Frieden and Soffer (Phys. Rev. A 74, 052108 (2006)) derived, from their Extreme Physical Information principle applied to a gedanken phase-hologram position measurement, a phase-space distribution p(x,μ)=g(μ∣x)⊗pW(x,μ)p(x,\mu)=g(\mu|x)\otimes p_W(x,\mu): a one-dimensional Gaussian convolution of the Wigner function with an instrument function of grain width σ\sigma, along either the momentum or the position direction. Nonnegativity is proven only in the σ→∞\sigma\to\infty limit where the distribution degenerates to a marginal, and the paper states that finding the minimum grain size achieving positivity for a given state "has yet to be done". The open problem: characterize the states that admit a finite one-dimensional grain (including whether σmin⁡(ψ)=+∞\sigma_{\min}(\psi)=+\infty occurs), and compute the minimal grain as a function of the state.

Background

The Wigner function is the standard quasi-probability representation of a quantum state on classical phase space: for a pure state with wavefunction ψ(x)\psi(x) it is pW(x,μ)=12π∫dz ψ∗(x−ℏz/2) ψ(x+ℏz/2) e−izμp_W(x,\mu)=\frac{1}{2\pi}\int dz\,\psi^{*}(x-\hbar z/2)\,\psi(x+\hbar z/2)\,e^{-iz\mu}, with μ\mu the momentum variable, a real function whose position and momentum marginals reproduce the correct probability densities but which generally takes negative values. Those negative regions are the standard signature of nonclassicality (their volume is a widely used nonclassicality measure, Kenfack and Życzkowski 2004), and they obstruct interpreting pWp_W itself as a joint probability density for simultaneously measured position and momentum.

The classical remedy is coarse graining: convolving pWp_W with a smoothing kernel. A classical result (Cartwright 1976; the ss-ordering formalism of Cahill and Glauber 1969 — a one-parameter family of Gaussian-smoothed quasiprobability distributions running from the raw Wigner function (s=0s=0) to the fully smoothed Husimi QQ function (s=−1s=-1)) — shows that convolution with a two-dimensional Gaussian whose variances saturate the Heisenberg uncertainty bound — i.e. with the Wigner function of a minimum-uncertainty wavepacket, producing the Husimi QQ distribution — yields a function nonnegative for every state; this is the sense in which the minimum coarse graining that suffices for all states is fixed by the uncertainty principle. That guarantee is two-dimensional and state-independent. The state-dependent question — how much smoothing a particular state needs, and what happens when the kernel is constrained to act in only one phase-space dimension — has no general answer: Keating, Steege, and Pattanayak (2008) coarse-grain Wigner functions with isotropic two-dimensional Gaussians and leave open what fixes the physical degree of coarse graining for a given state, and Diósi (2004) proved finite-time positivity of the Wigner function itself under one specific open-system decoherence dynamics.

Frieden's Extreme Physical Information (EPI) program treats a measurement as a flow of Fisher information (a functional measuring how sensitively an acquired probability law depends on an unknown parameter) from source to observer, and derives physical laws by extremizing the information loss I−JI-J (acquired minus source information) subject to the unitarity of the underlying amplitude-space transformation. Frieden and Soffer (2006) applied EPI to a gedanken measurement in which a particle in state ψ(x)\psi(x) illuminates a phase hologram of amplitude Ψ(x,z)\Psi(x,z) and then undergoes a single position measurement. The EPI solution is not the raw Wigner function but a smoothed phase-space law p(x,μ)=g(μ∣x)⊗pW(x,μ)≡∫dμ′ g(μ−μ′∣x) pW(x,μ′),p(x,\mu)=g(\mu|x)\otimes p_W(x,\mu)\equiv\int d\mu'\,g(\mu-\mu'|x)\,p_W(x,\mu'), a one-dimensional convolution (Eq. (19) of the source) along momentum, where the instrument function g(μ∣x)=F.T.(∣Z∣)g(\mu|x)=\mathrm{F.T.}(|Z|) is the Fourier transform of the modulus of the hologram spectrum Z(x,z)Z(x,z); carrying out the variational steps in the conjugate representation instead yields a one-dimensional convolution along xx (Eq. (26)). For a Gaussian instrument function of width σ\sigma (the "grain"), the smoothed distribution is the Wigner integrand damped by e−σ2z2/2e^{-\sigma^2 z^2/2} in its conjugate variable (cf. Eq. (32) of the source). Unlike the ad hoc two-dimensional postulate, the smoothing here is derived from the physics of the specific measurement.

