Quantum teleportation
|ψ⟩ ------------
|+⟩ ------------
|ψ⟩|+⟩ = (a|0⟩ + b|1⟩)(|0⟩ + |1⟩)
|ψ⟩ ----.-------
|
|+⟩ ----.---------------
a|00⟩ + a|01⟩ + b|10⟩ - b|11⟩
|ψ⟩ ----.---H---
|
|+⟩ ----.---------------
a|00⟩ + a|10⟩ + a|01⟩ + a|11⟩ + b|00⟩ - b|10⟩ - b|01⟩ + b|11⟩
|ψ⟩ ----.--------H---MZ
|
|+⟩ ----.---------------
If +1 then (a|+⟩ + b|-⟩); If -1 then (a|+⟩ - b|-⟩)
|ψ⟩ ----.--------H---MZ
|
|+⟩ ----.--------------- Xᵐ H|ψ⟩
“The identity above is easily generalized to the following identity:”
|ψ⟩ ----.---Zθ---H---MZ | |+⟩ ----.--------------- XᵐHZθ|ψ⟩
Proof: The output is the same as if Zθ|ψ⟩ had been input instead of |ψ⟩.
“By varying the basis in which the first qubit is measured, i.e., by varying θ, we can vary the unitary transformation effected on the second qubit.”
“Remarkable. Although the first qubit is measured, no quantum information is lost; The posterior state of the second qubit is related by a known unitary transformation to the original input, |ψ⟩, regardless of the measurement outcome.”
“It is tempting to regard this as unsurprising. After all, suppose we replaced the CZ by a SWAP, which merely interchanges the state of the two qubits. Then we would not expect a measurement on the first qubit to destroy any quantum information, since all the quantum information would have been transferred from qubit one to qubit two before the measurement on qubit one.” In this case, measuring the first qubit has no effect on the second qubit.
“But this is not what happens in the circuit above. There we can vary the transformation effected on the second qubit without destroying any quantum information.
Two instances:|ψ⟩ -----.---H---MZ | |+⟩ -----.-------.---H---MZ | |+⟩ -------------.-------------- Xᵐ²H Xᵐ¹H|ψ⟩
Can rewrite to expose a Bell measurement circuit on the first two qubits;
There are Things you can do in any non-classical GPT and things you can only do in quantum theory. Quantum teleportation is one of the latter.
- Teleporting a qubit is equivalent to sending it through a noiseless channel;