Symplectic and Lagrangian Polar Duality; Applications to Quantum Information Geometry

TL;DR

This paper introduces two symplectically covariant versions of polar duality tailored for quantum mechanics, enabling new insights into quantum states and their geometric properties within quantum information theory.

Contribution

It develops novel symplectic and Lagrangian polar dualities and applies them to define geometric quantum states and analyze quantum covariance matrices.

Findings

New symplectic polar duality framework for quantum phase space

Definition of geometric quantum states via Lagrangian polar duality

Enhanced understanding of quantum covariance matrices

Abstract

Polar duality is a well-known concept from convex geometry and analysis. In the present paper, we study two symplectically covariant versions of polar duality keeping in mind their applications to quantum mechanics. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The latter is a fundamental object in quantum information theory., The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes. It allows us to define the notion of "geometric quantum states" with are in bijection with generalized Gaussians.