I can't get my head around this. I'm trying to implement a simple CNOT-Gate.
I have to qubits:
c = $|0\rangle$ = [1, 0]
t = $|1\rangle$ = [0, 1]
For all combinations of states $|00\rangle$, $|10\rangle$, $|01\rangle$ and $|11\rangle$ we would now be in $|01\rangle$.
So CNOT should have no effect because qubit c (control) is in state $|0\rangle$.
Let's create the tensor product:
$c \otimes t$ = [1 * 0, 1 * 1, 0 * 0, 0 * 1] = [0, 1, 0, 0]
Now we apply CNOT:
$$\operatorname{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} $$
to the tensor product [0, 1, 0, 0]:
The result is: [0, 1, 0, 0]
So we have 0 * $|00\rangle$, 1 * $|10\rangle$, 0 * $|01\rangle$ and 0 * $|11\rangle$ .
So we are actually in state c = $|1\rangle$ and t = $|0\rangle$ now. That is a contradiction to the state we were in at the beginning $|01\rangle$ .
What am I missing here?