Optimal phase estimation for a Mach-Zehnder interferometer.

We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible to achieve the standard quantum limit *independently* from the true value of the phase shift and specific assumptions on the noise of the interferometer. We have been able to implement the protocol using parallel operation of two photon-number-resolving detectors and multiphoton coincidence logic electronics at the output ports of a weakly-illuminated Mach-Zehnder interferometer. This protocol is unbiased and saturates the Cramer-Rao phase uncertainty bound and, therefore, is an optimal phase estimation strategy.