CNOT and FTL communication

Question

Let's consider two qubits $A$ and $B$.

They are related by a CNOT gate, I call $U_{NOT}$ the matrix of the CNOT.

So, I have $|A^{out}\rangle |B^{out}\rangle = U_{NOT} |A^{in}\rangle |B^{in}\rangle$

In particular, I have:

$$ U_{NOT} |0\rangle |0\rangle =|0\rangle |0\rangle$$

and

$$ U_{NOT} |1\rangle |0\rangle = |1\rangle |1\rangle$$

The thing I don't understand with the CNOT gate is that for me it would allow FTL communication.

Indeed, if I call $d$ the distance between my two qubits, if the CNOT evolution is fast enough, I call $T$ the time of execution of the CNOT, $A$ could be able to send either the bit $1$ or $0$ to $B$ faster than light.

For example, $A$ wants to send the bit $1$ to B that is initially in the state $|0\rangle$, it applies $ U_{NOT} |1\rangle |0\rangle$, the final state will thus be $|1\rangle|1\rangle$ : if $T<d/c$, the communication would have been FTL.

Where is my mistake?

Answer 1

The reason is that the CNOT operation is not a separable operation, i.e., you cannot write $U_{NOT}=U_A\otimes U_B$ where $U_A$ acts only on the first subsystem and $U_B$ acts only on the second. In other words, in order to perform a CNOT operation, you need to act on both qubits "at the same time" or, equivalently, you need to have the qubits "in the same place".

In practice, this means that the CNOT time T must be greater than $d/c$.