Finite models of quantum light that preserve Wigner positivity and the energy budget

Zixuan He

Published Sep 28, 2026• 0 citations

Abstract

The state space of quantum light contains infinitely many photon-number levels, whereas numerical models usually retain only finitely many. Direct truncation can introduce Wigner negativity into an initially Wigner-positive state. We prove that every Wigner-positive state of finitely many modes admits physical approximations with finite photon-number support under the same mean-energy cap. For $m$ modes, mean total photon number at most $E$, and trace-norm error at most $\varepsilon\in(0,1]$, a sufficient cutoff is \[L=O\!\left((E+1)\varepsilon^{-2}\left[m+\ln\frac{E+2}{\varepsilon}\right]\right).\] The construction represents Wigner positivity by operator positivity in an enlarged system, where local photon-number projections preserve it. It yields finite semidefinite programmes whose optima $q_L(A,E)$, for any bounded Hermitian observable $A$, satisfy \[0\le\Omega_E(A)-q_L(A,E)\le\varepsilon\inf_{c\in\mathbb R}\|A-cI\|,\] where $\Omega_E(A)$ is the optimum over all Wigner-positive states under the same energy cap. At fixed mode number and energy, a continuity bound also provides finite approximations to Wigner entropy minima and finite witnesses for every strict entropy deficit. These bounds determine how much of the infinite state space a calculation must retain to certify its conclusions.

DOI: 10.5281/zenodo.23022267

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Finite models of quantum light that preserve Wigner positivity and the energy budget

Zixuan He

Published Sep 28, 2026• 0 citations

Publisher

Abstract

The state space of quantum light contains infinitely many photon-number levels, whereas numerical models usually retain only finitely many. Direct truncation can introduce Wigner negativity into an initially Wigner-positive state. We prove that every Wigner-positive state of finitely many modes admits physical approximations with finite photon-number support under the same mean-energy cap. For (m) modes, mean total photon number at most (E), and trace-norm error at most (\varepsilon\in(0,1]), a sufficient cutoff is [L=O!\left((E+1)\varepsilon^{-2}\left[m+\ln\frac{E+2}{\varepsilon}\right]\right).] The construction represents Wigner positivity by operator positivity in an enlarged system, where local photon-number projections preserve it. It yields finite semidefinite programmes whose optima (q_L(A,E)), for any bounded Hermitian observable (A), satisfy [0\le\Omega_E(A)-q_L(A,E)\le\varepsilon\inf_{c\in\mathbb R}|A-cI|,] where (\Omega_E(A)) is the optimum over all Wigner-positive states under the same energy cap. At fixed mode number and energy, a continuity bound also provides finite approximations to Wigner entropy minima and finite witnesses for every strict entropy deficit. These bounds determine how much of the infinite state space a calculation must retain to certify its conclusions.

DOI: 10.5281/zenodo.23022267

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Hidden quantum coherence at the Wigner entropy endpoint

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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 $\mathcal W_E$ 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 $\mathcal G_E$ 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 $P_L$ project onto photon numbers at most $L$ , and define the cutoff error by $e_L(\rho)=\tfrac12\|\rho-P_L\rho P_L\|_1.$ This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard $O_E(L^{-1/2})$ guarantee. At the exact Wigner endpoint, the article proves $\sup_{\rho\in\mathcal W_E}e_L(\rho)\le C_E(L+1)^{-3/4},\qquad \sup_{\rho\in\mathcal G_E}e_L(\rho)\le \frac{4E}{L+1},\qquad L\ge2.$ 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 $\tau_d\in\mathcal W_E$ has exact energy $E$ , covariance $(E+1/2)I$ , strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs $L_d=\Theta(d^3)$ , $e_{L_d}(\tau_d)\ge c_Ed^{-5/2},\qquad (L_d+1)e_{L_d}(\tau_d)\longrightarrow\infty.$ 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 $n_X(\delta)$ be the minimum worst-case number of copies for trace-distance accuracy $\delta$ over source class $X$ . As $\delta\downarrow0$ , the bounds are $\log n_{\mathcal G_E}(\delta)=\Theta(\delta^{-1}),\qquad c\delta^{-6/5}\le\log n_{\mathcal W_E}(\delta)\le C\delta^{-4/3}.$ 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.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 $\mathcal W_E$ 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 $\mathcal G_E$ 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 $P_L$ project onto photon numbers at most $L$, and define the cutoff error by \[e_L(\rho)=\tfrac12\|\rho-P_L\rho P_L\|_1.\] This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard $O_E(L^{-1/2})$ guarantee. At the exact Wigner endpoint, the article proves \[\sup_{\rho\in\mathcal W_E}e_L(\rho)\le C_E(L+1)^{-3/4},\qquad \sup_{\rho\in\mathcal G_E}e_L(\rho)\le \frac{4E}{L+1},\qquad L\ge2.\] 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 $\tau_d\in\mathcal W_E$ has exact energy $E$, covariance $(E+1/2)I$, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs $L_d=\Theta(d^3)$, \[e_{L_d}(\tau_d)\ge c_Ed^{-5/2},\qquad (L_d+1)e_{L_d}(\tau_d)\longrightarrow\infty.\] 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 $n_X(\delta)$ be the minimum worst-case number of copies for trace-distance accuracy $\delta$ over source class $X$. As $\delta\downarrow0$, the bounds are \[\log n_{\mathcal G_E}(\delta)=\Theta(\delta^{-1}),\qquad c\delta^{-6/5}\le\log n_{\mathcal W_E}(\delta)\le C\delta^{-4/3}.\] 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.

