Sharp entropy thresholds in multimode coherent-state approximation

Zixuan He

Published Oct 31, 2026• 0 citations

Abstract

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 $m$-mode state $\rho$, write $Q_\rho(\alpha)=\langle\alpha|\rho|\alpha\rangle$, with $d\mu_m=\pi^{-m}d^{2m}\alpha$, and define \[M(\rho)=2^m\int Q_\rho(\alpha)^2\,d\mu_m(\alpha),\qquad s=-\ln M(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1.\] Here $s$ is the second-order Wehrl entropy excess and $D$ is the full trace-norm distance, whose maximum is 2. The collision quantity $M$ 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 $\mathcal X$, with $F_{\mathcal X}(s)=\sup_{\rho\in\mathcal X,\,-\ln M(\rho)=s}D(\rho)^2$ and the supremum over all finite mode numbers, \[F_{\mathcal X}(s)=4\quad\Longleftrightarrow\quad s\ge\ln4.\] The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and $M\to1/4$ 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 $\mathcal D$, the approach from below has the exact corner \[\lim_{s\uparrow\ln4}\frac{4-F_{\mathcal D}(s)}{\ln4-s}=\frac4{\sqrt e}.\] 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.

DOI: 10.5281/zenodo.23828147

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Sharp entropy thresholds in multimode coherent-state approximation

Zixuan He

Published Oct 31, 2026• 0 citations

Publisher

Abstract

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 (m)-mode state (\rho), write (Q_\rho(\alpha)=\langle\alpha|\rho|\alpha\rangle), with (d\mu_m=\pi^{-m}d^{2m}\alpha), and define [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.] Here (s) is the second-order Wehrl entropy excess and (D) is the full trace-norm distance, whose maximum is 2. The collision quantity (M) 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 (\mathcal X), with (F{\mathcal X}(s)=\sup_{\rho\in\mathcal X,,-\ln M(\rho)=s}D(\rho)^2) and the supremum over all finite mode numbers, [F_{\mathcal X}(s)=4\quad\Longleftrightarrow\quad s\ge\ln4.] The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and (M\to1/4) 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 (\mathcal D), the approach from below has the exact corner [\lim_{s\uparrow\ln4}\frac{4-F_{\mathcal D}(s)}{\ln4-s}=\frac4{\sqrt e}.] 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.

DOI: 10.5281/zenodo.23828147

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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

From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability

Zixuan He

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 $p>0$ and every density operator on any finite number of modes, $D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,$ where $\delta_p$ is the total Rényi–Wehrl deficit and $C_p$ 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, $D\asymp a+b$ , where $a$ is the population outside the reference vacuum and $b$ 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: $C_p^{\rm opt}\sim4/[p\log(1/p)]$ as $p\downarrow0$ . 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 $C_1=64/\log 2$ and $C_2=16$ . 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 $p\log(1/p)C_p^{\rm opt}\to4$ . 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.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 $p>0$ and every density operator on any finite number of modes, $$D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,$$ where $\delta_p$ is the total Rényi–Wehrl deficit and $C_p$ 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, $D\asymp a+b$, where $a$ is the population outside the reference vacuum and $b$ 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: $C_p^{\rm opt}\sim4/[p\log(1/p)]$ as $p\downarrow0$. 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 $C_1=64/\log 2$ and $C_2=16$. 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 $p\log(1/p)C_p^{\rm opt}\to4$. 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.

Oct 6, 20260 citations

Hidden quantum coherence at the Wigner entropy endpoint

Zixuan He

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.

