ(τ ), C(0)) = −G(υ(τ ), C(0)).
dτ

@ B. Vacchini - Unimi & INFN

8 Weak coupling master equation 151

To obtain the solution of this equation we simply have to consider the previously
obtained results for ⟨Σz (t)⟩ and ⟨Σ+(t)⟩ and consider the corresponding initial condi-
tions (υ(0), C(0)). Since we are interested in the quantity ⟨σ+(τ )σ−(0)⟩eq, we take
C(0) → σ− and therefore need the initial conditions (υ(0), σ−(0)). We are considering
the stationarity state in resonance and at zero temperature, so that
⎛ ⎞ ⎛ ⎞
σ+ iγ0 Ω
⎝ σ− ⎠ = − 1 ⎜ ⎟
⎝ −iγ0 Ω ⎠.
γ 2
+ 2Ω 2
σz 0 γ02

We have therefore

(σ+(0), σ−(0)) = ⟨σ+(0)σ−(0)⟩eq − ⟨σ+⟩eq⟨σ−⟩eq

1
= ⟨σz + 1⟩eq − |⟨σ+⟩eq|2
2
Ω2 γ02Ω2
= 2 −
γ0 + 2Ω2 (γ02 + 2Ω2)2
2Ω4
= ,
(γ02 + 2Ω2)2

as well as

(σ−(0), σ−(0)) = ⟨σ−(0)σ−(0)⟩eq − ⟨σ−⟩eq⟨σ−⟩eq

= −⟨σ−⟩2eq
γ02Ω2
=
(γ02 + 2Ω2)2

and

(σz(0), σ−(0)) = ⟨σz (0)σ−(0)⟩eq − ⟨σz ⟩eq⟨σ−⟩eq

= −⟨σ−⟩eq − ⟨σz ⟩eq⟨σ−⟩eq
= −⟨σ−⟩eq⟨σz + 1⟩eq
2γ Ω3
= −i 2 0 2 2 .
(γ0 + 2Ω )

As a last step we use the results for ⟨Σz(t)⟩ and ⟨Σ+(t)⟩, multiplying them by a factor
eiω0t, which arises when moving σ+ to the Schrödinger picture, while the equations have
been solved on resonance corresponding to the interaction picture

e−iω0τ ⟨σ+(τ )σ−(0)⟩eq = ⟨σ+⟩eq⟨σ−⟩eq + (σ+(τ ), σ−(0))

= |⟨σ+⟩eq|2 + (σ+(τ ), σ−(0))
γ02Ω2 1 − 1 γτ
= + e 2 [(σ+(0), σ−(0)) + (σ−(0), σ−(0))]
(γ0 + 2Ω )
2 2 2 2
6 7
1 − 4 γτ
3
γ
+ e cos (µτ ) + sin(µτ ) [(σ+(0), σ−(0)) − (σ−(0),
2 4µ
σ−(0))]
Ω − 3 γτ
+i e 4 sin(µτ )(σz (0), σ−(0)).
2µ
◦
152 Table of contents

After evaluating this quantity one has to take its Fourier transform to come to S(ω).
We do not give the general expression, which is quite cumbersome, but consider two lim-
γ
iting situations. If the driving is weak, i.e. Ω ≪ 40 , the power spectrum of the emitted
field reads
4 52
Ω
S(ω) ≃ δ(ω − ω0),
γ
γ0
as predicted by elastic Rayleigh scattering. In the limit of strong driving Ω ≫ 4
we have
instead
γ0
1 2
S(ω) ≃ B C
2 γ0 2 + (ω − ω0)2
2
3 3
1 γ0 1 γ0
+ 8 92 4 + 8 92 4 ,
4 3γ + (ω − ω + Ω)2 4 3γ + (ω − ω − Ω) 2
4 0 0 4 0 0

so that the spectrum is given by the sum of three Lorentzians, describing three peaks
with heights in the ratio 1:3:1 and integrated area below the peaks in the ratio 1:2:1.
Besides the central peak at the resonant frequency, one has two side peaks at frequency
ω0 ± Ω. This spectrum is known as Mollow spectrum, a typical quantum feature which
has been experimentally observed.
◦
◦

@ B. Vacchini - Unimi & INFN

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Acknowledgments.
We are grateful to our students Marco Rabbiosi, Carlo Sparaciari, Michele Invernizzi,
Luca Fresta, Emanuele Albertinale, Cesare Paulin, Valeria Vento for careful reading and
suggestions on the manuscript.