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Distinct stability laws for Wigner entropy and non-Gaussian structure

Zixuan He

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 $\nu_m$ for the $m$ -mode vacuum and define $\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)$ , using natural logarithms; $H_1$ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, $E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.$ Here $\mathcal F_m$ is the trace-norm-closed convex hull of Gaussian states and $d_{\mathcal F_m}$ 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 $-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0$ , so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity $\eta$ and any fixed positive total input-energy budget $E$ , the sharp threshold is $\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}$ With $\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}$ , $\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}$ The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For $\alpha\ge2$ , 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 $\rho$ , define the half-loss output Shannon excess $s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.$ The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields $\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).$ 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 $\epsilon$ is guaranteed by $s\le(\ln2)\epsilon^2/16$ . Conversely, a surviving output Gaussian-hull gap enforces $s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.$ 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 $\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.$ At half transmission this lower bound is $g(c)=(c/8)e^{-7c/8}$ . Starting from one complete block, Gaussian processing requires $\Theta(m)$ output modes to retain any fixed positive gap below this bound at success probability at least a fixed $p_0>0$ . 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 $\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25$ within half trace distance $10^{-6}$ . Let $C_m$ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed $0<c\le1$ , every achieving Gaussian protocol obeys the asymptotic lower bound $\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).$ 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 $h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).$ The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy $E_A,E_B\sim s/2$ . At exactly equal energy, the two excesses instead share $s_i(E)=2E-(35/8)E^2+O(E^3)$ and differ by $s_A(E)-s_B(E)=15E^3+O(E^4)$ . 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 $\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K$ , trusted half loss and all-mode on/off detection with efficiency at least $\xi_{\min}$ , the click probability $q$ gives $s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.$ A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses $K=4$ , $\tau_K=0$ , $\xi=0.8$ 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 $E=1$ , $\alpha=1.9$ , $\eta=0.9505$ and $m=10^5$ , the certified bound $\Delta_\alpha<0.06942$ rules out a target deficit of $0.1$ nat. A separate finite single-mode preparation model predicts positive Shannon deficit near $\eta=0.97$ , negative deficit near $\eta=0.85$ , and a Gaussian-hull witness above $4.4\times10^{-3}$ 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 $\alpha=2-\delta$ and $L=\ln(1/\delta)$ . For the maximum deficit $D_{m,E}$ at fixed total energy, $D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),$ where $T_h$ is the first mode number allowing a fixed positive deficit $h$ . Beyond a sufficiently large fixed multiple of $L/\delta$ , $D_{m,E}(2-\delta)=\Theta_E(m\delta)$ . 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: $h(W_\rho)\ge m\ln(2\pi)+\varepsilon_m$ with $\varepsilon_m>0$ for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With $\mathcal C_{r,k}$ denoting the ordering-weighted norm of photon-number-difference block $k$ , $\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.$ For vacuum– $n$ -photon coherence and $1<\alpha<2$ , $\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.$ 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.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 $\nu_m$ for the $m$-mode vacuum and define $\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)$, using natural logarithms; $H_1$ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, $$E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.$$ Here $\mathcal F_m$ is the trace-norm-closed convex hull of Gaussian states and $d_{\mathcal F_m}$ 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 $-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0$, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity $\eta$ and any fixed positive total input-energy budget $E$, the sharp threshold is $$\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}$$ With $\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}$, $$\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}$$ The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For $\alpha\ge2$, 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 $\rho$, define the half-loss output Shannon excess $$s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.$$ The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields $$\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).$$ 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 $\epsilon$ is guaranteed by $s\le(\ln2)\epsilon^2/16$. Conversely, a surviving output Gaussian-hull gap enforces $$s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.$$ 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 $$\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.$$ At half transmission this lower bound is $g(c)=(c/8)e^{-7c/8}$. Starting from one complete block, Gaussian processing requires $\Theta(m)$ output modes to retain any fixed positive gap below this bound at success probability at least a fixed $p_0>0$. 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 $\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25$ within half trace distance $10^{-6}$. Let $C_m$ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed $0<c\le1$, every achieving Gaussian protocol obeys the asymptotic lower bound $$\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).$$ 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 $$h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).$$ The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy $E_A,E_B\sim s/2$. At exactly equal energy, the two excesses instead share $s_i(E)=2E-(35/8)E^2+O(E^3)$ and differ by $s_A(E)-s_B(E)=15E^3+O(E^4)$. 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 $\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K$, trusted half loss and all-mode on/off detection with efficiency at least $\xi_{\min}$, the click probability $q$ gives $$s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.$$ A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses $K=4$, $\tau_K=0$, $\xi=0.8$ 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 $E=1$, $\alpha=1.9$, $\eta=0.9505$ and $m=10^5$, the certified bound $\Delta_\alpha<0.06942$ rules out a target deficit of $0.1$ nat. A separate finite single-mode preparation model predicts positive Shannon deficit near $\eta=0.97$, negative deficit near $\eta=0.85$, and a Gaussian-hull witness above $4.4\times10^{-3}$ 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 $\alpha=2-\delta$ and $L=\ln(1/\delta)$. For the maximum deficit $D_{m,E}$ at fixed total energy, $$D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),$$ where $T_h$ is the first mode number allowing a fixed positive deficit $h$. Beyond a sufficiently large fixed multiple of $L/\delta$, $D_{m,E}(2-\delta)=\Theta_E(m\delta)$. 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: $h(W_\rho)\ge m\ln(2\pi)+\varepsilon_m$ with $\varepsilon_m>0$ for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With $\mathcal C_{r,k}$ denoting the ordering-weighted norm of photon-number-difference block $k$, $$\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.$$ For vacuum–$n$-photon coherence and $1<\alpha<2$, $$\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.$$ 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.

