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

Theoretical and Mathematical Physics

From the Wigner function to the $$s$$-ordered phase-space distribution via a Gaussian noise channel

Yue Zhang, Shunlong Luo

Published Mar 1, 2022• 2 citations
Publisher

Abstract

Various phase-space distributions, from the celebrated Wigner function, to the Husimi $$Q$$ function and the Glauber–Sudarshan $$P$$ distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the $$s$$ -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter $$s$$ , we interpret the $$s$$ -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the $$s$$ -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the $$s$$ -ordered phase-space distribution. In this connection, the parameter $$-s/2$$ (rather than the original $$s$$ ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter $$s$$ , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the $$s$$ -ordered phase-space distribution is derived.

DOI
10.1134/s0040577922030126

Related papers

Characterizing maximally singular phase-space distributions

Jan Sperling

Phase-space distributions are widely applied in quantum optics to access the nonclassical features of radiations fields. In particular, the inability to interpret the Glauber-Sudarshan distribution in terms of a classical probability density is the fundamental benchmark for quantum light. However, this phase-space distribution cannot be directly reconstructed for arbitrary states, because of its singular behavior. In this work, we perform a characterization of the Glauber-Sudarshan representation in terms of distribution theory. We address important features of such distributions: (i) the maximal degree of their singularities is studied, (ii) the ambiguity of representation is shown, and (iii) their dual space for nonclassicality tests is specified. In this view, we reconsider the methods for regularizing the Glauber-Sudarshan distribution for verifying its nonclassicality. This treatment is supported with comprehensive examples and counterexamples.

arXiv:1605.04425•Physical Review A•Jul 7, 2016•35 citations

Relating the Glauber-Sudarshan, Wigner, and Husimi quasiprobability distributions operationally through the quantum-limited amplifier and attenuator channels

Tomasz Linowski, Łukasz Rudnicki

The Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions are indispensable tools in quantum optics. However, although mathematical relations between them are well established, not much is known about their operational connection. In this paper, we prove that a single composition of finite-strength quantum limited amplifier and attenuator channels, known for their noise-adding properties, turns the Glauber-Sudarshan distribution of any input operator into its Wigner distribution, and its Wigner distribution into its Husimi distribution. As we dissect, the considered process, which can be performed in a quantum optical laboratory with relative ease, may be interpreted as realizing a quantum-to-classical transition.

arXiv:2312.08216•Physical Review A•Feb 20, 2024•5 citations

Spin nonclassicality via variance

Yue Zhang, Shunlong Luo

Although variance, as one of the most fundamental and ubiquitous quantities in quantifying uncertainty, has been widely used in both classical and quantum physics, there are still new applications awaiting exploration. In this work, by interchanging the roles of the state variable and the observable variable, i.e., by formally regarding any state as an observable (which is rational because any state is a priori a Hermitian operator ) and considering the average variance of this state (now in the position of an observable ) in all spin coherent states, we introduce a quantifier of spin nonclassicality with respect to a resolution of identity induced by spin coherent states. This quantifier is easy to compute and it admits various operational interpretations, such as the purity deficit, the Tsallis 2-entropy deficit, and the squared norm deficit between the Wigner function and the Husimi function. We reveal several intuitive properties of this quantifier, connect it to the phase-space distribution uncertainty, and illustrate it with some prototypical examples. Various extensions are further indicated.

Theoretical and Mathematical Physics•Jul 1, 2021•4 citations

Understanding Squeezed States of Light Through Wigner's Phase-Space

Sibel Baskal, Marilyn E. Noz

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.

quant-pharXiv:2606.23102•Mathematics•Jun 23, 2026•1 citation

Gaussian versus Non-Gaussian Filtering of Phase-Insensitive Nonclassicality

B. Kühn, W. Vogel, Valérian Thiel +2 more

Measures of quantum properties are essential to understanding the fundamental differences between quantum and classical systems as well as quantifying resources for quantum technologies. Here, two broad classes of bosonic phase-space functions, which are filtered versions of the Glauber-Sudarshan P function, are compared with regard to their ability to uncover nonclassical effects of light through their negativities. Gaussian filtering of the P function yields the family of s-parametrized quasiprobabilities, while more powerful regularized nonclassicality quasiprobabilities are obtained by non-Gaussian filtering. A method is proposed to directly sample such phase-space functions for the restricted case of phase-independent quantum states from balanced homodyne measurements. This overcomes difficulties of previous approaches that manually append uniformly distributed optical phases to the measured quadrature data. We experimentally demonstrate this technique for heralded single- and two-photon states using balanced homodyne detection with varying efficiency. The s-parametrized quasiprobabilities, which can be directly sampled, are non-negative for detection efficiencies below 0.5. By contrast, we show that significant negativities of non-Gaussian filtered quasiprobabilities uncover nonclassical effects for arbitrarily low efficiencies.

