A Result About the Classification of Quantum Covariance Matrices Based on Their Eigenspectra

TL;DR

This paper investigates the classification of quantum covariance matrices based on their eigenspectra, revealing a specific class where all matrices with a given eigenspectrum are related by orthogonal symplectic transformations, especially for thermal and squeezing parameters.

Contribution

It identifies a non-trivial class of eigenspectra for quantum covariance matrices where all matrices are related by orthogonal symplectic transformations, linking eigenspectra to physical parameters.

Findings

Characterization of eigenspectra for quantum covariance matrices.

Identification of a class where matrices are related by symplectic transformations.

Connection between eigenspectra and physical parameters of Gaussian states.

Abstract

The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices due to Heisenberg's uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix produces a positive-definite matrix that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a covariance matrix of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.