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

• URL: http://arxiv.org/abs/2412.19698v2
• Date: Wed, 06 Aug 2025 12:50:16 GMT
• Title: Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states
• Authors: Jan de Boer, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni,
• Abstract summary: We develop the theory of continuous majorization in the general $N$-mode case.<n>We also propose extensions to include states with finite Wigner negativity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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