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

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 []. In this work, we develop the theory of continuous majorization in the general <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-mode case. In addition, we propose extensions to include states with finite Wigner negativity. For the special case of the convex hull of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math>-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.