Oct 26, 20260 citations

The active entropy photon-number inequality: Gaussian channel entropy minima without entanglement

Zixuan He

Abstract How much does a nearly minimal output entropy reveal about the quantum states entering an amplifier? We prove the active entropy photon-number inequality for two independent multimode inputs, each with arbitrary internal correlations, and characterise every equality case. Writing 𝒩n(ρ) = g−1(S(ρ)/n), where g(x) = (x + 1) log(x + 1) − x log x, the output obeys 𝒩n(C) ≥ G𝒩n(A) + (G − 1)[𝒩n(B) + 1]. Equality requires Gaussian inputs with one common symplectic shape, reflected on the creation-operator port. At fixed finite mode number and input energy, a power of the actual output gap bounds the full trace distance to a compatible Gaussian pair with the original means and entropies. For a pure one-mode signal and a specified Gaussian idler, a Wehrl-entropy bridge gives a square-root bound. The inequality also fixes the exact constrained output-entropy minimum of every finite block of thermal amplifiers. For unequal one-mode Gaussian channels, the optimisation over all entangled and non-Gaussian block inputs reduces to scalar entropy allocation, with a sharp threshold at which a second mode becomes active. These results determine both the optimal output and the input structure that can approach it, turning an entropy benchmark into a quantitative test of source models. Version 1.2 — 8 October 2026 This revision develops the input-rigidity theory: near saturation of the active EPnI now constrains both independent inputs jointly. Complete equality classification. For two independent finite-energy multimode inputs, equality is characterised by a common symplectic Gaussian shape, with reflection on the creation-operator port. Each input may contain arbitrary internal correlations. Quantitative stability for arbitrary inputs. At fixed finite mode number, gain and input-energy bound, a power-law estimate controls the full trace distance to one compatible Gaussian pair matching the original means and entropies. The theorem includes pure and nonfaithful states and states with infinite Fock support. A stronger one-mode bound. For a pure signal and a specified Gaussian idler, which may be mixed, a Wehrl-entropy bridge gives a square-root estimate and an explicit sufficient tolerance for the input error. Expanded proofs and figures. New appendices provide the complete arguments and explicit constants. A new vector figure connects the equality geometry with the quantitative input-error bound. The active EPnI, exact finite-block thermal-amplifier minima and exact scalar entropy allocation for heterogeneous Gaussian channels are retained. The 49-page PDF contains the complete manuscript, proofs, four figures and references.Abstract How much does a nearly minimal output entropy reveal about the quantum states entering an amplifier? We prove the active entropy photon-number inequality for two independent multimode inputs, each with arbitrary internal correlations, and characterise every equality case. Writing 𝒩n(ρ) = g−1(S(ρ)/n), where g(x) = (x + 1) log(x + 1) − x log x, the output obeys 𝒩n(C) ≥ G𝒩n(A) + (G − 1)[𝒩n(B) + 1]. Equality requires Gaussian inputs with one common symplectic shape, reflected on the creation-operator port. At fixed finite mode number and input energy, a power of the actual output gap bounds the full trace distance to a compatible Gaussian pair with the original means and entropies. For a pure one-mode signal and a specified Gaussian idler, a Wehrl-entropy bridge gives a square-root bound. The inequality also fixes the exact constrained output-entropy minimum of every finite block of thermal amplifiers. For unequal one-mode Gaussian channels, the optimisation over all entangled and non-Gaussian block inputs reduces to scalar entropy allocation, with a sharp threshold at which a second mode becomes active. These results determine both the optimal output and the input structure that can approach it, turning an entropy benchmark into a quantitative test of source models. Version 1.2 — 8 October 2026 This revision develops the input-rigidity theory: near saturation of the active EPnI now constrains both independent inputs jointly. Complete equality classification. For two independent finite-energy multimode inputs, equality is characterised by a common symplectic Gaussian shape, with reflection on the creation-operator port. Each input may contain arbitrary internal correlations. Quantitative stability for arbitrary inputs. At fixed finite mode number, gain and input-energy bound, a power-law estimate controls the full trace distance to one compatible Gaussian pair matching the original means and entropies. The theorem includes pure and nonfaithful states and states with infinite Fock support. A stronger one-mode bound. For a pure signal and a specified Gaussian idler, which may be mixed, a Wehrl-entropy bridge gives a square-root estimate and an explicit sufficient tolerance for the input error. Expanded proofs and figures. New appendices provide the complete arguments and explicit constants. A new vector figure connects the equality geometry with the quantitative input-error bound. The active EPnI, exact finite-block thermal-amplifier minima and exact scalar entropy allocation for heterogeneous Gaussian channels are retained. The 49-page PDF contains the complete manuscript, proofs, four figures and references.

