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arXiv:2611.04009

Tight Quasiprobabilistic Entropic Uncertainty Relations for Discrete Quantum Systems

Jonathan J. Thio, Stephan De Bièvre, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur

quant-phSubmitted Nov 2, 2026• Announced Nov 4, 2026
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Abstract

It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.

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It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.
Wehrl entropy is an entropy associated with the Husimi quasiprobability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Bia\l{}ynicki-Birula--Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
Abstract We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the Rényi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb’s inequalities.
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.
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.
Abstract The Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in spin systems. We investigate the statistical interpretation of the Husimi function and the Wehrl entropy, taking the system of N spin-1/2 particles as an example. Due to the completeness of coherent states, the Husimi function and Wehrl entropy can be explained via the positive operator-valued measurement (POVM) theory, although the coherent states are not a set of orthonormal basis. Here, with the help of the Bayes’ theorem, we provide an alternative probabilistic interpretation for the Husimi function and the Wehrl entropy. This interpretation is based on direct measurements of the system, and thus does not require the introduction of an ancillary system as in the POVM theory. Moreover, under this interpretation the classical correspondences of the Husimi function and the Wehrl entropy are just phase-space probability distribution function of N classical tops, and its associated entropy, respectively. Therefore, this explanation contributes to a better understanding of the relationship between the Husimi function, Wehrl entropy, and classical-quantum correspondence. The generalization of this statistical interpretation to continuous-variable systems is also discussed.