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

Jan de Boer1, Giuseppe Di Giulio2, Esko Keski-Vakkuri3,4,5, and Erik Tonni6

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Phys. Rev. A 112, 032405 – Published 2 September, 2025

DOI: https://doi.org/10.1103/w561-h3z5

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) by Van Herstraeten et al. [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.

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Outline

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Phys. Rev. A 112, 032405– Published 2 September, 2025
  • Received 23 January 2025
  • Revised 29 July 2025
  • Accepted 7 August 2025
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