Sep 23, 20260 citations

Wigner entropy below vacuum: physical counterexamples and stability limits

Zixuan He

quant-pharXiv:2609.13312Sep 11, 20260 citations

Wigner entropy below vacuum: physical counterexamples and stability limits

Zixuan He

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.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.

Sep 24, 20261 citation

Coherence beyond Gaussian mixtures at low energy: sharp Wigner-positive bounds and phase-noise thresholds

Zixuan He

Wigner-positive light can carry more vacuum–one-photon coherence than every Gaussian mixture under the same mean photon-number budget. We determine the exact Gaussian benchmark and the sharp low-energy scales of this separation. Relative to the unconstrained ceiling $U(E)=\sqrt{E(1-E)}$ , the Gaussian coherence shortfall is $U(E)-M_G(E)\sim E^{5/2}$ , whereas the optimum over all Wigner-positive states satisfies $\lim_{E\downarrow0}E\log[U(E)-M_\infty(E)]=-1/2$ . A hard Fock cutoff changes the cost: with support on levels $0,\ldots,K$ , the shortfall is $(KK!/2)E^{K+1/2}[1+O(K^2E)]$ , uniformly for $K\geq2$ as $K^2E\to0$ . The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance $\Gamma$ , the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below $\Gamma_{\rm cert}(E)=2E^2+O(E^3)$ . Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.Wigner-positive light can carry more vacuum–one-photon coherence than every Gaussian mixture under the same mean photon-number budget. We determine the exact Gaussian benchmark and the sharp low-energy scales of this separation. Relative to the unconstrained ceiling $U(E)=\sqrt{E(1-E)}$, the Gaussian coherence shortfall is $U(E)-M_G(E)\sim E^{5/2}$, whereas the optimum over all Wigner-positive states satisfies $\lim_{E\downarrow0}E\log[U(E)-M_\infty(E)]=-1/2$. A hard Fock cutoff changes the cost: with support on levels $0,\ldots,K$, the shortfall is $(KK!/2)E^{K+1/2}[1+O(K^2E)]$, uniformly for $K\geq2$ as $K^2E\to0$. The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance $\Gamma$, the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below $\Gamma_{\rm cert}(E)=2E^2+O(E^3)$. Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.

