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arXiv:2603.19024

Exact Law of Quantum Reversibility under Gaussian Pure Loss

Ammar Fayad

quant-phmath-phphysics.opticsSubmitted Mar 19, 2026• Announced Mar 20, 2026• 0 citations
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Abstract

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics -- the canonical model of continuous-variable decoherence in optical attenuation channels, squeezed-light interferometric sensing, and superconducting bosonic architectures -- complete positivity, the requirement that the dynamics remain physical even for systems entangled with an ancilla, creates an exact phase boundary at which the minimum reverse cost vanishes, fixes the reverse-noise budget on both sides, and makes pure nonclassical targets dynamically singular. The minimum reverse cost vanishes exactly at a critical squeezing-to-thermal ratio and is strictly positive away from it, with a sharp asymmetry: below the boundary, standard reverse prescriptions such as the fixed-diffusion Bayes reverse remain feasible at mild cost; above it, these prescriptions become infeasible, the covariance-aligned generator remains CP-feasible and uniquely optimal, and the cost can be severe. The optimal reverse noise is locked to the state's own fluctuation geometry and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact reversal of a pure quantum state is dynamically unattainable. These results establish an exact law of quantum reversibility in the canonical pure-loss setting and provide a sharp benchmark for broader theories of quantum reverse diffusion.

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arXiv admin note: text overlap with arXiv:2603.06488

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Score Reversal Is Not Free for Quantum Diffusion Models

Ammar Fayad

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.

quant-pharXiv:2603.06488•Mar 9, 2026•0 citations

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.

•Sep 23, 2026•0 citations

Generating quantum ensembles via reverse-time quantum diffusions

Maël Bompais, Mădălin Guţă, Juan P. Garrahan

We establish a reverse-time denoising theory for quantum diffusions of continuously measured quantum systems. Starting from the stochastic Schrödinger equation of a forward noising dynamics, we derive the exact reverse-time dynamics for quantum trajectories, whose law coincides with the time-reversal of the original process. We prove that the denoising dynamics is a physically admissible quantum diffusion, with the same measurement-induced noise but a state-dependent feedback Hamiltonian, a direct analogue of the "score function" of generative classical diffusion models. This provides a principled framework for converting samples of a simple distribution into those of a more complex ensemble of quantum states. We show how the denoising dynamics can be directly learnt from forward trajectory data, and how to exploit purification to initialise the denoising process.

quant-pharXiv:2606.03848•Jun 3, 2026•0 citations

Classical capacity of squeezed-noise channels: optimal photon allocation without entangled encoding

Zixuan He

Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let \(\Phi=\bigotimes_{j=1}^{d}\Phi_j\), let \(E\ge0\) be the total average input photon number per block, and let \(C_j(E_j)\) be the explicitly attained one-channel rate specified in the paper. Then \[ C(\Phi,E)=\chi_\Phi(E)=\max_{E_j\ge0,\,\sum_jE_j\le E}\sum_{j=1}^{d} C_j(E_j), \] \[ \chi_{\Phi^{\otimes n}}(nE)=nC(\Phi,E)\qquad(n\ge1). \] The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by \(\ln2\) gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance \(I/2\), each channel acts as \(V\mapsto\eta V+\operatorname{diag}(y_q,y_p)\), where \(0<\eta<1\), \(y_q\ge y_p>0\), and \(y_qy_p\ge(1-\eta)^2/4\). Define \[ u_e=\frac12\sqrt{\frac{y_q}{y_p}},\qquad E_{\mathrm{th}}=u_e-\frac12+\frac{y_q-y_p}{2\eta}. \] Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing \(g(x)=(x+1)\ln(x+1)-x\ln x\), the high-energy rate is \[ C_j(E)=g\!\left(\eta E+\frac{\eta+y_q+y_p-1}{2}\right)-g\!\left(\frac{\eta}{2}+\sqrt{y_qy_p}-\frac12\right),\quad E\ge E_{\mathrm{th}}. \] Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: \[ C_j'(E_j^*)=\lambda\quad(E_j^*>0),\qquad C_j'(0+)\le\lambda\quad(E_j^*=0). \] A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately \(E=0.436\) photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.

