Geometry of Gaussian quantum states

TL;DR

This paper derives an analytical measure for the volume of Gaussian states in continuous variable quantum systems, enabling the study of their typical properties and entropy distributions.

Contribution

It introduces an explicit Hilbert-Schmidt volume element for multi-mode Gaussian states and analyzes their entropy and purity distributions under this measure.

Findings

Hilbert-Schmidt measure yields a normalizable distribution of symplectic eigenvalues.

Distribution of von Neumann entropy and purity for Gaussian states is characterized.

Bures measure does not produce a normalizable distribution for one-mode Gaussian states.

Abstract

We study the Hilbert-Schmidt measure on the manifold of mixed Gaussian states in multi mode continuous variable quantum systems. An analytical expression for the Hilbert-Schmidt volume element is derived. Its corresponding probability measure can be used to study typical properties of Gaussian states. It turns out that although the manifold of Gaussian states is unbounded, an ensemble of Gaussian states distributed according to this measure still has a normalizable distribution of symplectic eigenvalues, from which unitarily invariant properties can be obtained. By contrast, we find that for an ensemble of one-mode Gaussian states based on the Bures measure the corresponding distribution cannot be normalized. As important applications, we determine the distribution and the mean value of von Neumann entropy and purity for the Hilbert-Schmidt measure.