What can an imperfect destructive optical test guarantee about the light that remains unused? This article gives an explicit certificate of proximity to coherent states for the unmeasured members of an independent, possibly nonidentical batch. Its main optical contribution is a physical symmetrization whose error can be controlled by mean photon number alone, while retaining every photon-number sector. Cancel the optical error before estimating it Let \(P_e\) be the input-picture projector for vacuum in every relative output mode of a beam splitter at angle \(\pi/4+e\). The same port rotation acts on each paired internal mode. A fair random exchange of the complete effective inputs, after arm-specific passive transformations, replaces the event by \((P_e+P_{-e})/2\). For any normal two-arm state \(\Omega\) with total mean photons at most \(E\), and \(|e|\le\theta\le\pi/4\), \[\left|\operatorname{Tr}\Omega\left(\frac{P_e+P_{-e}}{2}-P_0\right)\right|\le\frac{1+\sqrt3}{2}E\theta^2.\] The exact projector curvature on the total \(k\)-photon sector is \(k+\sqrt{3k^2-2k}\). Its linear growth in \(k\) permits the infinite-tail return using a first energy moment, without a photon cutoff or a mode-count factor. The article also charges imperfect randomization and distinguishes passive mismatch from angular bias. A guarantee for the actual untested preparations Prepare \(B\) pairs in the product state \(\bigotimes_{i=1}^{B}(\rho_i\otimes\sigma_i)\): all \(2B\) preparations are independent, although their states may differ and may be mixed. Each pair obeys the total mean-photon bound \(E\). Select a uniform fixed-size sample of \(n<B\) pairs, independently of the batch and its potential test rates, including after conditioning on valid calibration information. Let \(U\) be the remaining indices and \[D(\rho)=\inf_\alpha\|\rho-|\alpha\rangle\langle\alpha|\|_1,\qquad \overline D_U^{\,2}=\frac{1}{2(B-n)}\sum_{i\in U}\bigl[D(\rho_i)^2+D(\sigma_i)^2\bigr].\] The norm is the full trace norm, with maximum distance two. For a predetermined zero-click protocol, observing no recorded clicks gives the frequentist certificate \[\overline D_U^{\,2}\le\min\left\{4,\frac4c\left[\delta+\frac\nu\eta+\frac{B\ln(1/\beta_{\rm sig})}{\eta n(B-n)}\right]\right\}.\] Here \(\delta\) bounds optical inflation of the passing probability, \(\eta>0\) is the retained detection-efficiency lower bound, and \(\nu\) is a separate absolute click-probability decrement. The coefficient is \(c=1-\exp(-4/\bar C)\), where \(\bar C\ge C_\star\) comes from the established sharp Wehrl theorem; the displayed examples use \(\bar C=4.114205\). Total failure probability is at most \(\beta_{\rm cal}+\beta_{\rm sig}\). The paper also supplies bounds for nonzero counts and a separately scoped, anytime-valid result for tested conditional streams. Resources, calibration and scope The source requirement is a product batch, not merely product structure conditional on an unobserved shared label. The conclusion controls average squared distance to the coherent-state set at the retained state plane; different preparations may have different coherent references. Detector bounds must cover all relevant modes and multiphoton inputs. Single-photon calibration estimates efficiencies within that trusted response model. Signal consumption is \(2B\) preparations, with \(2n\) tested and \(2(B-n)\) retained. Calibration is additional. A fully specified hypothetical single-mode example uses three million incident calibration photons and 20,000 signal preparations, with at most 3,200,000 mean incident photons in total. Source-generation attempts, control and storage costs are additional. The figures evaluate proved bounds and certificate floors; they do not report experimental measurements, demonstrated hardware performance or universal impossibility limits. The analysis builds on established coherent-state comparison, quantum sampling and the sharp Wehrl entropy-to-distance theorem. Its contribution is the full-Fock optical error control and its compatible return through calibration and counting to the actual unused product batch. The integrated 14-page article contains two figure groups, complete proofs and references.