•Oct 13, 2026•0 citations

Reverse quantum state diffusion from differential geometry

Dinh-Long Vu, Patrick Rebentrost

Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.

quant-pharXiv:2609.39861•Oct 1, 2026•0 citations

Approximate reversibility in the context of entropy gain, information gain, and complete positivity

Francesco Buscemi, Siddhartha Das, Mark M. Wilde

There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, entropic disturbance, and complete positivity of open quantum systems dynamics. Our first result is a remainder term for the entropy gain of a quantum channel. This result implies that a small increase in entropy under the action of a subunital channel is a witness to the fact that the channel's adjoint can be used as a recovery map to undo the action of the original channel. We apply this result to pure-loss, quantum-limited amplifier, and phase-insensitive quantum Gaussian channels, showing how a quantum-limited amplifier can serve as a recovery from a pure-loss channel and vice versa. Our second result regards the information gain of a quantum measurement, both without and with quantum side information. We find here that a small information gain implies that it is possible to undo the action of the original measurement if it is efficient. The result also has operational ramifications for the information-theoretic tasks known as measurement compression without and with quantum side information. Our third result shows that the loss of Holevo information caused by the action of a noisy channel on an input ensemble of quantum states is small if and only if the noise can be approximately corrected on average. We finally establish that the reduced dynamics of a system-environment interaction are approximately completely positive and trace preserving if and only if the data processing inequality holds approximately.

