Quantum Teleportation
#Mathematics/Quantum
Teleport your qubit by using one entangled pair and transmitting two bits of information
How it Works
Alice and Bob shares an entangled qubit pair, let's say |q 2q3⟩ =
1
(|00⟩ + |11⟩) where |q 2⟩
√2
belongs to Alice. Alice wants to teleport her qubit in state |q 1⟩ = α|0⟩ + β|1⟩ to Bob.
We can think about the three qubit system as
1
|q1q2q3⟩ = (α|0⟩ + β|1⟩) ⊗ (|00⟩ + |11⟩)
√2
1
= (α|000⟩ + α|011⟩ + β|100⟩ + β|111⟩)
√2
Using the transformation from computational basis to Bell basis, we can write this system in Bell
Basis as
+
|q1q2q3⟩ = |Φ ⟩ ⊗ (α|0⟩ + β|1⟩)
−
+ |Φ ⟩ ⊗ (α|0⟩ − β|1⟩)
+
+ |Ψ ⟩ ⊗ (α|1⟩ + β|0⟩)
−
+ |Ψ ⟩ ⊗ (α|1⟩ − β|0⟩)
See the relation between first two qubits and the third?
To teleport her qubit, Alice measures her two qubits |q q ⟩ in Bell basis. This measurement can
1 2
result in one of the four bell states. Depending on this measurement, Alice sends a two-bit
encoding to Bob, which is as follows:
If she measures
|Φ
+
⟩ , she sends 00,
|Φ
−
⟩ , she sends 01,
|Ψ
+
⟩ , she sends 10,
|Ψ
−
, she sends 11.
⟩
Depending on the two-bits he receives, Bob does the following:
If he receives
00 , he doesn't do anything
01 , he applies Z gate to his qubit
10 , he applies X gate to his qubit
11, he first applies Z gate and then X gate to his qubit.
This way, Bob ends up with α|0⟩ + β|1⟩.
Notice the similarity of encodings between teleportation and Superdense Coding.
