I know this question has been answered here, but the answers leave some things confusing to me. When broken down (and used to calculate), the idea that the states are entangled doesn't seem to be right. I'm sure there is a mistake in my reasoning, but I'm not sure where. Here goes:
Say we take qubits in states $a|0\rangle + b|1\rangle$ and send them through a CNOT gate, which looks like this, the gate on the bottom being an XOR (that outputs 1 if the inputs are not the same binary value).
First, we use a fanout, which, according to other answers, leaves us with two qubits in state $a|00\rangle + b|11\rangle$. The set of three qubits, before two are passed through the XOR gate, can be expressed as $a|00\rangle\otimes|0\rangle + b|11\rangle\otimes|0\rangle$.
When we put this through the XOR gate, there is probability $a^2$ that the qubit will be $0$ which would result in an output of $0$. This means that exiting the gate, the probability amplitude of $|0\rangle$ is $a$. There is probability $b^2$ that the qubit will be $1$ so the probability amplitude of getting $|1\rangle$ as the second ouput is $b$. So our qubit is in state $a|0\rangle + b|1\rangle$, a clone of the original state. We now have two qubits in this state, as the first one comes out of the first output unchanged. As far as I can tell, they are not entangled.
Obviously, there is something wrong in this logic, I just don't know where. Could someone help me out?