Invariant measures on multimode quantum Gaussian states

TL;DR

This paper derives an invariant measure on the space of multimode quantum Gaussian states, emphasizing the role of symplectic eigenvalues in quantum entanglement, with applications in quantum optics and information.

Contribution

It introduces a method to obtain the invariant measure on Gaussian states using Haar measure and energy constraints, highlighting nonlocal degrees of freedom.

Findings

Derived the invariant measure on Gaussian states manifold.

Obtained the distribution of symplectic eigenvalues under energy constraints.

Applicable to quantum optics and quantum information scenarios.

Abstract

We derive the invariant measure on the manifold of multimode quantum Gaussian states, induced by the Haar measure on the group of Gaussian unitary transformations. To this end, by introducing a bipartition of the system in two disjoint subsystems, we use a parameterization highlighting the role of nonlocal degrees of freedom -- the symplectic eigenvalues -- which characterize quantum entanglement across the given bipartition. A finite measure is then obtained by imposing a physically motivated energy constraint. By averaging over the local degrees of freedom we finally derive the invariant distribution of the symplectic eigenvalues in some cases of particular interest for applications in quantum optics and quantum information.