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Distinct stability laws for Wigner entropy and non-Gaussian structure

Zixuan He

Published Sep 23, 2026• 0 citations
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Abstract

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.

DOI
10.5281/zenodo.22908411

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•Oct 9, 2026•0 citations

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•Oct 31, 2026•0 citations
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\nu_mνm​ for the mmm-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ)\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)Δα,m​(ρ)=Hα​(Wνm​​)−Hα​(Wρ​), using natural logarithms; H1H_1H1​ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr⁡(Nmρm)→0,12∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞.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.Em​=Tr(Nm​ρm​)→0,21​∥ρm​−νm​∥1​→0,dFm​​(ρm​)→0,Δ1,m​(ρm​)→+∞. Here Fm\mathcal F_mFm​ is the trace-norm-closed convex hull of Gaussian states and dFmd_{\mathcal F_m}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-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0−Em​≤Δ1,m​(L1/2⊗m​ρm​)≤0, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η\etaη and any fixed positive total input-energy budget EEE, the sharp threshold is ηc(α)={1/2,0<α≤1,α/2,1<α<2.\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}ηc​(α)={1/2,α/2,​0<α≤1,1<α<2.​ With Pm,E={ρ:Wρ≥0, Tr⁡(Nmρ)≤E}\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}Pm,E​={ρ:Wρ​≥0, Tr(Nm​ρ)≤E}, sup⁡m≥1sup⁡ρ∈Pm,EΔα,m(Lη⊗mρ)={0,η≤ηc(α),+∞,η>ηc(α).\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}supm≥1​supρ∈Pm,E​​Δα,m​(Lη⊗m​ρ)={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\alpha\ge2α≥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\rhoρ, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−m(1+ln⁡π)≥0.s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.s(ρ)=h(WL1/2⊗m​ρ​)−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 inf⁡a∥ρ−∣a⟩⟨a∣∥12≤64ln⁡2 s(ρ).\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).infa​∥ρ−∣a⟩⟨a∣∥12​≤ln264​s(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\epsilonϵ is guaranteed by s≤(ln⁡2)ϵ2/16s\le(\ln2)\epsilon^2/16s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥ln⁡216 dFm(L1/2⊗mρ)2.s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.s(ρ)≥16ln2​dFm​​(L1/2⊗m​ρ)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/mc/mc/m has total input energy tending to 3c3c3c. Along the recovery boundary, its attenuated state obeys lim inf⁡m→∞dFm(ρ~m,ηc)≥cηc3exp⁡{−c[1−(1−ηc)3]}>0.\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.liminfm→∞​dFm​​(ρ​m,ηc​​)≥cηc3​exp{−c[1−(1−ηc​)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8g(c)=(c/8)e^{-7c/8}g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m)\Theta(m)Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>0p_0>0p0​>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\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25τ∗​=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25 within half trace distance 10−610^{-6}10−6. Let CmC_mCm​ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤10<c\le10<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound log⁡Cm≥1128(m/c)1/3−log⁡m−O(1).\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).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+ln⁡π)+s,log⁡CA(s)=Θ(s−1/3),log⁡CB(s)=Θ(s−1/2).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}).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/2E_A,E_B\sim s/2EA​,EB​∼s/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3)s_i(E)=2E-(35/8)E^2+O(E^3)si​(E)=2E−(35/8)E2+O(E3) and differ by sA(E)−sB(E)=15E3+O(E4)s_A(E)-s_B(E)=15E^3+O(E^4)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\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_KTr[N1N>K​σa​]≤τK​, trusted half loss and all-mode on/off detection with efficiency at least ξmin⁡\xi_{\min}ξmin​, the click probability qqq gives s(σ)≤AKq+τK,AK=K1−(1−ξmin⁡/2)K.s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.s(σ)≤AK​q+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=4K=4K=4, τK=0\tau_K=0τK​=0, ξ=0.8\xi=0.8ξ=0.8 and N=3,517,336N=3{,}517{,}336N=3,517,336 independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most 0.050.050.05 with false-acceptance probability at most 0.0010.0010.001 and missed acceptance at most 0.0010.0010.001 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=1E=1E=1, α=1.9\alpha=1.9α=1.9, η=0.9505\eta=0.9505η=0.9505 and m=105m=10^5m=105, the certified bound Δα<0.06942\Delta_\alpha<0.06942Δα​<0.06942 rules out a target deficit of 0.10.10.