# 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?

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.

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.