The Controlled Z-gate

When we apply the $Z$-gate on separate qubits, we usually see the phase shift for those states where the respective qubit is in state $|1\rangle$. When both qubits are in-state $|1\rangle$, the phase shift adds up. Since the $Z$-gate accounts for a phase shift by half a wavelength, applying it twice results in the initial phase. Hence, the state $|11\rangle$ has the same phase as a state $|00\rangle$.

Sometimes we don’t want this effect. We might want to switch the phase of a single state. This is where, the controlled $Z$ ...