Typical Gaussian Quantum Information

TL;DR

This paper explores the geometry and measures on the space of mixed Gaussian quantum states, establishing a unique volume measure under fixed purity and analyzing typical quantum correlations like entanglement and steerability.

Contribution

It introduces a unique invariant measure for mixed Gaussian states with fixed purity and studies typical quantum correlations within this framework.

Findings

Invariant measures coincide at fixed purity

Explicit computation of typical symplectic invariants

Numerical analysis of quantum correlations with energy constraints

Abstract

We investigate different geometries and invariant measures on the space of mixed Gaussian quan- tum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically.