Quantum Computing
Lecture 5
Applications of Quantum Information
Anuj Dawar
Some Applications
We look at some applications of the encoding of information in quantum
states.
• Quantum Cryptography, or more accurately Quantum Key
Distribution.
• Superdense Coding.
• Quantum Teleportation
These do not rely on quantum computation as such, but the properties of
information encoded in quantum states: superposition and entanglement.
Quantum Key Distribution
A protocol for quantum key distribution was described by Bennett and
Brassard in 1984 (and is known as BB84).
Eve
qubit channel The protocol does not provide
Alice Bob the means of transmitting an
classical channel arbitrary message.
At the end of the protocol, there is a random sequence of bits that is
shared between Alice and Bob but unknown to any third party.
Assumptions
The BB84 protocol relies on the following assumptions:
• Alice has a source of random (classical) bits.
• Alice can produce qubits in states |0i and |1i.
• Alice can apply a Hadamard operator H to the qubits.
• Bob can measure incoming qubits
• either in the basis |0i, |1i;
• or in the basis √1 (|0i + |1i), √1 (|0i − |1i).
2 2
These conditions are satisfied, for instance, by a system based on
polarised photons.
The Protocol
Alice sends Bob a stream of qubits.
For each qubit, before sending it, she
• randomly chooses a bit |0i or |1i;
• randomly either applies H to the qubit or not; and
• sends it to Bob.
So, Bob receives a random sequence of qubits, each of which is in one of
the four states:
|0i, |1i, H|0i, H|1i
The Protocol–contd.
• For each qubit, Bob randomly chooses either the basis |0i, |1i or the
basis H|0i, H|1i and measures the qubit in the chosen basis.
• Bob announces (over the classical channel) which basis he used for
each measurement.
• Alice tells Bob which measurements were made in the correct basis.
• The qubits which were measured in the wrong basis are discarded,
while the rest form a shared key.
Attacks
Why not announce the bases for all qubits before transmission, thus
avoiding the loss of half the bits?
This allows Eve to intercept, measure and re-transmit the bits.
Why not wait until Bob has received all the qubits, then have Alice
announce the basis for each one before Bob measures them?
• Requires Bob to store the qubits—currently technically difficult.
• If Bob can store the qubits, then Eve can too and then she can
retransmit after measurement.
If we could fix the basis before hand, this could be used to transmit a
fixed (rather than random) message.
Attack 2
What happens if Eve intercepts the qubits, measures each one randomly
in either the basis |0i, |1i or the basis H|0i, H|1i and then retransmits it?
For half of the bits that are shared between Alice and Bob, Eve will have
measured them in the wrong basis.
Moreover, these bits will have changed state, and so for approx. 14 of the
shared bits, the value measured by Bob will be different to the one
encoded by Alice.
Alice and Bob can choose a random sample of their shared bits and
publically check their values against each other and detect the presence
of an eavesdropper.
Attack 3
Could Eve intercept the qubits, make a copy without measuring them
and re-transmit to Bob and then wait for the basis to be announced?
No Cloning Theorem:
There is no unitary operation U which for an arbitrary state ψ
gives
U|ψ0i = |ψψi.
Exercise: Prove the no-cloning theorem.
Key Distribution
Quantum key distribution relies on nothing more than
• linear superposition of states; and
• change of basis.
In particular, it does not rely on entanglement.
We next look at some applications of entanglement.
Bell States
Entanglement based protocols generally rely on using the following four
states of a two-qubit system, known as the Bell states.
1 1
√ (|00i + |11i), √ (|01i + |10i)
2 2
1 1
√ (|00i − |11i), √ (|01i − |10i)
2 2
These form an orthonormal basis for C4 , known as the Bell basis.
Note that, in each of the states, measuring either qubit in the
computational basis yields |0i or |1i with equal probability, but after the
measurement, the other bit is determined.
Generating Bell States
We can generate the Bell states from the computational basis
|00i, |01i, |10i, |11i using the following circuit:
|00i √1 (|00i + |11i)
2
√1 (|00i + |10i)
2
Superdense Coding
In general, it is impossible to extract more than one classical bit of
information from a single qubit.
However, if Alice and Bob is each in possession of one qubit of a pair in a
known Bell state
1
√ (|00i + |11i)
2
Then Alice can perform an operation solely on her own qubit, and then
send it to Bob to convey two bits of information.
Superdense Coding 2
Generating Bell states from √1 (|00i + |11i) with only operations on the
2
first qubit.
1 1
(X ⊗ I ) √ (|00i + |11i) = √ (|01i + |10i)
2 2
1 1
(Z ⊗ I ) √ (|00i + |11i) = √ (|00i − |11i)
2 2
1 1
((ZX ) ⊗ I ) √ (|00i + |11i) = √ (|01i − |10i)
2 2
Superdense Coding 3
Once he has both qubits, Bob can convert back to the computational
basis using the circuit.
After this, a measurement in the computational basis yields the two bits
that Alice intended to convey.
Quantum Teleportation
The superdense coding protocol allows Alice to send Bob two classical
bits by transmitting a single qubit, provided they already share an
entangled pair.
Conversely, the quantum teleportation protocol allows Alice to send Bob
a qubit, by sending just two classical bits along a classical channel,
provided they already share an entangled pair.
Contrast this with the no-cloning theorem, which tells us that we cannot
make a copy of a qubit.
Quantum Teleportation 2
Alice has a state |φi that she wishes to transmit to Bob. The two already
share a pair of qubits in state √12 (|00i + |11i).
|φi H
M
Entangled Pair
Quantum Teleportation 3
Alice conveys to Bob the result of her measurement. Say the qubit in
Bob’s possession is in state |θi, then:
• If Alice measures |00i, then |φi = |θi.
• If Alice measures |01i, then |φi = X |θi.
• If Alice measures |10i, then |φi = Z |θi.
• If Alice measures |11i, then |φi = XZ |θi.
Thus, Bob performs the appropriate operation and now has a qubit
whose state is exactly |φi.
