I'm stuck on this passage from a book:
Suppose we try to copy a qubit in the unknown state $|ψ\rangle = a|0\rangle + b|1\rangle$ in the same manner by using a $\text{CNOT}$ gate. The input state of the two qubits may be written as $$ \big[a|0\rangle + b|1\rangle \big]\ |0\rangle = a |00\rangle + b |10\rangle $$
The function of $\text{CNOT}$ is to negate the second qubit when the first qubit is $1$, and thus the output is simply
$$ a |00\rangle + b |11\rangle $$
It seems $|0\rangle$ distributes over $\big[a|0\rangle + b|1\rangle \big]$ to get $a |00\rangle + b |10\rangle$, so the first part makes sense, but supposing the first qubit state is $|1\rangle$ then wouldn't the final state be
$$ \big[a|0\rangle + b|1\rangle \big]\ |1\rangle = a |01\rangle + b |11\rangle $$
since the second qubit is changed by the gate? Where does $a |00\rangle + b |11\rangle$ come from?
I've realized that
$$ |0\rangle |0\rangle = |0\rangle \otimes |0\rangle $$
is short hand for the tensor product.