Oct 7, 20260 citations

Finite-rank approximation with exact position records and energy

Zixuan He

Can a finite-rank quantum state approximate a given optical state while reproducing an entire continuous measurement record and its original mean energy exactly? This paper provides a constructive affirmative answer, with quantitative rank, trace-distance and entropy control. The main result concerns one oscillator $A=L^2(\mathbb R)$ , with $[Q,P]=i$ and photon operator $N=(Q^2+P^2-1)/2$ , coupled to a finite memory $B=\mathbb C^b$ . The input is any normal state $\rho_{AB}$ with mean photons $E=\operatorname{Tr}(N\rho_A)<\infty$ . Its complete position–memory record $G_\rho(q)$ retains both the position probability and the conditional quantum state of the memory. For every integer $L\ge0$ and $0<a\le1$ , there is a state $\omega$ satisfying $G_\omega=G_\rho,\qquad \operatorname{Tr}(N\omega_A)=E,\qquad \operatorname{rank}\omega\le b(L+2),\qquad \tfrac12\|\rho-\omega\|_1\le\frac{E}{L+1}+a.$ Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most $b(L+2)$ ; retaining both the original memory and this register costs total dimension at most $b^2(L+2)$ . The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most $r$ and the tail mass is $p$ , at most $b$ replacement vectors suffice, giving rank at most $r+b$ and half trace-distance error at most $p$ before energy restoration. A position-dependent unitary with $\|U-I\|\le a$ then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms, $|S(\rho)-S(\omega)|\le h_2(p)+p\,[g(E/p)+\log b],$ where $h_2$ is binary entropy, $g(x)=(1+x)\log(1+x)-x\log x$ , and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.Can a finite-rank quantum state approximate a given optical state while reproducing an entire continuous measurement record and its original mean energy exactly? This paper provides a constructive affirmative answer, with quantitative rank, trace-distance and entropy control. The main result concerns one oscillator $A=L^2(\mathbb R)$, with $[Q,P]=i$ and photon operator $N=(Q^2+P^2-1)/2$, coupled to a finite memory $B=\mathbb C^b$. The input is any normal state $\rho_{AB}$ with mean photons $E=\operatorname{Tr}(N\rho_A)<\infty$. Its complete position–memory record $G_\rho(q)$ retains both the position probability and the conditional quantum state of the memory. For every integer $L\ge0$ and $0<a\le1$, there is a state $\omega$ satisfying \[G_\omega=G_\rho,\qquad \operatorname{Tr}(N\omega_A)=E,\qquad \operatorname{rank}\omega\le b(L+2),\qquad \tfrac12\|\rho-\omega\|_1\le\frac{E}{L+1}+a.\] Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most $b(L+2)$; retaining both the original memory and this register costs total dimension at most $b^2(L+2)$. The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most $r$ and the tail mass is $p$, at most $b$ replacement vectors suffice, giving rank at most $r+b$ and half trace-distance error at most $p$ before energy restoration. A position-dependent unitary with $\|U-I\|\le a$ then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms, \[|S(\rho)-S(\omega)|\le h_2(p)+p\,[g(E/p)+\log b],\] where $h_2$ is binary entropy, $g(x)=(1+x)\log(1+x)-x\log x$, and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.

Oct 29, 20260 citations

[Q,P]=i

, mean photon number at most

E>0

, and a known parity centre. The full class

\mathcal W_E

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

\mathcal G_E

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

P_L

project onto photon numbers at most

L

, and define the cutoff error by

e_L(\rho)=\tfrac12\|\rho-P_L\rho P_L\|_1.

This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard

O_E(L^{-1/2})

guarantee. At the exact Wigner endpoint, the article proves

\sup_{\rho\in\mathcal W_E}e_L(\rho)\le C_E(L+1)^{-3/4},\qquad \sup_{\rho\in\mathcal G_E}e_L(\rho)\le \frac{4E}{L+1},\qquad L\ge2.

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

\tau_d\in\mathcal W_E

has exact energy

E

, covariance

(E+1/2)I

, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs

L_d=\Theta(d^3)

e_{L_d}(\tau_d)\ge c_Ed^{-5/2},\qquad (L_d+1)e_{L_d}(\tau_d)\longrightarrow\infty.

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

n_X(\delta)

be the minimum worst-case number of copies for trace-distance accuracy

\delta

over source class

X

. As

\delta\downarrow0

, the bounds are

\log n_{\mathcal G_E}(\delta)=\Theta(\delta^{-1}),\qquad c\delta^{-6/5}\le\log n_{\mathcal W_E}(\delta)\le C\delta^{-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.

\nu_m

for the

m

-mode vacuum and define

\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)

, using natural logarithms;

H_1

is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously,

E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.