Oct 7, 20260 citations

Sharp entropy bounds from thermal local records and photon fluctuations

Zixuan He

When an optical source reproduces the local records of a thermal state, how much microscopic mixedness do those records actually guarantee? This paper gives explicit entropy bounds and a sharp large-budget law for that inference when the source's photon fluctuations are bounded. The setting is one oscillator with $[Q,P]=i$ and $N=(Q^2+P^2-1)/2$ . Fix a mean photon number $E>0$ , and put $v=E+1/2$ and $A=v-1/(4v)$ . The complete position density, probability current and local kinetic density are required to be $n_v(q)=\frac{e^{-q^2/(2v)}}{\sqrt{2\pi v}},\qquad j_v(q)=0,\qquad k_v(q)=\left(\frac{q^2}{4v^2}+A\right)n_v(q).$ These are exact functions, with no factor $1/2$ in the definition of $k_v$ . They force mean photons $E$ . The additional resource is the actual expectation $\operatorname{Tr}(N^2\rho)\le M$ of the same source. At fixed mean, this is also a photon-number-variance cap. For every feasible finite $M$ , every normal state compatible with these records satisfies $S(\rho)\ge h_2\!\left(\min\{\eta_E(M),1/2\}\right)>0.$ $\eta_E(M)=\left[\frac{A}{\sqrt M+\sqrt{M+A(A+B)}}\right]^2,\qquad B=1-\frac1{2v}.$ Here $S(\rho)=-\operatorname{Tr}(\rho\log\rho)$ , $h_2(t)=-t\log t-(1-t)\log(1-t)$ , and logarithms are natural. The bound applies to all compatible normal states, with no Gaussian or finite-rank restriction. Let $S_{\min}(E,M)$ be the entropy infimum over that full compatible class. As $M\to\infty$ at fixed $E>0$ , $S_{\min}(E,M)=\frac{c(E)\log M+c(E)[1-\log c(E)]}{M}+O_E\!\left(\frac{\log M}{M^2}\right).$ The coefficient is $c(E)=E^2(E+1)^2/(2E+1)^2$ . A positive mixture of two pure Gaussian states preserves all three complete records and has an exactly computed photon-square cost. It supplies the matching upper bound, fixing both the leading coefficient and the next term. The same asymptotic law therefore holds within actual probability mixtures of pure Gaussian states. Finite-budget results are explicit bounds; the sharp equality is the stated fixed- $E$ , large- $M$ expansion. Without a photon-square cap, the exact compatible entropy range is $(0,g(E)]$ , where $g(E)=(E+1)\log(E+1)-E\log E$ . The thermal state uniquely attains the upper endpoint, while compatible Gaussian mixtures approach zero entropy. This separates a maximum-entropy model from the minimum entropy forced by the data. The proof uses established local Gram relations, projects onto an actual largest spectral component, and charges its positive spectral tail to the same source's $N^2$ budget. This connection retains the column correlations needed for the lower bound. The analysis concerns one mode and ideal complete records; finite-precision or sampled records require additional error control. The integrated 13-page PDF contains two figures, complete proofs and references.When an optical source reproduces the local records of a thermal state, how much microscopic mixedness do those records actually guarantee? This paper gives explicit entropy bounds and a sharp large-budget law for that inference when the source's photon fluctuations are bounded. The setting is one oscillator with $[Q,P]=i$ and $N=(Q^2+P^2-1)/2$. Fix a mean photon number $E>0$, and put $v=E+1/2$ and $A=v-1/(4v)$. The complete position density, probability current and local kinetic density are required to be \[n_v(q)=\frac{e^{-q^2/(2v)}}{\sqrt{2\pi v}},\qquad j_v(q)=0,\qquad k_v(q)=\left(\frac{q^2}{4v^2}+A\right)n_v(q).\] These are exact functions, with no factor $1/2$ in the definition of $k_v$. They force mean photons $E$. The additional resource is the actual expectation $\operatorname{Tr}(N^2\rho)\le M$ of the same source. At fixed mean, this is also a photon-number-variance cap. For every feasible finite $M$, every normal state compatible with these records satisfies \[S(\rho)\ge h_2\!\left(\min\{\eta_E(M),1/2\}\right)>0.\] \[\eta_E(M)=\left[\frac{A}{\sqrt M+\sqrt{M+A(A+B)}}\right]^2,\qquad B=1-\frac1{2v}.\] Here $S(\rho)=-\operatorname{Tr}(\rho\log\rho)$, $h_2(t)=-t\log t-(1-t)\log(1-t)$, and logarithms are natural. The bound applies to all compatible normal states, with no Gaussian or finite-rank restriction. Let $S_{\min}(E,M)$ be the entropy infimum over that full compatible class. As $M\to\infty$ at fixed $E>0$, \[S_{\min}(E,M)=\frac{c(E)\log M+c(E)[1-\log c(E)]}{M}+O_E\!\left(\frac{\log M}{M^2}\right).\] The coefficient is $c(E)=E^2(E+1)^2/(2E+1)^2$. A positive mixture of two pure Gaussian states preserves all three complete records and has an exactly computed photon-square cost. It supplies the matching upper bound, fixing both the leading coefficient and the next term. The same asymptotic law therefore holds within actual probability mixtures of pure Gaussian states. Finite-budget results are explicit bounds; the sharp equality is the stated fixed-$E$, large-$M$ expansion. Without a photon-square cap, the exact compatible entropy range is $(0,g(E)]$, where $g(E)=(E+1)\log(E+1)-E\log E$. The thermal state uniquely attains the upper endpoint, while compatible Gaussian mixtures approach zero entropy. This separates a maximum-entropy model from the minimum entropy forced by the data. The proof uses established local Gram relations, projects onto an actual largest spectral component, and charges its positive spectral tail to the same source's $N^2$ budget. This connection retains the column correlations needed for the lower bound. The analysis concerns one mode and ideal complete records; finite-precision or sampled records require additional error control. The integrated 13-page PDF contains two figures, complete proofs and references.