What is and is not proven: the source establishes nonnegativity of p(x,μ)p(x,\mu) only in the limit σ→∞\sigma\to\infty (Eqs. (33), (35)), where the Gaussian damping suppresses all z≠0z\neq 0 contributions and the law degenerates to the position (respectively momentum) marginal — positive but physically trivial. For finite σ\sigma no positivity statement is given, and Sec. VI.A records verbatim: "A minimum necessary grain size σ\sigma for achieving positivity for a given state ψ(x)\psi(x) could be found in this way. This has yet to be done." The derivation also carries two acknowledged approximations: the variational rotation was performed in probability-density-function (PDF)/Fourier space rather than strictly in amplitude space, and the zero-source-information ansatz J=0J=0 is self-consistent only for wavefunctions with symmetric or periodic boundary behavior — the class where the consistency condition I=0I=0 holds — leaving other states uncharacterized. The hologram measurement is a gedanken construction, and the instrument function was never tied to realistic detector physics. The open problem stated by the source is the finite-grain positivity question itself: determine for which states a finite grain suffices for positivity, and compute the minimal such grain.

Problem Statement

The problem is the quantitative completion of the Frieden–Soffer measurement-derived one-dimensional coarse graining of the Wigner distribution, with the following components under one common objective — a state-resolved, measurement-grounded account of how much one-dimensional coarse graining produces a legitimate joint phase-space probability:

(1) Existence. For a pure state ψ(x)\psi(x), consider the smoothed law pσ(x,μ)  ∝  ∫dz  e−σ2z2/2 ψ∗(x−ℏz/2) ψ(x+ℏz/2) e−izμ,p_\sigma(x,\mu)\;\propto\;\int dz\; e^{-\sigma^2 z^2/2}\,\psi^{*}(x-\hbar z/2)\,\psi(x+\hbar z/2)\,e^{-iz\mu}, the Wigner integrand Gaussian-damped in its conjugate variable — equivalently pσ=gσ(μ∣x)⊗pWp_\sigma=g_\sigma(\mu|x)\otimes p_W, Eq. (19) of the source, smoothing along μ\mu — together with the position-direction analogue (Eq. (26) of the source, smoothing along xx), pσ(x,μ)  ∝  ∫dk  e−σ2k2/2 ϕ∗(μ+k) ϕ(μ−k) e−2ikx/ℏ,p_\sigma(x,\mu)\;\propto\;\int dk\; e^{-\sigma^2 k^2/2}\,\phi^{*}(\mu+k)\,\phi(\mu-k)\,e^{-2ikx/\hbar}, where ϕ(μ)\phi(\mu) is the momentum-space wavefunction (the Fourier transform of ψ\psi; cf. Eq. (34) of the source). Determine, in each of the two one-dimensional variants separately, which states admit a finite grain σ\sigma such that pσ(x,μ)≥0p_\sigma(x,\mu)\ge 0 for all (x,μ)(x,\mu), and decide whether states exist for which no finite grain in a given direction suffices, i.e. whether σmin⁡(ψ)=+∞\sigma_{\min}(\psi)=+\infty occurs.

(2) Minimal grain. For states admitting a finite grain, compute the minimal grain σmin⁡(ψ)\sigma_{\min}(\psi) in each direction, as the source itself proposes to obtain from the small-zz expansion of ψ∗(x−ℏz/2)ψ(x+ℏz/2)\psi^{*}(x-\hbar z/2)\psi(x+\hbar z/2).