153
Index

................. . . . . . 16, 66 dual map . . . . . . . . . . . . . . . . . . . 56

B(H) . . . . . . . . . . . . .
. . . . . . .
. 12 duality relation . . . . . . . . . . . . . . . . 17
B ′(H) . . . . . . . . . . . . .
. . . . . . .
. 17 Duhamel’s formula . . . . . . . . . . . . . . 111
E(H) . . . . . . . . . . . . .
. . . . . . .
. 28 Dyson expansion . . . . . . . . . . . . . . . 112
S(H) . . . . . . . . . . . . .
. . . . . . .
. 17 effects . . . . . . . . . . . . . . . . . . . . . 25
T (H) . . . . . . . . . . . . .
. . . . . . .
. 15 entangled . . . . . . . . . . . . . . . . . 89, 92
T ′(H) . . . . . . . . . . . . .
. . . . . . .
. 16 entanglement entropy . . . . . . . . . . . . 91
a posteriori state . . . . . . .
. . . . . . .
. 73 environment . . . . . . . . . . . . . . . . . 85
a priori state . . . . . . . . .
. . . . . . .
. 73 equivalence class . . . . . . . . . . . . . 61, 75
action . . . . . . . . . . . . .
. . . . . . .
. 45 equivalence classes . . . . . . . . . . . . . . 25
adjoint map . . . . . . . . . .
. . . . . . .
. 55 events . . . . . . . . . . . . . . . . . . . . 9, 12
affine . . . . . . . . . . . . .
. . . . . . .
. 25 extremal points . . . . . . . . . . . . . . . . 18
σ-algebra . . . . . . . . . . .
. . . . . . .
.. 7 factorized . . . . . . . . . . . . . . . . . 89, 91
assignment map . . . . . . .
. . . . . . .
. 59 generalized probability measure . . . . . . . 29
automorphisms . . . . . . . .
. . . . . . .
. 45 Gibbs form . . . . . . . . . . . . . . . . . . 34
bath . . . . . . . . . . . . . .
. . . . . . .
. 85 Gleason . . . . . . . . . . . . . . . . . . . . 30
Bell basis . . . . . . . . . . .
. . . . . . .
. 90 Gorini . . . . . . . . . . . . . . . . . . . . . 108
Bloch basis . . . . . . . . . .
. . . . . . .
. 97 Heisenberg picture . . . . . . . . . . . . . . 62
Bloch equations . . . . . . .
. . . . . . .
. 137 Hilbert-Schmidt operators . . . . . . . . . . 16
Bloch sphere . . . . . . . . .
. . . . . . .
. 19 Hille-Yoshida . . . . . . . . . . . . . . . . . 107
Bochner’s theorem . . . . . .
. . . . . . .
. 130 ideality . . . . . . . . . . . . . . . . . . . . 63
Boolean algebra . . . . . . .
. . . . . . .
.. 9 idempotency . . . . . . . . . . . . . . . 13, 38
Born approximation . . . . .
. . . . . . .
. 126 idempotent . . . . . . . . . . . . . . . . . . 120
boundary . . . . . . . . . . .
. . . . . . .
. 19 incompatible . . . . . . . . . . . . . . . . . 25
Breuer . . . . . . . . . . . .
. . . . . . .
. 121 instrument . . . . . . . . . . . . . . . . . . 72
Busch . . . . . . . . . . . . .
. . . . . . .
. 29 integral kernel . . . . . . . . . . . . . . . . 124
canonical basis . . . . . . . .
. . . . . . .
. 97 Klein’s inequality . . . . . . . . . . . . . . . 21
channel . . . . . . . . . . . .
. . . . . . .
. 58 β-KMS condition . . . . . . . . . . . . . . . 131
Choi . . . . . . . . . . . . . .
. . . . . 103, 103 Kossakowski . . . . . . . . . . . . . . . . . 108
Choi matrix representation .
. . . . . . . . 105 Kossakowski matrix . . . . . . . . . . . . . 129
Choi-Jamiolkowski . . . . . .
. . . . . . . . 105 lattice . . . . . . . . . . . . . . . . . . . . .. 9
coexistence . . . . . . . . . .
. . . . . . . . 44 Lindblad . . . . . . . . . . . . . . . . . . . 108
coherent states . . . . . . . .
. . . . . . . . 53 linear matrix representation . . . . . . . . . 95
commutativity . . . . . . . .
. . . . . . . . 44 Liouville von Neumann equation . . . . . . 107
compact operators . . . . . .
. . . . . . . . 16 Markov approximation . . . . . . . . . . . . 126
compatible . . . . . . . . . .
. . . . . . . . 81 master equation . . . . . . . . . . . . . . . 107
complete positivity . . . . . .
. . . . . . . . 55 χ-matrix representation . . . . . . . . . . . 100
composite system . . . . . . .
. . . . . . . . 85 maximum information principle . . . . . . . 34
convexity . . . . . . . . . . .
. . . . . . . . 17 measure space . . . . . . . . . . . . . . . .. 7
.
correlated projection operator . . . . . . . 123 microsystem . . . . . . . . . . . . . . . . . 12
covariant . . . . . . . . . . .
. . . . . . . . 45 moments problem . . . . . . . . . . . . . . 39
decoherence . . . . . . . . . .
. . . . . 112, 114 momentum observable . . . . . . . . . . . . 49
decoherence function . . . . .
. . . . . . . . 118 Naimark . . . . . . . . . . . . . . . . . . . 42
detailed balance condition . .
. . . . . . . . 131 Nakajima . . . . . . . . . . . . . . . . . . . 120
dipole interaction . . . . . . .
. . . . . . . . 134 Nakajima-Zwanzig master equation . . . . . 124
dispersion free states . . . . .
. . . . . . . . 30 noise . . . . . . . . . . . . . . . . . . . . . 70
displacement operator . . . .
. . . . . . . . 115 non-orthogonal decomposition . . . . . . . . 23
distinguishability . . . . . . .
. . . . . . . . 70 operations . . . . . . . . . . . . . . . . . . 55
disturbance . . . . . . . . . .
. . . . . . . . 70 optical Bloch equations . . . . . . . . . . . 140

