Mathematical Expression of Quantum Teleportation

In our previous article, we had an article about quantum teleportation. In today's article, we will talk about how quantum teleportation is mathematically possible. To briefly restate quantum teleportation;

Quantum teleportation allows a state vector to be transmitted somewhere without any quantum channels, even at the speed of light distance between them.

For this, the following protocol is applied.

A long time ago, Alice and Bob shared two qubits in an entangled state with each other.

Let Alice's state vector be:

The operations on the state vector |ψ> in the above protocol are |ψ₁>, |ψ₂>, |ψ₃> and |ψ₄>.

Let's examine the state vectors one by one.

Let's pass the state vector |ψ₁> through the CNOT and get |ψ₂>.

The first two qubits belong to Alice, and the last qubits to Bob. Now let's go through Hadamard.

Now Alice can measure over her two qubits. As a result of the measurements, four results can be obtained with a 25% probability.

According to Alice's measurement results, Bob's actions are as follows:

There are two points to note here. First of all, this process does not happen at the speed of light. Because, as a result of Alice's measurement, she sends her qubit to Bob with classical channels (there are two channels), not quantum channels.

The second situation may seem to violate the no-cloning theory on my blog, but this is also not true. Because when Alice measures the qubits in her hand, the initial state vector collapses and Bob rebuilds it.

Reference

https://www.youtube.com/watch?v=fJb6qcHtN8M

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