Quantum Teleportation

Below is mainly taken from:

Alice wants to send quantum information to Bob. Specifically, suppose she wants to send the qubit state |ψ⟩=α|0⟩+β|1⟩. This entails passing on information about α and β to Bob.

However, according to the non-clone theorem, Alice can’t simply generate a copy of |ψ⟩ and give the copy to Bob. So we need to use Quantum Teleportation to build the tunnel.

A third party Telamon has been introduced in this protocol:

Step 1:

Alice needs to transmit |ψ⟩=α|0⟩+β|1⟩(a random qubit) to Bob. She doesn’t know the state of the qubit. For this, Alice and Bob take the help of a third party (Telamon). Telamon prepares a pair of entangled qubits for Alice and Bob. The entangled qubits could be written in Dirac Notation as:

Alice and Bob each possess one qubit of the entangled pair (denoted as A and B respectively):

This creates a three qubit quantum system where Alice has the first two qubits and Bob the last one.

Step 2:

Now according to the protocol Alice applies CNOT gate on her two qubits followed by Hadamard gate on the first qubit. This results in the state:

Which can then be separated and written as:

Step 3:

Alice measures the first two qubit (which she owns) and sends them as two classical bits to Bob. The result she obtains is always one of the four standard basis states |00⟩,|01⟩,|10⟩, and |11⟩with equal probability.

On the basis of her measurement, Bob’s state will be projected to,

Step 4:

Bob, on receiving the bits from Alice, knows he can obtain the original state |ψ⟩ by applying appropriate transformations on his qubit that was once part of the entangled pair.

The transformations he needs to apply are:

After this step Bob will have successfully reconstructed Alice’s state.