Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states

TL;DR

This paper develops the theory of continuous majorization in quantum phase space for multiple modes and extends it to include Wigner-negative states, providing new tools for quantum resource comparison.

Contribution

It generalizes continuous majorization to N-mode systems and proves a conjecture for Gaussian states, also extending majorization concepts to Wigner-negative states.

Findings

Proved a conjecture for convex hulls of Gaussian states.

Developed phase space Uhlmann's theorem.

Extended majorization to Wigner-negative states.

Abstract

In quantum resource theory, one is often interested in identifying which states serve as the best resources for particular quantum tasks. If a relative comparison between quantum states can be made, this gives rise to a partial order, where states are ordered according to their suitability to act as a resource. In the literature, various different partial orders for a variety of quantum resources have been proposed. In discrete variable systems, vector majorization of Wigner functions in discrete phase space provides a natural partial order between quantum states. In the continuous variable case, a natural counterpart would be continuous majorization of Wigner functions in quantum phase space. Indeed, this concept was recently proposed and explored (mostly restricting to the single-mode case) in Van Herstraeten, Jabbour, Cerf, Quantum 7, 1021 (2023). In this work, we develop the theory of continuous majorization in the general $N$-mode case. In addition, we propose extensions to include states with finite Wigner negativity. For the special case of the convex hull of $N$-mode Gaussian states, we prove a conjecture made by Van Herstraeten, Jabbour and Cerf. We also prove a phase space counterpart of Uhlmann's theorem of majorization.