Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Conference Series

Gaussianity bounds for quantum mixed states with a positive Wigner function

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

Published Nov 1, 2010• 7 citations
Publisher

Abstract

According to Hudson's theorem, the only quantum pure states with a positive Wigner function are Gaussian states. We summarize and compare some recent attempts at extending this theorem to the space of quantum mixed states, and complement them with new results obtained from numerical observations. Specifically, we look for upper bounds on the admissible non-Gaussianity of mixed states with a positive Wigner function. © 2010 IOP Publishing Ltd.

DOI
10.1088/1742-6596/254/1/012011

Related papers

Extending Hudson’s theorem to mixed quantum states

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions.

arXiv:0808.2501•Physical Review A•Jun 3, 2009•55 citations

Towards an extension of the Hudson's theorem to mixed quantum states

Aikaterini Mandilara, Evgueni Karpov, Nicolas J. Cerf

In this work, we make a step towards the extension of the Hudson's theorem to mixed states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity, the purity of the states and the purity of Gaussian states determined by the same covariance matrix as of these states. Even though the bounds are not tight, they permit for a preliminary visualization of the space of states with positive Wigner functions and for the derivation of a bound on the purity of a state with strictly positive Wigner function and a given covariance matrix.

•Aug 18, 2008•0 citations

Hudson’s theorem for finite-dimensional quantum systems

D. Gross

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

arXiv:quant-ph/0602001•Journal of Mathematical Physics•Dec 1, 2006•366 citations

Hudson's theorem fails for the SU(1,1) discrete series

Chon-Fai Kam

Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of $SU(1,1)$, realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of $24.93^\circ$ at Bargmann index $k=1$, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by $\arctan(1/\sqrt{2k})$ and leaves the threshold itself fixed at intermediate hyperbolic distance.

quant-pharXiv:2609.07242•Sep 8, 2026•0 citations

Mixed states with positive Wigner functions

T. Bröcker, Reinhard F. Werner

The Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low-dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.

Journal of Mathematical Physics•Jan 1, 1995•70 citations

Wigner-positive quantum states can have lower entropy than the vacuum

Zacharie Van Herstraeten, Nicolas J. Cerf, Ulysse Chabaud

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 $10^{-3}$. Such states with sub-vacuum Wigner entropy can be found arbitrarily close to the vacuum and can also exhibit sub-vacuum Wigner-Rényi $α$-entropy for $0<α<2$. 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.

quant-pharXiv:2609.24670•Sep 22, 2026•0 citations
According to Hudson's theorem, the only quantum pure states with a positive Wigner function are Gaussian states. We summarize and compare some recent attempts at extending this theorem to the space of quantum mixed states, and complement them with new results obtained from numerical observations. Specifically, we look for upper bounds on the admissible non-Gaussianity of mixed states with a positive Wigner function. © 2010 IOP Publishing Ltd.
According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions.
In this work, we make a step towards the extension of the Hudson's theorem to mixed states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity, the purity of the states and the purity of Gaussian states determined by the same covariance matrix as of these states. Even though the bounds are not tight, they permit for a preliminary visualization of the space of states with positive Wigner functions and for the derivation of a bound on the purity of a state with strictly positive Wigner function and a given covariance matrix.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.
Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of SU(1,1)SU(1,1)SU(1,1), realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of 24.93∘24.93^\circ24.93∘ at Bargmann index k=1k=1k=1, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by arctan⁡(1/2k)\arctan(1/\sqrt{2k})arctan(1/2k​) and leaves the threshold itself fixed at intermediate hyperbolic distance.
The Wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent, according to a result of Hudson. The characterization of mixed states with a positive Wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters (i.e., functions with given ‘‘Wigner spectrum’’ in the sense of Narcowich). If a state is a convex combination of coherent states, it has the property that the Wigner spectrum contains the unit interval, which in turn implies that the Wigner function is positive. It is shown by explicit examples that the converses of both implications are false. The examples are taken from a low-dimensional section of the state space, in which all Wigner spectra can be computed. In this set counterexamples to a conjecture by Narcowich concerning the Wigner spectrum of products are also found, as well as a state whose Wigner spectrum is a convergent sequence of discrete points.
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.