Oct 29, 20260 citations

Brightness limits in quantum channel certification with coherent light

Zixuan He

How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least $\varepsilon$ away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget $E_s$ . This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys $\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p$ under every channel in the testing promise. The probe budget $E_p$ is distinct from the signal budget $E_s$ defining the certification guarantee. Optimal mean-energy query laws. For fixed positive $E_s,E_p$ , fixed finite mode number $m$ , and both testing errors at most $\delta$ , the minimax number of calls satisfies $T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),$ $T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.$ The bounds hold for sufficiently small $\varepsilon$ , uniformly over $0<\delta\le1/4$ . Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as $t\log t$ for one mode and as $t^m$ for fixed $m\ge2$ . A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which $\log T_{\mathrm G}=\Theta(\varepsilon^{-2})$ at fixed confidence. A fixed brightness cap gives $\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))$ . The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least $\varepsilon$ away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget $E_s$. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys \[\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p\] under every channel in the testing promise. The probe budget $E_p$ is distinct from the signal budget $E_s$ defining the certification guarantee. Optimal mean-energy query laws. For fixed positive $E_s,E_p$, fixed finite mode number $m$, and both testing errors at most $\delta$, the minimax number of calls satisfies \[T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),\] \[T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.\] The bounds hold for sufficiently small $\varepsilon$, uniformly over $0<\delta\le1/4$. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as $t\log t$ for one mode and as $t^m$ for fixed $m\ge2$. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which $\log T_{\mathrm G}=\Theta(\varepsilon^{-2})$ at fixed confidence. A fixed brightness cap gives $\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))$. The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.

Oct 25, 20260 citations

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 (m)-mode state (\rho), write (Q_\rho(\alpha)=\langle\alpha|\rho|\alpha\rangle), with (d\mu_m=\pi^{-m}d^{2m}\alpha), and define [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.] Here (s) is the second-order Wehrl entropy excess and (D) is the full trace-norm distance, whose maximum is 2. The collision quantity (M) 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 (\mathcal X), with (F{\mathcal X}(s)=\sup_{\rho\in\mathcal X,,-\ln M(\rho)=s}D(\rho)^2) and the supremum over all finite mode numbers, [F_{\mathcal X}(s)=4\quad\Longleftrightarrow\quad s\ge\ln4.] The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and (M\to1/4) 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 (\mathcal D), the approach from below has the exact corner [\lim_{s\uparrow\ln4}\frac{4-F_{\mathcal D}(s)}{\ln4-s}=\frac4{\sqrt e}.] 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.

