Phase variance of squeezed vacuum states
E. Bagán, Alex Monràs, R. Muñoz-Tapia · Physical Review A · 2008
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.