Symplectic Polar Duality, Quantum Blobs, and Generalized Gaussians

TL;DR

This paper explores the use of convex geometric polar duality to characterize quantum covariance regions called quantum blobs, providing new insights into Gaussian states and symplectic tomography in quantum phase space.

Contribution

It introduces a novel geometric characterization of quantum blobs via polar duality, enhancing understanding of quantum covariance ellipsoids and Gaussian states.

Findings

Quantum blobs are characterized by reflexivity under polar duality.

The approach improves previous geometric descriptions of quantum covariance regions.

Applications to symplectic tomography and Gaussian state characterization.

Abstract

We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of "quantum blobs" introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson--Schr\"odinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Gaussian states in terms of partial information on the covariance ellipsoid, which allows us to formulate statements related to symplectic tomography.