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97\eta=0.97η=0.97, negative deficit near η=0.85\eta=0.85η=0.85, and a Gaussian-hull witness above 4.4×10−34.4\times10^{-3}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\alpha=2-\deltaα=2−δ and L=ln⁡(1/δ)L=\ln(1/\delta)L=ln(1/δ). For the maximum deficit Dm,ED_{m,E}Dm,E​ at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ),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),Dm,E​(2−δ)=Θm,E​(δ/L)for fixed m,Th​(δ,E)=ΘE,h​(L/δ), where ThT_hTh​ is the first mode number allowing a fixed positive deficit hhh. Beyond a sufficiently large fixed multiple of L/δL/\deltaL/δ, Dm,E(2−δ)=ΘE(mδ)D_{m,E}(2-\delta)=\Theta_E(m\delta)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)+εmh(W_\rho)\ge m\ln(2\pi)+\varepsilon_mh(Wρ​)≥mln(2π)+εm​ with εm>0\varepsilon_m>0εm​>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k\mathcal C_{r,k}Cr,k​ denoting the ordering-weighted norm of photon-number-difference block kkk, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−1−rη.\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.Cr,k​(Lη​ρ)=CTη​(r),k​(ρ),Tη​(r)=1−η1−r​. For vacuum–nnn-photon coherence and 1<α<21<\alpha<21<α<2, C1−α/2(Lη∣n⟩⟨0∣)=(2ηα)n.\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.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.
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 2×10−212\times10^{-21}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)]βE^γ/[\ln(1/E)]^βEγ/[ln(1/E)]β, with γ≃0.7412γ\simeq0.7412γ≃0.7412 and β≃0.5861β\simeq0.5861β≃0.5861. Because γ<1γ<1γ<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.
Abstract When can a quantum optical source be replaced by a coherent state with controlled error in every subsequent experiment? We turn its Husimi phase-space entropy excess into an error bound for the entire quantum state. For every fixed Rényi order p>0p>0p>0 and every density operator on any finite number of modes, D(ρ)2≤Cpδp(ρ),D(ρ)=inf⁡α∥ρ−∣α⟩⟨α∣∥1,D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,D(ρ)2≤Cp​δp​(ρ),D(ρ)=infα​∥ρ−∣α⟩⟨α∣∥1​, where δp\delta_pδp​ is the total Rényi–Wehrl deficit and CpC_pCp​ is independent of mode count, energy, rank and photon-number support. One coherent replacement then bounds every measurement-probability error after any quantum channel. We identify the source structure that determines the finer error near the coherent manifold: after optimising displacement, D≍a+bD\asymp a+bD≍a+b, where aaa is the population outside the reference vacuum and bbb its coherence with the remaining state. This distinguishes linear thermal response from square-root squeezing response and yields intermediate rates. The guarantee holds throughout the positive-order range, but its optimal coefficient necessarily diverges at small order: Cpopt∼4/[plog⁡(1/p)]C_p^{\rm opt}\sim4/[p\log(1/p)]Cpopt​∼4/[plog(1/p)] as p↓0p\downarrow0p↓0. A reduction of fractional Husimi moments to one active mode proves this sharp limit. The results specify both when entropy certifies a multimode coherent approximation and how its accuracy depends on the structure of the source. Version 1.1 — 9 October 2026 This revision extends the original mode-uniform Wehrl theorem to every positive Rényi order and develops its use as a coherent-state approximation guarantee. All positive orders and arbitrary density operators. The full trace-norm bound is uniform over every finite mode number, without a finite-energy assumption. Explicit sufficient coefficients include C1=64/log⁡2C_1=64/\log 2C1​=64/log2 and C2=16C_2=16C2​=16. Source-dependent accuracy. A local population–coherence law separates thermal, squeezed and intermediate responses. The same coherent reference controls subsequent quantum channels and measurement probabilities. Sharp small-order limit. The optimal global coefficient satisfies plog⁡(1/p)Cpopt→4p\log(1/p)C_p^{\rm opt}\to4plog(1/p)Cpopt​→4. The complete proof uses a one-active-mode reduction of fractional Husimi moments. Expanded presentation. Two vector figures explain source-dependent error, the role of total entropy deficit and the order-dependent bounds. The 19-page PDF includes the complete manuscript, proofs and updated references. This is an expanded revision of Uniform Wehrl stability across all bosonic mode numbers: total entropy deficit controls full trace-norm distance to coherent states, first published on 6 October 2026.