\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.

p>0

and every density operator on any finite number of modes,

D(\rho)^2\le C_p\delta_p(\rho),\qquad D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,

where

\delta_p

is the total Rényi–Wehrl deficit and

C_p

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\asymp a+b

, where

a

is the population outside the reference vacuum and

b

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:

C_p^{\rm opt}\sim4/[p\log(1/p)]

p\downarrow0

. 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

C_1=64/\log 2

and

C_2=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

p\log(1/p)C_p^{\rm opt}\to4

. 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.

[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.

[Q,P]=i

and

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

. Fix a mean photon number

E>0

, and put

v=E+1/2

and

A=v-1/(4v)

. The complete position density, probability current and local kinetic density are required to be

n_v(q)=\frac{e^{-q^2/(2v)}}{\sqrt{2\pi v}},\qquad j_v(q)=0,\qquad k_v(q)=\left(\frac{q^2}{4v^2}+A\right)n_v(q).

These are exact functions, with no factor

1/2

in the definition of

k_v

. They force mean photons

E

. The additional resource is the actual expectation

\operatorname{Tr}(N^2\rho)\le M

of the same source. At fixed mean, this is also a photon-number-variance cap. For every feasible finite

M

, every normal state compatible with these records satisfies

S(\rho)\ge h_2\!\left(\min\{\eta_E(M),1/2\}\right)>0.

\eta_E(M)=\left[\frac{A}{\sqrt M+\sqrt{M+A(A+B)}}\right]^2,\qquad B=1-\frac1{2v}.

Here

S(\rho)=-\operatorname{Tr}(\rho\log\rho)

h_2(t)=-t\log t-(1-t)\log(1-t)

, and logarithms are natural. The bound applies to all compatible normal states, with no Gaussian or finite-rank restriction. Let

S_{\min}(E,M)

be the entropy infimum over that full compatible class. As

M\to\infty

at fixed

E>0

S_{\min}(E,M)=\frac{c(E)\log M+c(E)[1-\log c(E)]}{M}+O_E\!\left(\frac{\log M}{M^2}\right).

The coefficient is

c(E)=E^2(E+1)^2/(2E+1)^2

. A positive mixture of two pure Gaussian states preserves all three complete records and has an exactly computed photon-square cost. It supplies the matching upper bound, fixing both the leading coefficient and the next term. The same asymptotic law therefore holds within actual probability mixtures of pure Gaussian states. Finite-budget results are explicit bounds; the sharp equality is the stated fixed-

E

, large-

M

expansion. Without a photon-square cap, the exact compatible entropy range is

(0,g(E)]

, where

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

. The thermal state uniquely attains the upper endpoint, while compatible Gaussian mixtures approach zero entropy. This separates a maximum-entropy model from the minimum entropy forced by the data. The proof uses established local Gram relations, projects onto an actual largest spectral component, and charges its positive spectral tail to the same source's

N^2

budget. This connection retains the column correlations needed for the lower bound. The analysis concerns one mode and ideal complete records; finite-precision or sampled records require additional error control. The integrated 13-page PDF contains two figures, complete proofs and references.

\varepsilon

away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget

E_s

. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys

\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p

under every channel in the testing promise. The probe budget

E_p

is distinct from the signal budget

E_s

defining the certification guarantee. Optimal mean-energy query laws. For fixed positive

E_s,E_p

, fixed finite mode number

m

, and both testing errors at most

\delta

, the minimax number of calls satisfies

T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),

T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.

The bounds hold for sufficiently small

\varepsilon

, uniformly over

0<\delta\le1/4

. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as

t\log t

for one mode and as

t^m

for fixed

m\ge2

. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which

\log T_{\mathrm G}=\Theta(\varepsilon^{-2})

at fixed confidence. A fixed brightness cap gives

\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))

. The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.