CNOT gate (Controlled NOT gate)

The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.

This gate is the first example of a controlled quantum gate that you should study.

To understand why the gate is called a CNOT gate, you should think as follows.

First let's produce a generic quantum state vector where the control qubit is certain to be 0.

On the standard basis: we see that this means that only $|00⟩$ and $|01⟩$ should be possible. Therefore, the state must be of the form: where $x$ and $y$ are two complex numbers such that $|x|+|y|=1.0$

If we operate the CNOT gate on that state, we obtain: and so the input is unchanged as desired, because the control qubit is 0.

If however we take only states where the control qubit is for sure 1:

Therefore, in that case, what happened is that the probabilities of $|10⟩$ and $|11⟩$ were swapped from $x$ and $y$ to $y$ and $x$ respectively, which is exactly what the quantum NOT gate does.

So from this we understand more concretelly what "the gate only operates if the first qubit is set to one" means.

Now go and study the Bell state and understand intuitively how this gate is used to produce it.