155
156 Index

orthogonal decompositions . . . . . . . . . . 23 relaxing . . . . . . . . . . . . . .. . . . . .

131
orthogonal projections . . . . . . . . . . . . . 9 relevant observables . . . . . . .. . . . . .
31
outcome space . . . . . . . . . . . . . . . . 35 relevant variable . . . . . . . . .. . . . . .
120
overcomplete . . . . . . . . . . . . . . . . . 53 repeatability . . . . . . . . . . .. . . . . .
63
Ozawa . . . . . . . . . . . . . . . . . . . . 79 reservoir . . . . . . . . . . . . .. . . . . .
85
P(H) . . . . . . . . . . . . . . . . . . . . . . 9 Schmidt decomposition . . . . .. . . . . .
88
partial trace . . . . . . . . . . . . . . . . . 86 Schrödinger picture . . . . . . .. . . . . .
62
Pauli master equation . . . . . . . . . . . . 133 self-adjoint operators . . . . . . .. . . . . .
38
Poincarè sphere . . . . . . . . . . . . . . . . 19 separable . . . . . . . . . . . . .. . . . 89, 92
pointer basis . . . . . . . . . . . . . . . . . 80 Shannon entropy . . . . . . . . .. . . . . . 20
position and momentum observable . . . . . 50 sharp observables . . . . . . . . .. . . . . . 38
position observable . . . . . . . . . . . . . . 46 simplex . . . . . . . . . . . . . .. . . . . . 10
n-positive . . . . . . . . . . . . . . . . . . . 103 G-space . . . . . . . . . . . . . .. . . . . . 45
positive operator-valued measure . . . . . . 36 spectral density . . . . . . . . . .. . . 118, 136
post-measurement state . . . . . . . . . . . 61 spectral representation . . . . . .. . . . . . 17
POVM . . . . . . . . . . . . . . . . . . . . 36 standard projection operator . .. . . . . . 122
pre-measurement state . . . . . . . . . . . . 73 statistical operator . . . . . . . .. . . . . . 15
preparation apparatus . . . . . . . . . . . . 12 Stinespring . . . . . . . . . . . .. . . . . . 64
preparation procedure . . . . . . . . . . . . 12 subcollections . . . . . . . . . . .. . . . . . 55
Prob(Ω) . . . . . . . . . . . . . . . . . . . . 36 Sudarshan . . . . . . . . . . . .. . . . . . 108
probability formula . . . . . . . . . . . . . . . 7 system of imprimitivity . . . . .. . . . . . 47
probability space . . . . . . . . . . . . . . . . 7 trace distance . . . . . . . . . . .. . . . . . 70
projection operator . . . . . . . . . . . . . . 120 trace operation . . . . . . . . . .. . . . . . 15
projection-valued measure . . . . . . . . . . 37 trace-class operators . . . . . . .. . . . . . 15
pure states . . . . . . . . . . . . . . . . . . 18 transition rates . . . . . . . . . .. . . . . . 132
purification . . . . . . . . . . . . . . . . . . 93 transposition . . . . . . . . . . .. . . . . . 93
purity . . . . . . . . . . . . . . . . . . . . . 18 unital . . . . . . . . . . . . . . .. . . . . . 58
PVM . . . . . . . . . . . . . . . . . . . . . 38 unsharpness .......... .. . . . . . 52
quantum dynamical map ..... . . . . . 107 von Neumann entropy . . . . . .. . . . . . 20
quantum dynamical semigroup . . . . . . . 107 von Neumann-Lüders instrument . . . . . . 75
range of an observable . . . . . . . . . . . . 43 Weyl operators . . . . . . . . . .. . . . . . 50
registration apparatus . . . . . . . . . . . . 12 Wigner’s formula . . . . . . . . .. . . . . . 77
registration procedure . . . . . . . . . . . . 12 Zwanzig . . . . . . . . . . . . . .. . . . . . 120

@ B. Vacchini - Unimi & INFN

Epilogue
“One of the principal objects of theoretical research in any department of
knowledge is to find the point of view from which the subject appears in
the greatest simplicity.”
(Josiah Willard Gibbs)

157