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[Q,P]=i[Q,P]=i, mean photon number at most E>0E>0E>0, and a known parity centre. The full class WE\mathcal W_EWE​ 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\mathcal G_EGE​ 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 PLP_LPL​ project onto photon numbers at most LLL, and define the cutoff error by eL(ρ)=12∥ρ−PLρPL∥1.e_L(\rho)=\tfrac12\|\rho-P_L\rho P_L\|_1.eL​(ρ)=21​∥ρ−PL​ρPL​∥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)O_E(L^{-1/2})OE​(L−1/2) guarantee. At the exact Wigner endpoint, the article proves sup⁡ρ∈WEeL(ρ)≤CE(L+1)−3/4,sup⁡ρ∈GEeL(ρ)≤4EL+1,L≥2.\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.supρ∈WE​​eL​(ρ)≤CE​(L+1)−3/4,supρ∈GE​​eL​(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\tau_d\in\mathcal W_Eτd​∈WE​ has exact energy EEE, covariance (E+1/2)I(E+1/2)I(E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3)L_d=\Theta(d^3)Ld​=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞.e_{L_d}(\tau_d)\ge c_Ed^{-5/2},\qquad (L_d+1)e_{L_d}(\tau_d)\longrightarrow\infty.eLd​​(τd​)≥cE​d−5/2,(Ld​+1)eLd​​(τd​)⟶∞. Thus the Gaussian-mixture L−1L^{-1}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)n_X(\delta)nX​(δ) be the minimum worst-case number of copies for trace-distance accuracy δ\deltaδ over source class XXX. As δ↓0\delta\downarrow0δ↓0, the bounds are log⁡nGE(δ)=Θ(δ−1),cδ−6/5≤log⁡nWE(δ)≤Cδ−4/3.\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}.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.
Detecting non-Gaussian light establishes that a resource is present. Its structure and preparation record determine the cost of turning it into a specified output. This paper gives matching conversion-cost laws for weak optical sources and a general input certificate that can exclude every allowed Gaussian receiver before a circuit is designed. Sources with identical energy, covariance and total photon statistics can have different cost exponents; losing a classical phase bit can make a fixed-source conversion impossible. Source, target and accounting. The Wigner-positive two-mode sources are obtained by half loss on each mode: ρt,e,q=L1/2⊗2 ⁣[∣00⟩⟨00∣+t2{(1−q)∣p+⟩⟨p+∣+q∣p−⟩⟨p−∣}1+t2],\rho_{t,e,q}=\mathcal L_{1/2}^{\otimes2}\!\left[\frac{|00\rangle\langle00|+t^2\{(1-q)|p_+\rangle\langle p_+|+q|p_-\rangle\langle p_-|\}}{1+t^2}\right],ρt,e,q​=L1/2⊗2​[1+t2∣00⟩⟨00∣+t2{(1−q)∣p+​⟩⟨p+​∣+q∣p−​⟩⟨p−​∣}​], ∣p±⟩=∣22⟩±e(∣40⟩+∣04⟩)/21+e2,0≤e≤14,0≤q≤12,E=2t21+t2.|p_\pm\rangle=\frac{|22\rangle\pm e(|40\rangle+|04\rangle)/\sqrt2}{\sqrt{1+e^2}},\quad 0\le e\le\tfrac14,\quad 0\le q\le\tfrac12,\quad E=\frac{2t^2}{1+t^2}.∣p±​⟩=1+e2​∣22⟩±e(∣40⟩+∣04⟩)/2​​,0≤e≤41​,0≤q≤21​,E=1+t22t2​. Here eee selects the source, qqq is an unflagged phase-record error probability, and EEE is delivered mean photon number. Conversion targets τ∗=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25,12∥σout−τ∗∥1≤δ,0<δ<10−5.\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25,\qquad \tfrac12\|\sigma_{\mathrm{out}}-\tau_*\|_1\le\delta,\quad 0<\delta<10^{-5}.τ∗​=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25,21​∥σout​−τ∗​∥1​≤δ,0<δ<10−5. Joint processing of independent supplied copies, Gaussian ancillas, squeezing, memory, Gaussian measurements, feedback, discarding and postselection are allowed. Cost is C=EK/psC=\mathbb E K/p_sC=EK/ps​: every source request counts, including vacuum requests and failed attempts. The task uses ideal Gaussian operations and charges supplied copies. 1. A source-selection law controlled by record accuracy. Optimizing both the source parameter eee and the allowed Gaussian protocol gives, uniformly in q∈[0,1/2]q\in[0,1/2]q∈[0,1/2] as E→0E\to0E→0, log⁡Cbest(E,q)=Θδ ⁣(max⁡{E−1/3,q/E}).