Phase variance of squeezed vacuum states

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−2, only for N 4. For N 4 this fundamental scaling breaks down and it becomes n−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−2 is recovered for large samples.