Together, (1)–(2) constitute the source's own stated open task, posed "for a given state ψ(x)\psi(x)"; the derivation's two acknowledged approximations and the gedanken character of the hologram measurement are scope caveats of the derivation, not additional components of the problem.

References

  1. B. Roy Frieden and Bernard H. Soffer, "Information-theoretic significance of the Wigner distribution", Physical Review A 74, 052108 (2006). DOI: 10.1103/PhysRevA.74.052108; arXiv:quant-ph/0609157, https://arxiv.org/abs/quant-ph/0609157
  2. N. D. Cartwright, "A non-negative Wigner-type distribution", Physica A 83, 210 (1976). DOI: 10.1016/0378-4371(76)90145-X, https://doi.org/10.1016/0378-4371(76)90145-X
  3. K. E. Cahill and R. J. Glauber, "Density Operators and Quasiprobability Distributions", Physical Review 177, 1882 (1969). DOI: 10.1103/PhysRev.177.1882, https://doi.org/10.1103/PhysRev.177.1882
  4. Anatole Kenfack and Karol Życzkowski, "Negativity of the Wigner function as an indicator of non-classicality", Journal of Optics B: Quantum and Semiclassical Optics 6, 396 (2004). DOI: 10.1088/1464-4266/6/10/003; arXiv:quant-ph/0406015, https://arxiv.org/abs/quant-ph/0406015
  5. Lajos Diósi, "Wigner Centennial: His Function, and Its Environmental Decoherence", Acta Physica Hungarica B 20, 29 (2004). arXiv:quant-ph/0212103, https://arxiv.org/abs/quant-ph/0212103
  6. Tyler E. Keating, Adam T. C. Steege, and Arjendu K. Pattanayak, "Negativity of the Coarse Grained Wigner Function as a Measure of Quantal Behavior" (2008). arXiv:0811.2924, https://arxiv.org/abs/0811.2924
  7. B. Nugmanov, N. Zunikov, and F. Ya. Khalili, "Robustness of negativity of the Wigner function to dissipation" (2022). arXiv:2201.03257, https://arxiv.org/abs/2201.03257
  8. Wojciech Broniowski and Enrique Ruiz Arriola, "Wigner and Husimi Partonic Distributions of the Pion in a Chiral Quark Model", Acta Physica Polonica B 54, 7-A4 (2023). DOI: 10.5506/aphyspolb.54.7-a4; arXiv:2307.14167, https://arxiv.org/abs/2307.14167
  9. Bouchra El Alaoui, A. Slaoui, Abderrahim Lakhfif, and Rachid Ahl Laamara, "Parity symmetry breaking of spin-j coherent state superpositions in Gaussian noise channel", Physica Scripta 100, 015113 (2024). DOI: 10.1088/1402-4896/ad9dbf, https://doi.org/10.1088/1402-4896/ad9dbf
  10. B. Roy Frieden and Bernard H. Soffer, "de Broglie's wave hypothesis from Fisher information", Physica A: Statistical Mechanics and its Applications 388, 1315 (2009). DOI: 10.1016/j.physa.2008.11.040, https://doi.org/10.1016/j.physa.2008.11.040
  11. B. Frieden, "Estimating a Repeatable Statistical Law by Requiring Its Stability During Observation", Entropy 17, 7453 (2015). DOI: 10.3390/e17117453, https://doi.org/10.3390/e17117453
  12. B. Roy Frieden and Robert A. Gatenby, "Derivation of Principle of Extreme Physical Information" (2014). arXiv:1406.3615, https://arxiv.org/abs/1406.3615
  13. A. V. Luzanov, "Spectral and entropic characterizations of Wigner functions: Applications to model vibrational systems", The Journal of Chemical Physics 129, 094103 (2008). DOI: 10.1063/1.2968607, https://doi.org/10.1063/1.2968607
  14. F. Pennini and A. Plastino, "Localization estimation and global vs. local information measures", Physics Letters A 365, 263 (2007). DOI: 10.1016/j.physleta.2007.01.025, https://doi.org/10.1016/j.physleta.2007.01.025