We construct physical quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value $1+\lnπ$. Besides an explicit finite-energy counterexample, we obtain rank-two finite-Fock-support families with analytic positivity and entropy bounds. For fixed Fock level $n$ and coherence fraction $0\leλ<1$, the entropy difference satisfies $h(W)-(1+\lnπ)=(2n-2^nλ^2)t^2+O_{n,λ}(t^4)$, yielding finite-support counterexamples for every $n\ge3$ above an explicit coherence threshold. We also show that the absence of a negative quadratic term does not preclude entropy descent: a fully coherent vacuum--one-photon core with a vanishing positive thermal repair gives $h(W)-(1+\lnπ)=-4t^6/3+o(t^6)$. For the vacuum--three-photon construction, we determine the logarithmic asymptotic of the minimum fixed-thermal mixing weight required for Wigner nonnegativity. Finally, optimizing over all one-mode Wigner-nonnegative states with mean photon number at most $E$, we prove that the maximal entropy deficit has the sharp scale $E^γ/[\ln(1/E)]^β$, where $γ\simeq0.7412033679$ and $β\simeq0.5861054961$. Thus the vacuum entropy is recovered as $E\to0$, but the optimal deficit decays much more slowly than any universal linear correction in the mean energy.
We construct physical quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value 1+lnπ. Besides an explicit finite-energy counterexample, we obtain rank-two finite-Fock-support families with analytic positivity and entropy bounds. For fixed Fock level n and coherence fraction 0≤λ<1, the entropy difference satisfies h(W)−(1+lnπ)=(2n−2nλ2)t2+On,λ(t4), yielding finite-support counterexamples for every n≥3 above an explicit coherence threshold. We also show that the absence of a negative quadratic term does not preclude entropy descent: a fully coherent vacuum--one-photon core with a vanishing positive thermal repair gives h(W)−(1+lnπ)=−4t6/3+o(t6). For the vacuum--three-photon construction, we determine the logarithmic asymptotic of the minimum fixed-thermal mixing weight required for Wigner nonnegativity. Finally, optimizing over all one-mode Wigner-nonnegative states with mean photon number at most E, we prove that the maximal entropy deficit has the sharp scale Eγ/[ln(1/E)]β, where γ≃0.7412033679 and β≃0.5861054961. Thus the vacuum entropy is recovered as E→0, but the optimal deficit decays much more slowly than any universal linear correction in the mean energy.
Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of 2 × 10−21 preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale Eγ/[ln(1/E)]β, with γ ≃ 0.7412 and β ≃ 0.5861. Because γ < 1, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound. Version history. This is an expanded revision of arXiv:2609.13312v1, submitted on 10 September 2026 under the title "Physical Counterexamples to the Wigner Shannon Entropy Conjecture". That version already presented physical Wigner-positive counterexamples, finite-support families, a sixth-order one-photon-core construction, minimum thermal-repair asymptotics and the sharp one-mode low-energy deficit scale. The present revision develops the dimensional and loss-dependent consequences, supplies explicit rational entropy-sign certificates and a smaller thermal-repair example, and connects these results to the companion studies cited in the manuscript.
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.
Loss can restore the vacuum entropy bound without erasing non-Gaussian structure. This paper determines when an entropy statement justifies a coherent-state approximation, when non-Gaussian structure survives optical loss, and what mode and source-request budgets are needed to use that structure. The results give quantum-optics researchers separate criteria for entropy recovery, state approximation, Gaussian concentration and conversion to a specified output. Version 1.5 — revised 10 October 2026. The 96-page PDF contains the main article, Methods, Supplementary Information and references. This revision connects the exact loss boundary to a mode-uniform near-equality certificate, finite-data validation, a conversion-cost lower bound for the same thermally repaired source, and an exact equal-entropy comparison of sources with different conversion-cost exponents. The central figures are updated accordingly. 1. Near-vacuum states can have an unbounded entropy deficit Write νm for the m-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ), using natural logarithms; H1 is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr(Nmρm)→0,21∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞. Here Fm is the trace-norm-closed convex hull of Gaussian states and dFm uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives −Em≤Δ1,m(L1/2⊗mρm)≤0, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η and any fixed positive total input-energy budget E, the sharp threshold is ηc(α)={1/2,α/2,0<α≤1,1<α<2. With Pm,E={ρ:Wρ≥0,Tr(Nmρ)≤E}, supm≥1supρ∈Pm,EΔα,m(Lη⊗mρ)={0,+∞,η≤ηc(α),η>ηc(α). The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For α≥2, established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input ρ, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−m(1+lnπ)≥0. The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields infa∥ρ−∣a⟩⟨a∣∥12≤ln264s(ρ). The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most ϵ is guaranteed by s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥16ln2dFm(L1/2⊗mρ)2. Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation c/m has total input energy tending to 3c. Along the recovery boundary, its attenuated state obeys liminfm→∞dFm(ρm,ηc)≥cηc3exp{−c[1−(1−ηc)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>0. Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target τ∗=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25 within half trace distance 10−6. Let Cm be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound logCm≥1281(m/c)1/3−logm−O(1). This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become h(WωA)=h(WωB)=2(1+lnπ)+s,logCA(s)=Θ(s−1/3),logCB(s)=Θ(s−1/2). The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy EA,EB∼s/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3) and differ by sA(E)−sB(E)=15E3+O(E4). Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound Tr[N1N>Kσa]≤τK, trusted half loss and all-mode on/off detection with efficiency at least ξmin, the click probability q gives s(σ)≤AKq+τK,AK=1−(1−ξmin/2)KK. A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses K=4, τK=0, ξ=0.8 and N=3,517,336 independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most 0.05 with false-acceptance probability at most 0.001 and missed acceptance at most 0.001 on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at E=1, α=1.9, η=0.9505 and m=105, the certified bound Δα<0.06942 rules out a target deficit of 0.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97, negative deficit near η=0.85, and a Gaussian-hull witness above 4.4×10−3 at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put α=2−δ and L=ln(1/δ). For the maximum deficit Dm,E at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ), where Th is the first mode number allowing a fixed positive deficit h. Beyond a sufficiently large fixed multiple of L/δ, Dm,E(2−δ)=ΘE(mδ). The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds: h(Wρ)≥mln(2π)+εm with εm>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k denoting the ordering-weighted norm of photon-number-difference block k, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−η1−r. For vacuum–n-photon coherence and 1<α<2, C1−α/2(Lη∣n⟩⟨0∣)=(α2η)n. This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.
Quantum uncertainty limits how strongly Wigner functions can concentrate in phase space. It is commonly conjectured that the vacuum Wigner function continuously majorizes that of any Wigner-positive state. However, by constructing Wigner-positive states whose localized quantum coherence lowers their Wigner entropy below the vacuum value, we show that this conjecture is incorrect. We restore vacuum majorization at finite resolution by integrating single-mode Wigner functions over concentric radial energy shells, obtaining probability vectors majorized by that of the vacuum for all Wigner-positive states and for Wigner-negative states with non-negative shell weights. Anchoring these shells to a fixed oscillator frame breaks affine symplectic invariance, rendering this relation strict for every non-vacuum Gaussian pure state. For Wigner-negative states, the persistence of negative shell weights defines a radial negativity scale with Airy-edge semiclassical asymptotics for highly excited Fock states. Operationally, we combine the majorization relation with partial transposition to obtain a two-mode entanglement criterion directly evaluable from radially binned joint homodyne outcomes without state reconstruction.
How much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with [Q,P]=i, mean photon number at most E>0, and a known parity centre. The full class WE consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass GE consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let PL project onto photon numbers at most L, and define the cutoff error by eL(ρ)=21∥ρ−PLρPL∥1. This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard OE(L−1/2) guarantee. At the exact Wigner endpoint, the article proves supρ∈WEeL(ρ)≤CE(L+1)−3/4,supρ∈GEeL(ρ)≤L+14E,L≥2. The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family τd∈WE has exact energy E, covariance (E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞. Thus the Gaussian-mixture L−1 law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let nX(δ) be the minimum worst-case number of copies for trace-distance accuracy δ over source class X. As δ↓0, the bounds are lognGE(δ)=Θ(δ−1),cδ−6/5≤lognWE(δ)≤Cδ−4/3. These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.
We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value 1+lnπ. Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound πW≤1, we construct an explicit hierarchy of lower bounds Bn on S[W] by combining a truncated series lower bound for −lnx with moment identities of the Wigner function. This yields closed-form sufficient conditions, expressed in terms of the state purity μ:=Tr(ρ2)=2π∫dqdpW(q,p)2, ensuring S[W]≥1+lnπ. In particular, we first prove that all physical Wigner-non-negative states with μ≤4−23 satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, whose limiting sufficient condition is μ≤2/e. Finally, we show that the threshold 2/e is sharp under the relaxed constraints considered here, thereby identifying the need for additional quantum-realizability information in the remaining high-purity regime.