Estimation of Gaussian quantum states

TL;DR

This paper derives comprehensive formulas for the quantum Fisher information matrix in multi-mode Gaussian states, enabling optimal parameter estimation and addressing issues with pure states.

Contribution

It introduces new expressions for QFIM applicable to mixed and pure Gaussian states, including cases with pure modes, enhancing quantum parameter estimation techniques.

Findings

Derived formulas for QFIM for mixed and pure states

Provided methods to handle pure modes in Gaussian states

Discussed conditions for saturability of quantum Cramér-Rao bound

Abstract

We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cram\'{e}r-Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.