Hermitian operator of a Mach–Zehnder interferometer combined with parity detection and its applications

In quantum metrology, parity detection is often used to investigate the phase estimation based on optical interferometer. Different from previous work, we consider a Mach–Zehnder interferometer (MZI) and parity detection as a whole operation in Heisenberg picture, and obtain the corresponding Hermitian operator. By this alternative operator method, one can derive the parity signal directly in terms of input states. Our results clearly show that the phase sensitivity obtained by the parity detection is independent of both configurations of the estimated phases and the output ports of the MZI. As applications, we analytically demonstrate that the no-go theorem indeed holds for both parity detection and single-mode intensity detection. We also investigate the phase estimation of the MZI seeded by the two-mode squeezed coherent state, and obtain the more concise and illuminating results.