Knowledge-dependent optimal Gaussian strategies for phase estimation

TL;DR

This paper identifies how the optimal Gaussian probe states for phase estimation depend on prior knowledge, showing a transition from coherent to squeezed vacuum states as the estimate becomes more precise, with implications for adaptive measurement strategies.

Contribution

It provides a detailed analysis of the optimal Gaussian probe states based on prior knowledge, revealing a sudden transition to squeezed vacuum states for high precision estimates.

Findings

Optimal probe states depend on prior knowledge of the phase.

Transition from coherent to squeezed vacuum states as estimation improves.

Adaptive strategies can enhance phase estimation accuracy.

Abstract

When estimating an unknown phase rotation of a continuous-variable system with homodyne detection, the optimal probe state strongly depends on the value of the estimated parameter. In this article, we identify the optimal pure single-mode Gaussian probe states depending on the knowledge of the estimated phase parameter before the measurement. We find that for a large prior uncertainty, the optimal probe states are close to coherent states, a result in line with findings from noisy parameter estimation. But with increasingly precise estimates of the parameter it becomes beneficial to put more of the available energy into the squeezing of the probe state. Surprisingly, there is a clear jump, where the optimal probe state changes abruptly to a squeezed vacuum state, which maximizes the Fisher information for this estimation task. We use our results to study repeated measurements and compare different methods to adapt the probe state based on the changing knowledge of the parameter according to the previous findings.