Symmetry and Asymmetry in Bosonic Gaussian Systems: A Resource-Theoretic Framework

TL;DR

This paper develops a resource theory for Gaussian asymmetry in bosonic systems, characterizing free operations, monotones, and their behavior under various dynamics, with implications for quantum information and optics.

Contribution

It introduces a framework for Gaussian asymmetry resource theory, including realizations of free operations, monotones, and technical extensions of fundamental theorems.

Findings

Gaussian symmetry-respecting operations can be implemented via symmetry-preserving Hamiltonians and environments.

Identifies monotones that are non-increasing under Gaussian symmetry-respecting dynamics.

Provides new theoretical tools, including a novel approach to the Stinespring dilation theorem and an extension of Williamson's theorem.

Abstract

We study the interplay of symmetries and Gaussianity in bosonic systems, under closed and open dynamics, and develop a resource theory of Gaussian asymmetry. Specifically, we focus on Gaussian symmetry-respecting (covariant) operations, which serve as the free operations in this framework. We prove that any such operation can be realized via Gaussian Hamiltonians that respect the symmetry under consideration, coupled to an environment prepared in a symmetry-respecting pure Gaussian state. We further identify a family of tractable monotone functions of states that remain non-increasing under Gaussian symmetry-respecting dynamics, and are exactly conserved in closed systems. We demonstrate that these monotones are not generally respected under non-Gaussian symmetry-respecting dynamics. Along the way, we provide several technical results of independent interest to the quantum information and optics communities, including a new approach to the Stinespring dilation theorem, and an extension of Williamson's theorem for the simultaneous normal mode decomposition of Gaussian systems and conserved charges.