Optimal Gaussian measurements for phase estimation in single-mode Gaussian metrology

TL;DR

This paper investigates whether Gaussian measurements can reach the ultimate precision bound in single-mode Gaussian metrology for phase estimation, identifying optimal Gaussian strategies and the necessity of non-Gaussian measurements for full optimality.

Contribution

It demonstrates the conditions under which Gaussian measurements are optimal and shows that non-Gaussian measurements are needed for complete optimality in phase estimation.

Findings

Homodyne measurement attains the ultimate bound for displaced thermal and squeezed vacuum states.

Optimized Gaussian measurements are nearly optimal but not fully optimal for some states.

Non-Gaussian measurement based on product quadratures is required for full optimality.

Abstract

The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators XP+PX, i.e., a non-Gaussian measurement, is required to be fully optimal.