# Why don't quantum gates get entangled with the qubits they operate on?

I am currently taking a first course in quantum information theory and having trouble understanding how quantum gates are possible in practice. How does a unitary quantum gate evolve the qubit without interacting with the state and entangling it?

If the qubit is system A in the state $|\phi(t_0)\rangle$, and the quantum gate is system B in the state $|\psi(t_0)\rangle$, then the composite system AB is in a pure unentangled state $|\Omega(t_0)\rangle=|\phi(t_0)\rangle \otimes |\psi(t_0)\rangle$. The state evolves unitarily by $|\Omega(t)\rangle=U(t)^{AB}|\Omega(t_0)\rangle$. We only end up with an unentangled state if the interaction Hamiltonian is very weak.

How should I think of the system AB?

• I asked a related question here, in case it helps you. – knzhou Feb 14 '18 at 16:23

If you put the control qubit into a superposition state like $|+\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$, then you do risk the control becoming entangled with the target. But if the control qubit is in a computational basis state, either $|0\rangle$ or $|1\rangle$, then you will find that applying the CNOT operation never entangles the control with the target.