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arXiv:2412.19698 · Physical Review A

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

Jan de Boer, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni C

Published Aug 7, 2025• 1 citation
arXivHTMLPDF

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 []. 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.

DOI
10.1103/w561-h3z5

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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 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.
We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.
Quantum uncertainty limits how strongly Wigner functions can concentrate in phase space. It is commonly conjectured that the vacuum Wigner function continuously majorizes that of any Wigner-positive state. However, by constructing Wigner-positive states whose localized quantum coherence lowers their Wigner entropy below the vacuum value, we show that this conjecture is incorrect. We restore vacuum majorization at finite resolution by integrating single-mode Wigner functions over concentric radial energy shells, obtaining probability vectors majorized by that of the vacuum for all Wigner-positive states and for Wigner-negative states with non-negative shell weights. Anchoring these shells to a fixed oscillator frame breaks affine symplectic invariance, rendering this relation strict for every non-vacuum Gaussian pure state. For Wigner-negative states, the persistence of negative shell weights defines a radial negativity scale with Airy-edge semiclassical asymptotics for highly excited Fock states. Operationally, we combine the majorization relation with partial transposition to obtain a two-mode entanglement criterion directly evaluable from radially binned joint homodyne outcomes without state reconstruction.
We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set---i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone---the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states---e.g., photon-added, photon-subtracted, cubic-phase, and cat states---and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to subuniversal and universal quantum information processing over continuous variables.
The use of quantum information in technology promises to supersede the so-called classical devices used nowadays. Understanding what features are inherently non-classical is crucial for reaching better-than-classical performance. This thesis focuses on two nonclassical behaviours: quantum contextuality and Wigner negativity. The former is a notion superseding nonlocality that can be exhibited by quantum systems. To date, it has mostly been studied in discrete-variable scenarios. In those scenarios, contextuality has been shown to be necessary and sufficient for advantages in some cases. On the other hand, negativity of the Wigner function is another unsettling non-classical feature of quantum states that originates from phase-space formulation in continuous-variable quantum optics. Continuous-variable scenarios offer promising candidates for implementing quantum computations. Wigner negativity is known to be a necessary resource for quantum speedup with continuous variables. However contextuality has been little understood and studied in continuous-variable scenarios. We first set out a robust framework for properly treating contextuality in continuous variables. We also quantify contextuality in such scenarios by using tools from infinite-dimensional optimisation theory. Building upon this, we show that Wigner negativity is equivalent to contextuality in continuous variables with respect to Pauli measurements thus establishing a continuous-variable analogue of a celebrated result by Howard et al. We then introduce experimentally-friendly witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, using again tools from infinite-dimensional optimisation theory. We further extend the range of previously known discrete-variable results linking contextuality and advantage into a new territory of information retrieval.
The question of how to characterize disorder arises in a wide variety of settings, including information theory, economics, and quantum thermodynamics. The theory of majorization provides an elegant answer, playing a central role in these fields. However, the existing majorization framework is inapplicable to functions that both take negative values and are defined on infinite spaces. Yet such functions, in the form of quasiprobability distributions, are ubiquitous in fields such as quantum optics, signal analysis, and bosonic quantum computation. Here we develop a notion of majorization that is applicable to such functions, proving that it admits four equivalent characterizations that naturally reduce to the finite case, thereby generalizing a seminal theorem by Hardy, Littlewood, and Pólya. Moreover, we extend this equivalence to the setting where majorization is considered relative to an arbitrary positive distribution. We give several applications of our results in the context of quantum resource theories. These include deriving families of resource monotones and constraining quantum state conversions. We analytically and numerically study examples using the Wigner and Husimi functions, which feature prominently in quantum optics. Our results provide a comprehensive majorization framework for assessing the disorder of integrable functions over infinite measure spaces. Quasiprobability distributions, crucial in quantum optics and computation, challenge existing majorization frameworks due to their negative values and infinite domains. Here, the authors establish a novel majorization concept for such functions, offering four equivalent characterizations and extending applications to quantum resource theories, enhancing our understanding of quantum state transformations.
The Wigner entropy conjecture posits that pure Gaussian states minimize the Shannon entropy of non-negative Wigner functions, known as the Wigner entropy. We show that mixing a specific pure Wigner-negative state with another suitably chosen quantum state can restore Wigner positivity while retaining a Wigner entropy that is slightly - but strictly - below the vacuum entropy. As a consequence, we disprove the Wigner entropy conjecture, as well as the stronger Wigner majorization conjecture, by deriving simple analytical counterexamples with violations of the entropy bound no larger than 10−310^{-3}10−3. Such states with sub-vacuum Wigner entropy can be found arbitrarily close to the vacuum and can also exhibit sub-vacuum Wigner-Rényi ααα-entropy for 0<α<20<α<20<α<2. We identify the physical mechanism behind the existence of such states, which arises from a subtle interplay between the uncertainty principle and non-Gaussianity. The key idea of this paper was developed with the help of AI tools, building on the characterization of extreme non-negative Wigner functions.