arXiv:2010.02173•Physical Review Letters•Apr 30, 2021•8 citations

Theory of Quantum Phase Space: Foundations and Applications

Demin Huang, Biao Wu

This article provides a concise review of quantum phase space theory, beginning with its foundational principles and the properties of standard quantum quasi-probability distributions, specifically the Wigner, Husimi Q, and Glauber--Sudarshan P functions. We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring. A significant portion of this review highlights recent theoretical developments, particularly the quantum Wannier basis. This approach establishes a unitary mapping between the Hilbert space and a discretized phase space, yielding a genuine probability distribution in phase space and thereby providing a basis-dependent entropy for pure quantum states. Furthermore, we examine Bourgain's nonperiodic basis as a theoretical framework to circumvent the constraints imposed by the Balian--Low theorem. These developments provide practical tools for numerical studies based on the quantum Wannier basis, as well as conceptual benchmarks for understanding the localization limits of orthonormal phase-space representations.

quant-pharXiv:2606.02675•Jun 2, 2026•0 citations
From the Wigner function to the sss-ordered phase-space distribution via a Gaussian noise channel
Various phase-space distributions, from the celebrated Wigner function, to the Husimi QQQ function and the Glauber–Sudarshan PPP distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the sss -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter sss , we interpret the sss -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the sss -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the sss -ordered phase-space distribution. In this connection, the parameter −s/2-s/2−s/2 (rather than the original sss ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter sss , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the sss -ordered phase-space distribution is derived.
Phase-space distributions are widely applied in quantum optics to access the nonclassical features of radiations fields. In particular, the inability to interpret the Glauber-Sudarshan distribution in terms of a classical probability density is the fundamental benchmark for quantum light. However, this phase-space distribution cannot be directly reconstructed for arbitrary states, because of its singular behavior. In this work, we perform a characterization of the Glauber-Sudarshan representation in terms of distribution theory. We address important features of such distributions: (i) the maximal degree of their singularities is studied, (ii) the ambiguity of representation is shown, and (iii) their dual space for nonclassicality tests is specified. In this view, we reconsider the methods for regularizing the Glauber-Sudarshan distribution for verifying its nonclassicality. This treatment is supported with comprehensive examples and counterexamples.
The Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions are indispensable tools in quantum optics. However, although mathematical relations between them are well established, not much is known about their operational connection. In this paper, we prove that a single composition of finite-strength quantum limited amplifier and attenuator channels, known for their noise-adding properties, turns the Glauber-Sudarshan distribution of any input operator into its Wigner distribution, and its Wigner distribution into its Husimi distribution. As we dissect, the considered process, which can be performed in a quantum optical laboratory with relative ease, may be interpreted as realizing a quantum-to-classical transition.
Although variance, as one of the most fundamental and ubiquitous quantities in quantifying uncertainty, has been widely used in both classical and quantum physics, there are still new applications awaiting exploration. In this work, by interchanging the roles of the state variable and the observable variable, i.e., by formally regarding any state as an observable (which is rational because any state is a priori a Hermitian operator ) and considering the average variance of this state (now in the position of an observable ) in all spin coherent states, we introduce a quantifier of spin nonclassicality with respect to a resolution of identity induced by spin coherent states. This quantifier is easy to compute and it admits various operational interpretations, such as the purity deficit, the Tsallis 2-entropy deficit, and the squared norm deficit between the Wigner function and the Husimi function. We reveal several intuitive properties of this quantifier, connect it to the phase-space distribution uncertainty, and illustrate it with some prototypical examples. Various extensions are further indicated.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
Measures of quantum properties are essential to understanding the fundamental differences between quantum and classical systems as well as quantifying resources for quantum technologies. Here, two broad classes of bosonic phase-space functions, which are filtered versions of the Glauber-Sudarshan P function, are compared with regard to their ability to uncover nonclassical effects of light through their negativities. Gaussian filtering of the P function yields the family of s-parametrized quasiprobabilities, while more powerful regularized nonclassicality quasiprobabilities are obtained by non-Gaussian filtering. A method is proposed to directly sample such phase-space functions for the restricted case of phase-independent quantum states from balanced homodyne measurements. This overcomes difficulties of previous approaches that manually append uniformly distributed optical phases to the measured quadrature data. We experimentally demonstrate this technique for heralded single- and two-photon states using balanced homodyne detection with varying efficiency. The s-parametrized quasiprobabilities, which can be directly sampled, are non-negative for detection efficiencies below 0.5. By contrast, we show that significant negativities of non-Gaussian filtered quasiprobabilities uncover nonclassical effects for arbitrarily low efficiencies.
This article provides a concise review of quantum phase space theory, beginning with its foundational principles and the properties of standard quantum quasi-probability distributions, specifically the Wigner, Husimi Q, and Glauber--Sudarshan P functions. We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring. A significant portion of this review highlights recent theoretical developments, particularly the quantum Wannier basis. This approach establishes a unitary mapping between the Hilbert space and a discretized phase space, yielding a genuine probability distribution in phase space and thereby providing a basis-dependent entropy for pure quantum states. Furthermore, we examine Bourgain's nonperiodic basis as a theoretical framework to circumvent the constraints imposed by the Balian--Low theorem. These developments provide practical tools for numerical studies based on the quantum Wannier basis, as well as conceptual benchmarks for understanding the localization limits of orthonormal phase-space representations.