Mach-Zehnder interferometry based on mixing the coherent and the squeezed vacuum states of light has Heisenberg limited capabilities for phase estimation. This is also, because the quantum Cramer-Rao bound on sensitivity of phase estimation with the above interferometric scheme reaches the Heisenberg limit when the inputs are mixed in hear equal proportions. We show that a detection strategy based on the measurement of parity of photon number in one of the output modes of the interferometer saturates the quantum Cramer-Rao bound of the interferometric scheme, and therefore- as a consequence- hits the Heisenberg limit when the inputs are mixed in equal intensities.
We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number
N of uncorrelated copies. We find that it scales with the mean photon number
n, as dictated by the Heisenberg limit, i.e., as ${n}^{\ensuremath{-}2}$, only for
N>4. For $N\ensuremath{\leqslant}4$ this fundamental scaling breaks down and it becomes ${n}^{\ensuremath{-}N∕2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also compute the variance for repeated individual measurements (without classical communication or adaptivity) and find that the standard Heisenberg-limited scaling ${n}^{\ensuremath{-}2}$ is recovered for large samples.