Here

\mathcal F_m

is the trace-norm-closed convex hull of Gaussian states and

d_{\mathcal F_m}

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

-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0

, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity

\eta

and any fixed positive total input-energy budget

E

, the sharp threshold is

\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}

With

\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}

\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}

The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For

\alpha\ge2

, 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

\rho

, define the half-loss output Shannon excess

s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.

The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields

\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).

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

\epsilon

is guaranteed by

s\le(\ln2)\epsilon^2/16

. Conversely, a surviving output Gaussian-hull gap enforces

s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^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

\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.

At half transmission this lower bound is

g(c)=(c/8)e^{-7c/8}

. Starting from one complete block, Gaussian processing requires

\Theta(m)

output modes to retain any fixed positive gap below this bound at success probability at least a fixed

p_0>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

\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25

within half trace distance

10^{-6}

. Let

C_m

be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed

0<c\le1

, every achieving Gaussian protocol obeys the asymptotic lower bound

\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-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(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).

The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy

E_A,E_B\sim s/2

. At exactly equal energy, the two excesses instead share

s_i(E)=2E-(35/8)E^2+O(E^3)

and differ by

s_A(E)-s_B(E)=15E^3+O(E^4)

. 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

\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K

, trusted half loss and all-mode on/off detection with efficiency at least

\xi_{\min}

, the click probability

q

gives

s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.

A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses

K=4

\tau_K=0

\xi=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

\alpha=1.9

\eta=0.9505

and

m=10^5

, the certified bound

\Delta_\alpha<0.06942

rules out a target deficit of

0.1

nat. A separate finite single-mode preparation model predicts positive Shannon deficit near

\eta=0.97

, negative deficit near

\eta=0.85

, and a Gaussian-hull witness above

4.4\times10^{-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

\alpha=2-\delta

and

L=\ln(1/\delta)

. For the maximum deficit

D_{m,E}

at fixed total energy,

D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),

where

T_h

is the first mode number allowing a fixed positive deficit

h

. Beyond a sufficiently large fixed multiple of

L/\delta

D_{m,E}(2-\delta)=\Theta_E(m\delta)

. 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(W_\rho)\ge m\ln(2\pi)+\varepsilon_m

with

\varepsilon_m>0

for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With

\mathcal C_{r,k}

denoting the ordering-weighted norm of photon-number-difference block

k

\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.

For vacuum–

n

-photon coherence and

1<\alpha<2

\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^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.

2\times10^{-21}

preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale

E^γ/[\ln(1/E)]^β

, with

γ\simeq0.7412

and

β\simeq0.5861

. 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.

U(E)=\sqrt{E(1-E)}

, the Gaussian coherence shortfall is

U(E)-M_G(E)\sim E^{5/2}

, whereas the optimum over all Wigner-positive states satisfies

\lim_{E\downarrow0}E\log[U(E)-M_\infty(E)]=-1/2

. A hard Fock cutoff changes the cost: with support on levels

0,\ldots,K

, the shortfall is

(KK!/2)E^{K+1/2}[1+O(K^2E)]

, uniformly for

K\geq2

K^2E\to0

. The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance

\Gamma

, the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below

\Gamma_{\rm cert}(E)=2E^2+O(E^3)

. Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.

A=L^2(\mathbb R)

, with

[Q,P]=i

and photon operator

N=(Q^2+P^2-1)/2

, coupled to a finite memory

B=\mathbb C^b

. The input is any normal state

\rho_{AB}

with mean photons

E=\operatorname{Tr}(N\rho_A)<\infty

. Its complete position–memory record

G_\rho(q)

retains both the position probability and the conditional quantum state of the memory. For every integer

L\ge0

and

0<a\le1

, there is a state

\omega

satisfying

G_\omega=G_\rho,\qquad \operatorname{Tr}(N\omega_A)=E,\qquad \operatorname{rank}\omega\le b(L+2),\qquad \tfrac12\|\rho-\omega\|_1\le\frac{E}{L+1}+a.

Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most

b(L+2)

; retaining both the original memory and this register costs total dimension at most

b^2(L+2)

. The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most

r

and the tail mass is

p

, at most

b

replacement vectors suffice, giving rank at most

r+b

and half trace-distance error at most

p

before energy restoration. A position-dependent unitary with

\|U-I\|\le a

then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms,

|S(\rho)-S(\omega)|\le h_2(p)+p\,[g(E/p)+\log b],

where

h_2

is binary entropy,

g(x)=(1+x)\log(1+x)-x\log x

, and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.