Are there any single-qubit states $|a\rangle$ and $|b\rangle$ so that applying a CNOT to combined state $|ab\rangle$ changes the control qubit?

Question

When a CNOT gate is applied to 2 qubits, one is known as the control qubit while the other is called target qubit.

According to what I understand, if the control qubit is in state $|1\rangle$ then it flips the target qubit otherwise it does nothing.

I want to know whether there is some state of 2 qubits $|a\rangle$ and $|b\rangle$ such that when we apply CNOT on the two of them by which we can flip the control qubit when control qubit is in state $|1\rangle$ and do nothing to the target qubit.

Answer 1

Yes. What you want to make use of, essentially, is the identity meaning that single-qubit rotations can effectively reverse the controlled-not.

In particular, if we input the state $|+\rangle|-\rangle$ to controlled-not, the output is $|-\rangle|-\rangle$, so the control qubit has flipped its value to an orthogonal state.