Geometry of perturbed Gaussian states and quantum estimation

TL;DR

This paper explores the geometric structure of non-Gaussian states created by weak perturbations of Gaussian states and their implications for quantum estimation, revealing how non-Gaussianity relates to quantum Fisher information and state distinguishability.

Contribution

It provides a detailed analysis of the relationship between non-Gaussianity and quantum estimation, highlighting the geometric interpretation of non-Gaussian states near Gaussian states.

Findings

Non-Gaussianity can be expressed as a quantum Fisher information distance minus an energy term.

On isoenergetic surfaces, non-Gaussianity quantifies statistical distinguishability.

Non-Gaussianity bounds the quantum Fisher information depending on the type of perturbation.

Abstract

We address the nonGaussianity (nG) of states obtained by weakly perturbing a Gaussian state and investigate the relationships with quantum estimation. For classical perturbations, i.e. perturbations to eigenvalues, we found that nG of the perturbed state may be written as the quantum Fisher information (QFI) distance minus a term depending on the infinitesimal energy change, i.e. it provides a lower bound to statistical distinguishability. Upon moving on isoenergetic surfaces in a neighbourhood of a Gaussian state, nG thus coincides with a proper distance in the Hilbert space and exactly quantifies the statistical distinguishability of the perturbations. On the other hand, for perturbations leaving the covariance matrix unperturbed we show that nG provides an upper bound to the QFI. Our results show that the geometry of nonGaussian states in the neighbourhood of a Gaussian state is definitely not trivial and cannot be subsumed by a differential structure. Nevertheless, the analysis of perturbations to a Gaussian state reveals that nG may be a resource for quantum estimation. The nG of specific families of perturbed Gaussian states is analyzed in some details with the aim of finding the maximally non Gaussian state obtainable from a given Gaussian one.