The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.

arXiv:quant-ph/0406238Optics and SpectroscopySep 1, 20053 citations

Simulation of quantum entangled states by classical probabilistic models

Andrey Akhmeteli

We propose a classical probabilistic plasma-like model of bipartite quantum states, including entangled states. In a finite-dimensional approximation, a bipartite density operator can be written as a finite signed sum of product-state projectors, $\rho=\sum_k q_k |a_k,b_k\rangle\langle a_k,b_k|,$ where an entangled state necessarily requires negative coefficients in every such decomposition. By linearity and factorization of the Wigner transform on product states, the corresponding bipartite Wigner function has the form $W(\mathbf{x}_1,\mathbf{p}_1,\mathbf{x}_2,\mathbf{p}_2)=\sum_{k=1}^{K}q_kW_{1k}(\mathbf{x}_1,\mathbf{p}_1)W_{2k}(\mathbf{x}_2,\mathbf{p}_2).$ Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret $|q_k|$ as the expected value of the number of copies of the $k$ -th plasma pair in a realization of the ensemble, while the sign of $q_k$ distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. Averaging over the ensemble of plasma copies reproduces the bipartite Wigner function and therefore the corresponding quantum state.Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret the coefficients of the sum as the expected values of the number of copies of the bipartite plasmas in a realization of the ensemble, while the sign of the coefficients distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. The ensemble-averaged signed phase-space distribution reproduces the bipartite Wigner function and therefore the corresponding quantum state.We propose a classical probabilistic plasma-like model of bipartite quantum states, including entangled states. In a finite-dimensional approximation, a bipartite density operator can be written as a finite signed sum of product-state projectors,$$ \rho=\sum_k q_k |a_k,b_k\rangle\langle a_k,b_k|, $$where an entangled state necessarily requires negative coefficients in every such decomposition. By linearity and factorization of the Wigner transform on product states, the corresponding bipartite Wigner function has the form$$ W(\mathbf{x}_1,\mathbf{p}_1,\mathbf{x}_2,\mathbf{p}_2)=\sum_{k=1}^{K}q_kW_{1k}(\mathbf{x}_1,\mathbf{p}_1)W_{2k}(\mathbf{x}_2,\mathbf{p}_2). $$Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret $|q_k|$ as the expected value of the number of copies of the $k$-th plasma pair in a realization of the ensemble, while the sign of $q_k$ distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. Averaging over the ensemble of plasma copies reproduces the bipartite Wigner function and therefore the corresponding quantum state.Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret the coefficients of the sum as the expected values of the number of copies of the bipartite plasmas in a realization of the ensemble, while the sign of the coefficients distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. The ensemble-averaged signed phase-space distribution reproduces the bipartite Wigner function and therefore the corresponding quantum state.

Sep 15, 20260 citations

The Fourier-Wigner Transform

Springer eBooksMar 31, 20060 citations

Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics

Aritra Ghosh, Adam Miranowicz, Miloslav Znojil

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|ψ_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$ . One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $Ω: H \to \mathfrak{h}$ . Naturally, the well known ambiguity of these $H-$ dependent Dyson-map transformations implies also an ambiguity of the physical, $Ω-$ dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$ dependent Dyson maps $Ω=Ω(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$ .In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates $|ψ_n \rangle$ of a Hamiltonian operator $H$ with real spectrum which is non-Hermitian, $H \neq H^\dagger$. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of $H$ into its isospectral Hermitian ``twin" $\mathfrak{h}= \mathfrak{h}^\dagger$ via one of the so-called Dyson maps $Ω: H \to \mathfrak{h}$. Naturally, the well known ambiguity of these $H-$dependent Dyson-map transformations implies also an ambiguity of the physical, $Ω-$dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible $H-$dependent Dyson maps $Ω=Ω(H)$ is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected $H$.

quant-pharXiv:2511.20404Nov 26, 20250 citations

The Wigner distribution function

GERALD J. IAFRATE, H. L. Grubin, David K. Ferry

Physics Letters AJan 1, 198227 citations

\rho=\sum_k q_k |a_k,b_k\rangle\langle a_k,b_k|,

where an entangled state necessarily requires negative coefficients in every such decomposition. By linearity and factorization of the Wigner transform on product states, the corresponding bipartite Wigner function has the form

W(\mathbf{x}_1,\mathbf{p}_1,\mathbf{x}_2,\mathbf{p}_2)=\sum_{k=1}^{K}q_kW_{1k}(\mathbf{x}_1,\mathbf{p}_1)W_{2k}(\mathbf{x}_2,\mathbf{p}_2).

Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret

|q_k|

as the expected value of the number of copies of the

k

-th plasma pair in a realization of the ensemble, while the sign of

q_k

distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. Averaging over the ensemble of plasma copies reproduces the bipartite Wigner function and therefore the corresponding quantum state.Building on an earlier plasma-like model of one-particle Wigner functions in terms of classical particles and antiparticles, we interpret the coefficients of the sum as the expected values of the number of copies of the bipartite plasmas in a realization of the ensemble, while the sign of the coefficients distinguishes particle- and antiparticle-type contributions. Negative quasiprobabilities therefore enter the construction at two distinct levels: through possible negativity of the one-particle Wigner functions and through negative coefficients in the product-state decomposition associated with entanglement. The ensemble-averaged signed phase-space distribution reproduces the bipartite Wigner function and therefore the corresponding quantum state.

In the framework of the so-called quasi-Hermitian quantum mechanics of stationary unitary systems, bound states are usually constructed as eigenstates

|ψ_n \rangle

of a Hamiltonian operator

H

with real spectrum which is non-Hermitian,

H \neq H^\dagger

. One of the ways of the standard probabilistic interpretation of such systems consists in a transformation of

H

into its isospectral Hermitian ``twin"

\mathfrak{h}= \mathfrak{h}^\dagger

via one of the so-called Dyson maps

Ω: H \to \mathfrak{h}

. Naturally, the well known ambiguity of these

H-

dependent Dyson-map transformations implies also an ambiguity of the physical,

Ω-

dependent probabilistic and experimental interpretation of the system in question. In the present paper, an exhaustive classification of all of the eligible

H-

dependent Dyson maps

Ω=Ω(H)

is provided, implying also a systematic framework for a specification of all of the possible probabilistic interpretations of the quantum system characterized by a preselected

H