Phase space formalism for quantum estimation of Gaussian states

TL;DR

This paper develops a comprehensive asymptotic estimation theory for Gaussian quantum states using their moments, deriving the quantum Fisher information and optimal measurement strategies, including for pure states.

Contribution

It introduces a general framework expressing quantum Fisher information and SLD in terms of Gaussian states' moments, extending classical Fisher information concepts to the quantum domain.

Findings

Derived explicit formulas for QFI and SLD in Gaussian states.

Identified optimal homodyne detection schemes for parameter estimation.

Showed that pure state models can reach fundamental quantum limits.

Abstract

We formulate, with full generality, the asymptotic estimation theory for Gaussian states in terms of their first and second moments. By expressing the quantum Fisher information (QFI) and the elusive symmetric logarithmic derivative (SLD) in terms of the state's moments (and their derivatives) we are able to obtain the noncommutative extension of the well known expression for the Fisher information of a Gaussian probability distribution. Focusing on models with fixed first moments and identical Williamson 'diagonal' states --which include pure state models--, we obtain their SLD and QFI, and elucidate what features of the Wigner function are fundamentally accessible, and at what rates. In addition, we find the optimal homodyne detection scheme for all such models, and show that for pure state models they attain the fundamental limit.