\log C_{\mathrm{best}}(E,q)=\Theta_\delta\!\left(\max\{E^{-1/3},\sqrt{q/E}\}\right).logCbest​(E,q)=Θδ​(max{E−1/3,q/E​}). Errors of order E1/3E^{1/3}E1/3 or smaller preserve the fastest logarithmic cost. Above that scale, record quality controls the optimum. Equivalently, for q=q0Eaq=q_0E^aq=q0​Ea, fixed 0<q0≤1/20<q_0\le1/20<q0​≤1/2 and a≥0a\ge0a≥0, log⁡Cbest=Θδ(E−β),β=max⁡{1/3,(1−a)/2}.\log C_{\mathrm{best}}=\Theta_\delta(E^{-\beta}),\qquad \beta=\max\{1/3,(1-a)/2\}.logCbest​=Θδ​(E−β),β=max{1/3,(1−a)/2}. At fixed nonzero error, choosing the phase-insensitive source keeps conversion possible at logarithmic cost of order E−1/2E^{-1/2}E−1/2. 2. The same elementary diagnostics, exponentially different costs. The normalized excitation polynomials pA(z)=z14−z2448,pB(z)=(z12−z22)28p_A(z)=\frac{z_1^4-z_2^4}{\sqrt{48}},\qquad p_B(z)=\frac{(z_1^2-z_2^2)^2}{8}pA​(z)=48​z14​−z24​​,pB​(z)=8(z12​−z22​)2​ give calibrated half-loss sources with identical energy, covariance and total photon-number statistics, but for the same target and tolerance, log⁡CA(E)=Θδ(E−1/3),log⁡CB(E)=Θδ(E−1/2).\log C_A(E)=\Theta_\delta(E^{-1/3}),\qquad \log C_B(E)=\Theta_\delta(E^{-1/2}).logCA​(E)=Θδ​(E−1/3),logCB​(E)=Θδ​(E−1/2). The distinction is the local structure of their excitation-polynomial zeros. Full Gaussian-protocol lower bounds and matching heterodyne constructions establish both orders. 3. Certification survives errors that obstruct conversion. In the symmetric quartic family, the full first-mode photon distribution is independent of qqq. In particular, Px(1)=E(4+e2)16(1+e2)≥E/5,Px(N≥5)=0.P_x(1)=\frac{E(4+e^2)}{16(1+e^2)}\ge E/5,\qquad P_x(N\ge5)=0.Px​(1)=16(1+e2)E(4+e2)​≥E/5,Px​(N≥5)=0. Every state in the full one-mode Gaussian convex hull instead satisfies P(N≥5)≥P(1)64608000.P(N\ge5)\ge\frac{P(1)^6}{4608000}.P(N≥5)≥4608000P(1)6​. Photon counting therefore supplies a sufficient O(E−6)O(E^{-6})O(E−6) request budget at fixed false-positive and missed-detection probabilities against independent repetitions of any one unknown Gaussian-hull state. Yet, in the joint limit t→0t\to0t→0, e/t→∞e/t\to\inftye/t→∞, finite-cost conversion has the threshold qc(t,e;δ)=Θδ ⁣((E/e2)1/3).q_c(t,e;\delta)=\Theta_\delta\!\left((E/e^2)^{1/3}\right).qc​(t,e;δ)=Θδ​((E/e2)1/3). For fixed e>0e>0e>0, any fixed positive unflagged error thus eventually obstructs this output. Flagged erasures behave differently: independently retaining a correct bit with probability ar>0a_r>0ar​>0 adds at most a factor 1/ar1/a_r1/ar​ to a protocol that discards erasures. 4. A transferable exclusion certificate and an explicit receiver. A verified positive phase-space minorant with a physical pure Gaussian curvature reference supplies a source budget bκb_\kappabκ​. For a requested one-mode output whose trace-distance gap from the Gaussian convex hull is at least ggg, every allowed Gaussian protocol obeys C≥g2−κ/(2−κ)bκ,κ2−κ<g2.C\ge\frac{g^2-\kappa/(2-\kappa)}{b_\kappa},\qquad \frac{\kappa}{2-\kappa}<g^2.C≥bκ​g2−κ/(2−κ)​,2−κκ​<g2. The manuscript constructs the certificate and explicit incomplete-gamma budgets for the quartic sources. It also gives a centred-root heterodyne receiver. In its specified joint limit, with scaled error λ=deq(e/t)2/3\lambda=d_e q(e/t)^{2/3}λ=de​q(e/t)2/3, its optimized limiting one-photon population is sup⁡w=12[1+4 3−3/4λ3/4].\sup w=\frac1{2[1+4\,3^{-3/4}\lambda^{3/4}]}.supw=2[1+43−3/4λ3/4]1​. The paper supplies ded_ede​, the receiver action, and a photon-statistical output witness. These receiver formulas connect the desired output to tolerable record error; the general certificate supplies the protocol-independent cost bound. Together they let source designers compare preparation structure, record quality and total request budget for the same optical task.
