Phase variance of squeezed vacuum states

TL;DR

This paper analyzes the phase estimation limits of squeezed vacuum states, revealing how the variance scales with photon number depending on the number of copies, and compares different measurement strategies within a Bayesian framework.

Contribution

It derives bounds on phase variance for multiple copies of squeezed vacuum states, showing a transition in scaling behavior and identifying optimal energy distribution strategies.

Findings

Variance scales as n^{-2} for N>4 copies

For N≤4, variance scales as n^{-N/2}

Standard Heisenberg scaling recovered with repeated measurements

Abstract

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 $N\leq 4$ 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.