Distinct stability laws for Wigner entropy and non-Gaussian structure

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 1.7 — revised 25 October 2026. The 136-page integrated PDF contains the article, figures, Methods, complete supplementary proofs and references. This version strengthens the relation between entropy recovery, source structure and measurable response while retaining the full energy and photon-tail accounting.

What version 1.7 adds

Collective photon-sector constructions improve the achievable activation exponent under a fixed total-energy budget. Their input von Neumann entropy can tend to zero. Two legal sources with identical complete joint photon-number distributions and the same total energy can nevertheless have opposite output entropy responses. Thus even complete photon-counting statistics do not determine the finite-order entropy response of a growing-mode source.

A separate forward-response theorem concerns fixed mode number and bounded total energy. For an input with a near-maximal Wigner peak, counting statistics at a qualified center predict the global peak after pure loss, with error linear in the peak deficit. Explicit finite-sample and spatial-calibration bounds account for statistical and centering uncertainty. This qualified prediction regime is distinct from the growing-mode entropy comparison.

On the Shannon and higher-order recovery boundary below order two, proximity to equality controls distance to one input coherent state independently of mode number. Near order two, the optimal stability constant grows inversely with the distance to that order. The recovery, activation, structural-survival and resource-cost results are detailed below.

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,

\[\begin{gathered}E_m=\operatorname{Tr}(N_m\rho_m)\to0,\\[6pt] \tfrac12\|\rho_m-\nu_m\|_1\to0,\\[6pt] d_{\mathcal F_m}(\rho_m)\to0,\\[6pt] \Delta_{1,m}(\rho_m)\to+\infty.\end{gathered}\]

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. The exact activation exponent above recovery

The loss boundary tells us when an entropy advantage is allowed. The new activation theorem determines how weak even the best possible advantage must be just above that boundary. Fix a Rényi order \(1<p<2\), let \(r=2\eta-p\downarrow0\), and let the mode number satisfy \(mr\to0\). Define

\[\Delta_{\max}(p,r;m,E)=\sup_{\rho\in\mathcal P_{m,E}}\Delta_{p,m}(\mathcal L_{(p+r)/2}^{\otimes m}\rho).\]

Then

\[\begin{gathered}\Delta_{\max}=\exp\!\left[-\frac{A_*(p)+o(1)}{r}\right],\\[6pt] A_*(p)=\kappa p\ln\frac1{p-1},\\[6pt] \kappa=\ln(2e\kappa)>1,\quad\kappa\simeq2.678346990.\end{gathered}\]

This exact leading exponent holds under every fixed positive total-energy budget and without a common energy cap. The upper bound covers mixed and correlated Wigner-positive inputs with infinite photon tails; a physical thermally repaired source gives the matching exponent. For unequal losses, the same law uses \(r=\max_j(2\eta_j-p)_+\) when \(mr\to0\).

Increasing the energy budget alone therefore leaves the leading activation exponent unchanged in this regime. The matching source uses coherent Fock level \(n=\Theta(r^{-1})\) and vanishing occupation: low mean energy, preparation complexity and certification cost are different resources. Each point of the activation curve describes a fixed-order limit; its quantifiers hold the order fixed.

4. Source-independent finite-budget exclusions

A positive-input profile bounds the normalized output Wigner \(p\)-norm across unequal losses. With \(\beta_j=2\eta_j\) and \(\mathcal J\) equal to the output Wigner \(p\)-power integral divided by its vacuum value,

\[\log\mathcal J_{p,\boldsymbol\beta}(\rho)\le\sum_{j=1}^m\mathcal B_p(\beta_j).\]

The explicit profile vanishes throughout recovery, stays finite at zero loss, and improves on the unrestricted quantum norm rate above the boundary. A complementary positive-filter estimate incorporates the total energy of the complete input state, including its photon tail. These bounds let a researcher exclude a desired entropy advantage before choosing a source.

For fixed \(1<p<2\), \(E>0\) and target deficit \(h>0\), let \(M_{\rm crit}(p,r;E,h)\) be the least mode number that can attain that target. The energy-filter bound gives

\[\begin{gathered}\liminf_{r\downarrow0}\frac{r^2 M_{\rm crit}(p,r;E,h)}{\ln(1/r)}\ge C_{\rm new}(p),\\[6pt] C_{\rm new}(p)=\frac{p^3(2-p)[p+(p-1)(2-p)]}{(1+\sqrt{3-p})^2}.\end{gathered}\]

This is a necessary mode-budget bound. Equivalently, for every fixed \(c<C_{\rm new}(p)\), the optimal deficit vanishes when \(mr^2\le c\ln(1/r)\). This necessary bound and the constructive sufficient fronts have different scopes.

At \(p=1.9\), total energy \(E=1\) and transmissivity \(\eta=0.9505\), the certified deficit upper bounds are \(0.01542\) nat for \(m=10^4\) and \(0.06942\) nat for \(m=10^5\). Both exclude a \(0.1\)-nat target for every allowed input. At \(10^4\) modes, the new profile improves the earlier \(0.03268\)-nat bound. These finite-budget results apply independently of the small-\(mr\) activation theorem: here \(r=0.001\) and the two examples have \(mr=10\) and \(100\).

5. 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.

6. 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 universal source-request lower bound for protocols that attain the target. This is a necessary cost bound, not an achievability statement. 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.

7. 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

\[\begin{gathered}h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\\[6pt] \log C_A(s)=\Theta(s^{-1/3}),\\[6pt] \log C_B(s)=\Theta(s^{-1/2}).\end{gathered}\]

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.

8. 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

\[\begin{gathered}s(\sigma)\le A_Kq+\tau_K,\\[6pt] A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.\end{gathered}\]

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.

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.

9. 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,

\[\begin{gathered}D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\\[6pt] T_h(\delta,E)=\Theta_{E,h}(L/\delta),\end{gathered}\]

where \(T_h\) is the first mode number allowing a fixed positive deficit \(h\). Beyond a sufficiently large fixed multiple of \(L/\delta\), \(D_{m,E}(2-\delta)=\Theta_E(m\delta)\). These estimates determine the near-order-two crossing scale up to constants. At every fixed mode number, an energy-unrestricted uniform 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 common positive gap excludes sequences approaching the purity-based floor at fixed mode number, strengthening pointwise nonattainment. It imposes no energy or covariance bound; its constant depends on the mode number and is not claimed optimal.

The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With ordering coordinate \(\zeta\) and \(\mathcal C_{\zeta,k}\) denoting the ordering-weighted norm of photon-number-difference block \(k\),

\[\begin{gathered}\mathcal C_{\zeta,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(\zeta),k}(\rho),\\[6pt] T_\eta(\zeta)=1-\frac{1-\zeta}{\eta}.\end{gathered}\]

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.

Source families and tasks. The finite single-mode preparation model supplies loss-dependent measurement predictions; the distributed thermally repaired three-photon family supplies structural survival, retention and conversion-cost lower bounds; the separate entropy-matched two-mode quartic families show different conversion-cost exponents at exactly equal entropy. Each result retains its own source and resource assumptions.

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.