When does finite phase-space entropy stop guaranteeing proximity to coherent light? This article determines sharp thresholds for two broad optical source classes when the number of modes may grow. It separates the roles of entropy, mode number and photon energy in approximating an optical state by a single pure coherent state. For an mmm-mode state ρ\rhoρ, write Qρ(α)=⟨α∣ρ∣α⟩Q_\rho(\alpha)=\langle\alpha|\rho|\alpha\rangleQρ​(α)=⟨α∣ρ∣α⟩, with dμm=π−md2mαd\mu_m=\pi^{-m}d^{2m}\alphadμm​=π−md2mα, and define M(ρ)=2m∫Qρ(α)2 dμm(α),s=−ln⁡M(ρ),D(ρ)=inf⁡α∥ρ−∣α⟩⟨α∣∥1.M(\rho)=2^m\int Q_\rho(\alpha)^2\,d\mu_m(\alpha),\qquad s=-\ln M(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1.M(ρ)=2m∫Qρ​(α)2dμm​(α),s=−lnM(ρ),D(ρ)=infα​∥ρ−∣α⟩⟨α∣∥1​. Here sss is the second-order Wehrl entropy excess and DDD is the full trace-norm distance, whose maximum is 2. The collision quantity MMM is also the output purity after half loss and can be accessed using interference of two independent, identically prepared copies. Exact source-class thresholds. The two classes are states diagonal in a common passive-mode number basis, and arbitrary states supported on the vacuum together with configurations whose photons occupy only one mode at a time in a common passive basis. The latter class retains arbitrary complex coherences, including vacuum coherences. For either class X\mathcal XX, with FX(s)=sup⁡ρ∈X, −ln⁡M(ρ)=sD(ρ)2F_{\mathcal X}(s)=\sup_{\rho\in\mathcal X,\,-\ln M(\rho)=s}D(\rho)^2FX​(s)=supρ∈X,−lnM(ρ)=s​D(ρ)2 and the supremum over all finite mode numbers, FX(s)=4⟺s≥ln⁡4.F_{\mathcal X}(s)=4\quad\Longleftrightarrow\quad s\ge\ln4.FX​(s)=4⟺s≥ln4. The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and M→1/4M\to1/4M→1/4 force the one-photon probability to approach one, without an energy-bound assumption. This probability rigidity does not assert convergence of the mean energy. Sharp boundary geometry and energy. For the full number-diagonal class D\mathcal DD, the approach from below has the exact corner lim⁡s↑ln⁡44−FD(s)ln⁡4−s=4e.\lim_{s\uparrow\ln4}\frac{4-F_{\mathcal D}(s)}{\ln4-s}=\frac4{\sqrt e}.lims↑ln4​ln4−s4−FD​(s)​=e​4​. A small vacuum component changes the optimal coherent peak and produces the sharp slope. Separate all-state energy bounds show that vanishing coherent overlap requires at least one photon asymptotically, or two photons for pure states. Explicit sources attain these limiting energy costs, and exact-entropy constructions include their additional photon costs. The proofs retain arbitrary photon tails and constants uniform in the mode number. The earliest-threshold and corner results apply to the source classes stated above. The integrated article contains the complete proofs and two figure groups.