arXiv:1601.01207•Physical Review A•Jun 13, 2016•60 citations
Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics -- the canonical model of continuous-variable decoherence in optical attenuation channels, squeezed-light interferometric sensing, and superconducting bosonic architectures -- complete positivity, the requirement that the dynamics remain physical even for systems entangled with an ancilla, creates an exact phase boundary at which the minimum reverse cost vanishes, fixes the reverse-noise budget on both sides, and makes pure nonclassical targets dynamically singular. The minimum reverse cost vanishes exactly at a critical squeezing-to-thermal ratio and is strictly positive away from it, with a sharp asymmetry: below the boundary, standard reverse prescriptions such as the fixed-diffusion Bayes reverse remain feasible at mild cost; above it, these prescriptions become infeasible, the covariance-aligned generator remains CP-feasible and uniquely optimal, and the cost can be severe. The optimal reverse noise is locked to the state's own fluctuation geometry and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every t>0t>0t>0, but the optimum diverges as 2/t2/t2/t: exact reversal of a pure quantum state is dynamically unattainable. These results establish an exact law of quantum reversibility in the canonical pure-loss setting and provide a sharp benchmark for broader theories of quantum reverse diffusion.
Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every t>0t>0t>0, but the optimum diverges as 2/t2/t2/t: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.
Loss can restore the vacuum entropy bound without erasing non-Gaussian structure. This paper determines when an entropy statement justifies a coherent-state approximation, when non-Gaussian structure survives optical loss, and what mode and source-request budgets are needed to use that structure. The results give quantum-optics researchers separate criteria for entropy recovery, state approximation, Gaussian concentration and conversion to a specified output. Version 1.5 — revised 10 October 2026. The 96-page PDF contains the main article, Methods, Supplementary Information and references. This revision connects the exact loss boundary to a mode-uniform near-equality certificate, finite-data validation, a conversion-cost lower bound for the same thermally repaired source, and an exact equal-entropy comparison of sources with different conversion-cost exponents. The central figures are updated accordingly. 1. Near-vacuum states can have an unbounded entropy deficit Write νm\nu_mνm​ for the mmm-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ)\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)Δα,m​(ρ)=Hα​(Wνm​​)−Hα​(Wρ​), using natural logarithms; H1H_1H1​ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr⁡(Nmρm)→0,12∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞.E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.Em​=Tr(Nm​ρm​)→0,21​∥ρm​−νm​∥1​→0,dFm​​(ρm​)→0,Δ1,m​(ρm​)→+∞. Here Fm\mathcal F_mFm​ is the trace-norm-closed convex hull of Gaussian states and dFmd_{\mathcal F_m}dFm​​ uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives −Em≤Δ1,m(L1/2⊗mρm)≤0-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0−Em​≤Δ1,m​(L1/2⊗m​ρm​)≤0, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η\etaη and any fixed positive total input-energy budget EEE, the sharp threshold is ηc(α)={1/2,0<α≤1,α/2,1<α<2.\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}ηc​(α)={1/2,α/2,​0<α≤1,1<α<2.​ With Pm,E={ρ:Wρ≥0, Tr⁡(Nmρ)≤E}\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}Pm,E​={ρ:Wρ​≥0, Tr(Nm​ρ)≤E}, sup⁡m≥1sup⁡ρ∈Pm,EΔα,m(Lη⊗mρ)={0,η≤ηc(α),+∞,η>ηc(α).\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}supm≥1​supρ∈Pm,E​​Δα,m​(Lη⊗m​ρ)={0,+∞,​η≤ηc​(α),η>ηc​(α).​ The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For α≥2\alpha\ge2α≥2, established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input ρ\rhoρ, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−m(1+ln⁡π)≥0.s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.s(ρ)=h(WL1/2⊗m​ρ​)−m(1+lnπ)≥0. The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields inf⁡a∥ρ−∣a⟩⟨a∣∥12≤64ln⁡2 s(ρ).\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).infa​∥ρ−∣a⟩⟨a∣∥12​≤ln264​s(ρ). The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most ϵ\epsilonϵ is guaranteed by s≤(ln⁡2)ϵ2/16s\le(\ln2)\epsilon^2/16s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥ln⁡216 dFm(L1/2⊗mρ)2.s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.s(ρ)≥16ln2​dFm​​(L1/2⊗m​ρ)2. Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation c/mc/mc/m has total input energy tending to 3c3c3c. Along the recovery boundary, its attenuated state obeys lim inf⁡m→∞dFm(ρ~m,ηc)≥cηc3exp⁡{−c[1−(1−ηc)3]}>0.\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.liminfm→∞​dFm​​(ρ​m,ηc​​)≥cηc3​exp{−c[1−(1−ηc​)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8g(c)=(c/8)e^{-7c/8}g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m)\Theta(m)Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>0p_0>0p0​>0. Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target τ∗=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25τ∗​=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25 within half trace distance 10−610^{-6}10−6. Let CmC_mCm​ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤10<c\le10<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound log⁡Cm≥1128(m/c)1/3−log⁡m−O(1).\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).logCm​≥1281​(m/c)1/3−logm−O(1). This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become h(WωA)=h(WωB)=2(1+ln⁡π)+s,log⁡CA(s)=Θ(s−1/3),log⁡CB(s)=Θ(s−1/2).h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).h(WωA​​)=h(WωB​​)=2(1+lnπ)+s,logCA​(s)=Θ(s−1/3),logCB​(s)=Θ(s−1/2). The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy EA,EB∼s/2E_A,E_B\sim s/2EA​,EB​∼s/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3)s_i(E)=2E-(35/8)E^2+O(E^3)si​(E)=2E−(35/8)E2+O(E3) and differ by sA(E)−sB(E)=15E3+O(E4)s_A(E)-s_B(E)=15E^3+O(E^4)sA​(E)−sB​(E)=15E3+O(E4). Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound Tr⁡[N1N>Kσa]≤τK\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_KTr[N1N>K​σa​]≤τK​, trusted half loss and all-mode on/off detection with efficiency at least ξmin⁡\xi_{\min}ξmin​, the click probability qqq gives s(σ)≤AKq+τK,AK=K1−(1−ξmin⁡/2)K.s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.s(σ)≤AK​q+τK​,AK​=1−(1−ξmin​/2)KK​. A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses K=4K=4K=4, τK=0\tau_K=0τK​=0, ξ=0.8\xi=0.8ξ=0.8 and N=3,517,336N=3{,}517{,}336N=3,517,336 independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most 0.050.050.05 with false-acceptance probability at most 0.0010.0010.001 and missed acceptance at most 0.0010.0010.001 on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at E=1E=1E=1, α=1.9\alpha=1.9α=1.9, η=0.9505\eta=0.9505η=0.9505 and m=105m=10^5m=105, the certified bound Δα<0.06942\Delta_\alpha<0.06942Δα​<0.06942 rules out a target deficit of 0.10.10.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97\eta=0.97η=0.97, negative deficit near η=0.85\eta=0.85η=0.85, and a Gaussian-hull witness above 4.4×10−34.4\times10^{-3}4.4×10−3 at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put α=2−δ\alpha=2-\deltaα=2−δ and L=ln⁡(1/δ)L=\ln(1/\delta)L=ln(1/δ). For the maximum deficit Dm,ED_{m,E}Dm,E​ at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ),D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),Dm,E​(2−δ)=Θm,E​(δ/L)for fixed m,Th​(δ,E)=ΘE,h​(L/δ), where ThT_hTh​ is the first mode number allowing a fixed positive deficit hhh. Beyond a sufficiently large fixed multiple of L/δL/\deltaL/δ, Dm,E(2−δ)=ΘE(mδ)D_{m,E}(2-\delta)=\Theta_E(m\delta)Dm,E​(2−δ)=ΘE​(mδ). The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds: h(Wρ)≥mln⁡(2π)+εmh(W_\rho)\ge m\ln(2\pi)+\varepsilon_mh(Wρ​)≥mln(2π)+εm​ with εm>0\varepsilon_m>0εm​>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k\mathcal C_{r,k}Cr,k​ denoting the ordering-weighted norm of photon-number-difference block kkk, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−1−rη.\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.Cr,k​(Lη​ρ)=CTη​(r),k​(ρ),Tη​(r)=1−η1−r​. For vacuum–nnn-photon coherence and 1<α<21<\alpha<21<α<2, C1−α/2(Lη∣n⟩⟨0∣)=(2ηα)n.\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.C1−α/2​(Lη​∣n⟩⟨0∣)=(α2η​)n. This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.
We establish a reverse-time denoising theory for quantum diffusions of continuously measured quantum systems. Starting from the stochastic Schrödinger equation of a forward noising dynamics, we derive the exact reverse-time dynamics for quantum trajectories, whose law coincides with the time-reversal of the original process. We prove that the denoising dynamics is a physically admissible quantum diffusion, with the same measurement-induced noise but a state-dependent feedback Hamiltonian, a direct analogue of the "score function" of generative classical diffusion models. This provides a principled framework for converting samples of a simple distribution into those of a more complex ensemble of quantum states. We show how the denoising dynamics can be directly learnt from forward trajectory data, and how to exploit purification to initialise the denoising process.
Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let Φ=⨂j=1dΦj\Phi=\bigotimes_{j=1}^{d}\Phi_jΦ=⨂j=1d​Φj​, let E≥0E\ge0E≥0 be the total average input photon number per block, and let Cj(Ej)C_j(E_j)Cj​(Ej​) be the explicitly attained one-channel rate specified in the paper. Then C(Φ,E)=χΦ(E)=max⁡Ej≥0, ∑jEj≤E∑j=1dCj(Ej), C(\Phi,E)=\chi_\Phi(E)=\max_{E_j\ge0,\,\sum_jE_j\le E}\sum_{j=1}^{d} C_j(E_j), C(Φ,E)=χΦ​(E)=maxEj​≥0,∑j​Ej​≤E​∑j=1d​Cj​(Ej​), χΦ⊗n(nE)=nC(Φ,E)(n≥1). \chi_{\Phi^{\otimes n}}(nE)=nC(\Phi,E)\qquad(n\ge1). χΦ⊗n​(nE)=nC(Φ,E)(n≥1). The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by ln⁡2\ln2ln2 gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance I/2I/2I/2, each channel acts as V↦ηV+diag⁡(yq,yp)V\mapsto\eta V+\operatorname{diag}(y_q,y_p)V↦ηV+diag(yq​,yp​), where 0<η<10<\eta<10<η<1, yq≥yp>0y_q\ge y_p>0yq​≥yp​>0, and yqyp≥(1−η)2/4y_qy_p\ge(1-\eta)^2/4yq​yp​≥(1−η)2/4. Define ue=12yqyp,Eth=ue−12+yq−yp2η. u_e=\frac12\sqrt{\frac{y_q}{y_p}},\qquad E_{\mathrm{th}}=u_e-\frac12+\frac{y_q-y_p}{2\eta}. ue​=21​yp​yq​​​,Eth​=ue​−21​+2ηyq​−yp​​. Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing g(x)=(x+1)ln⁡(x+1)−xln⁡xg(x)=(x+1)\ln(x+1)-x\ln xg(x)=(x+1)ln(x+1)−xlnx, the high-energy rate is Cj(E)=g ⁣(ηE+η+yq+yp−12)−g ⁣(η2+yqyp−12),E≥Eth. C_j(E)=g\!\left(\eta E+\frac{\eta+y_q+y_p-1}{2}\right)-g\!\left(\frac{\eta}{2}+\sqrt{y_qy_p}-\frac12\right),\quad E\ge E_{\mathrm{th}}. Cj​(E)=g(ηE+2η+yq​+yp​−1​)−g(2η​+yq​yp​​−21​),E≥Eth​. Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: Cj′(Ej∗)=λ(Ej∗>0),Cj′(0+)≤λ(Ej∗=0). C_j'(E_j^*)=\lambda\quad(E_j^*>0),\qquad C_j'(0+)\le\lambda\quad(E_j^*=0). Cj′​(Ej∗​)=λ(Ej∗​>0),Cj′​(0+)≤λ(Ej∗​=0). A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately E=0.436E=0.436E=0.436 photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.
Quantum state diffusion unravels a Lindblad master equation into stochastic trajectories of pure states. The ensemble of trajectories is described by a probability density on the manifold of pure states, which contains much more information than the density matrix, its first moment. We derive the time reversal of this diffusion. Writing the reversal of a general diffusion on a manifold in Stratonovich form, and passing through the manifold of pure states, we obtain a backward stochastic differential equation for the state vector, valid for any Hilbert space dimension and any Lindbladian. The backward stochastic equation provides a physical, nonlinear unravelling of the inverse Lindbladian that holds only for the ensemble from which it was constructed. Following score-based generative models, we then use the backward equation to generate a new ensemble close to the original. For the depolarizing channel the unravelling is a Brownian motion on complex projective space, so the uniform distribution of pure states is its stationary law in any dimension. Starting the backward equation from that prior, we bound the error of the resulting distribution compared to the original one in different settings: exact and learned scores, continuous and discrete time.
There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, entropic disturbance, and complete positivity of open quantum systems dynamics. Our first result is a remainder term for the entropy gain of a quantum channel. This result implies that a small increase in entropy under the action of a subunital channel is a witness to the fact that the channel's adjoint can be used as a recovery map to undo the action of the original channel. We apply this result to pure-loss, quantum-limited amplifier, and phase-insensitive quantum Gaussian channels, showing how a quantum-limited amplifier can serve as a recovery from a pure-loss channel and vice versa. Our second result regards the information gain of a quantum measurement, both without and with quantum side information. We find here that a small information gain implies that it is possible to undo the action of the original measurement if it is efficient. The result also has operational ramifications for the information-theoretic tasks known as measurement compression without and with quantum side information. Our third result shows that the loss of Holevo information caused by the action of a noisy channel on an input ensemble of quantum states is small if and only if the noise can be approximately corrected on average. We finally establish that the reduced dynamics of a system-environment interaction are approximately completely positive and trace preserving if and only if the data processing inequality holds approximately.