What can an imperfect destructive optical test guarantee about the light that remains unused? This article gives an explicit certificate of proximity to coherent states for the unmeasured members of an independent, possibly nonidentical batch. Its main optical contribution is a physical symmetrization whose error can be controlled by mean photon number alone, while retaining every photon-number sector. Cancel the optical error before estimating it Let (P_e) be the input-picture projector for vacuum in every relative output mode of a beam splitter at angle (\pi/4+e). The same port rotation acts on each paired internal mode. A fair random exchange of the complete effective inputs, after arm-specific passive transformations, replaces the event by ((P_e+P_{-e})/2). For any normal two-arm state (\Omega) with total mean photons at most (E), and (|e|\le\theta\le\pi/4), [\left|\operatorname{Tr}\Omega\left(\frac{P_e+P_{-e}}{2}-P_0\right)\right|\le\frac{1+\sqrt3}{2}E\theta^2.] The exact projector curvature on the total (k)-photon sector is (k+\sqrt{3k^2-2k}). Its linear growth in (k) permits the infinite-tail return using a first energy moment, without a photon cutoff or a mode-count factor. The article also charges imperfect randomization and distinguishes passive mismatch from angular bias. A guarantee for the actual untested preparations Prepare (B) pairs in the product state (\bigotimes_{i=1}^{B}(\rho_i\otimes\sigma_i)): all (2B) preparations are independent, although their states may differ and may be mixed. Each pair obeys the total mean-photon bound (E). Select a uniform fixed-size sample of (n<B) pairs, independently of the batch and its potential test rates, including after conditioning on valid calibration information. Let (U) be the remaining indices and [D(\rho)=\inf_\alpha|\rho-|\alpha\rangle\langle\alpha||1,\qquad \overline D_U^{,2}=\frac{1}{2(B-n)}\sum{i\in U}\bigl[D(\rho_i)^2+D(\sigma_i)^2\bigr].] The norm is the full trace norm, with maximum distance two. For a predetermined zero-click protocol, observing no recorded clicks gives the frequentist certificate [\overline D_U^{,2}\le\min\left{4,\frac4c\left[\delta+\frac\nu\eta+\frac{B\ln(1/\beta_{\rm sig})}{\eta n(B-n)}\right]\right}.] Here (\delta) bounds optical inflation of the passing probability, (\eta>0) is the retained detection-efficiency lower bound, and (\nu) is a separate absolute click-probability decrement. The coefficient is (c=1-\exp(-4/\bar C)), where (\bar C\ge C_\star) comes from the established sharp Wehrl theorem; the displayed examples use (\bar C=4.114205). Total failure probability is at most (\beta_{\rm cal}+\beta_{\rm sig}). The paper also supplies bounds for nonzero counts and a separately scoped, anytime-valid result for tested conditional streams. Resources, calibration and scope The source requirement is a product batch, not merely product structure conditional on an unobserved shared label. The conclusion controls average squared distance to the coherent-state set at the retained state plane; different preparations may have different coherent references. Detector bounds must cover all relevant modes and multiphoton inputs. Single-photon calibration estimates efficiencies within that trusted response model. Signal consumption is (2B) preparations, with (2n) tested and (2(B-n)) retained. Calibration is additional. A fully specified hypothetical single-mode example uses three million incident calibration photons and 20,000 signal preparations, with at most 3,200,000 mean incident photons in total. Source-generation attempts, control and storage costs are additional. The figures evaluate proved bounds and certificate floors; they do not report experimental measurements, demonstrated hardware performance or universal impossibility limits. The analysis builds on established coherent-state comparison, quantum sampling and the sharp Wehrl entropy-to-distance theorem. Its contribution is the full-Fock optical error control and its compatible return through calibration and counting to the actual unused product batch. The integrated 14-page article contains two figure groups, complete proofs and references.
This work gives a resource law for engineering optical noise: how many photons must a single environment mode contain to prevent any signal from producing an output that is too spectrally concentrated? It provides an explicit optimal bath preparation, a matched Gaussian benchmark, and separately proved guarantees for correlated repeated use. Exact photon cost of a spectral noise guarantee A signal mixes with an independently prepared bath σ at a beam splitter of signal transmissivity η. The requirement is Φη,σ(ρ)≤cIfor every normal input state ρ,0<c<1. Equivalently, every rank-one output projection has probability at most c and the receiver's output min-entropy is at least −lnc. Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let d=⌈1/c⌉−1. Throughout 0<η≤1/⌈1/c⌉, the minimum mean photon numbers of the actual bath are Nall(c,η)=1−ηd−cd(d+1)/2.NG(c,η)=1−ηc−1−1. The first optimization includes all normal bath states and has a unique finite-support minimizer. The second includes all Gaussian density operators, with arbitrary squeezing and displacement; a centered thermal bath attains it. The coupling and spectral requirement are identical in both comparisons. At c=1/j, for every integer j≥2 and 0<η≤1/j, NG(1/j,η)=2Nall(1/j,η). Between reciprocal budgets the ratio varies. The cost counts mean photons in the physical bath, not laboratory preparation work or the energy of an auxiliary purification. A physical preparation and its complete single-use spectrum Writing r=1−dc, the attaining vacuum-output spectrum is λ∗=(d entriesc,…,c,r). It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly Smin(Φη,σ)=−dclnc−rlnr. The other positive-order Rényi minima are fixed by the same spectrum. Coherent inputs attain these minima. The proof identifies an actual optical dilation whose port antisymmetry enforces the all-input occupation bound. Exact examples also show that baths with identical energy and covariance can have different optimized entropy floors. What survives entangled repeated use For the same resource-optimal flat-output families of ranks j=2,3,4, separate estimates prove Sout≥mlnj for every finite number m of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with η along their specified families. Further results give sharp tensorized purity bounds for the entire three-level diagonal bath simplex and for every Fock environment at balanced coupling. The paper then treats a different, fixed bath, σ=∣1⟩⟨1∣, at every 0<η<1. With h(η)=−ηlnη−(1−η)ln(1−η), it proves: Exact Sout≥mh(η) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by 2π/q, q≥4. Coherence and entanglement within residue sectors and infinite tails remain allowed. The residue-zero restricted channel also has strong minimum-output-entropy additivity with arbitrary normal partner channels. The same exact entropy floor for up to three globally total-phase-invariant input modes, without a mean-energy limit. Sout≥Mh(η)−o(M) for physical-mode permutation-invariant inputs of finite mean energy density, with explicit errors and full infinite-tail control. For every fixed m and interior η, all outputs remain a uniformly positive trace distance from every state of rank below 2m, without any input-energy bound. This covers nonbalanced parameters where the older purity-based distance bound vanishes; the new positive constant is spectrally implicit. Interpretation and scope The input classes above are not nested, and an entropy rate, a finite-block entropy floor and a low-rank distance are distinct guarantees. The fixed single-photon bath is not generally the resource-optimal bath. Complete unrestricted nonbalanced single-photon entropy additivity and exact classical capacity remain open. A spectral noise guarantee constrains output mixing; it does not mean that all input information or reference correlations are erased. The article separates this issue through actual resolved-erasure qubit restrictions, antidegradability of the specified optimal reciprocal families, and a classical capacity bracket under a true signal-energy budget. The integrated 52-page article contains six scientific figures, the complete proofs, and the mathematical data needed to reconstruct its finite certificates. Logarithms and entropy values use natural units unless bits are explicitly stated.
Can a measurement that records only one global vacuum event recover an optical mode spectrum? This article identifies when increasing the number of simultaneously interfering copies supplies exactly the spectral information lost by a two-copy test. It provides a finite algebraic reconstruction without first determining the generating mode basis. Source and measurement. Consider a pure source with known total photon number N≥2 and at most m occupied orthogonal modes: ∣Ψ⟩=∑j=1mwjeiθj∣Nj⟩,wj≥0,∑jwj=1. Here ∣Nj⟩ has all photons in mode j, and the target is the unordered spectrum {wj} of the normalized one-particle reduced state. The source phases and a common passive mode basis may be unknown. This source class includes every pure two-photon state; for N≥3, the generalized NOON/GHZ form is an explicit preparation promise. Interfere ℓ complete, independent, identically prepared copies in a Fourier multiport, acting identically on every physical signal mode. The probability that all relative output modes are vacuum is Mℓ=ℓd∫QΨ(z)ℓdμd(z),QΨ(z)=∣⟨z∣Ψ⟩∣2,dμd=π−dd2dz, where d is the physical mode number. Spectator vacuum modes do not change this probability. Exact reconstruction and its mechanism. The probabilities M2,…,Mm, together with normalization, determine every mode weight, including zero and repeated weights. The new connection is from these optical probabilities to spectral power sums. A formal logarithm separates connected photon pairings; their strictly positive coefficients make the map triangular and invertible at every order. Standard Newton identities then recover the spectrum. For generic positive, distinct weights, the lower prefix through Mm−1 leaves even the largest weight undetermined. A concrete pair of three-photon sources has spectra (1/2,1/2,0) and (2/3,1/6,1/6). Both give M2=11/64, while their three-copy probabilities are 155/6561 and 183/6561. The reconstructed largest weight also determines the coherent peak and full trace-norm distance to a pure coherent state: q=N!NNe−Nmaxjwj,DC2=4(1−q). Physical resource accounting. Highest interference order is distinct from total sample consumption. With nℓ independent trials at order ℓ, the total number of input copies is Cin=∑ℓ=2mℓnℓ, and the incident photon count is NCin. A finite two-hypothesis counting example is included. The exact inverse is not a uniform noisy-spectrum confidence theorem, and the order-necessity result concerns this specified vacuum-event hierarchy rather than arbitrary optical measurements. Mode-resolved one-particle measurements provide a different route to the spectrum. The integrated 13-page article contains two figure groups, complete proofs and references. It distinguishes the all-order optical-to-spectral connection from established replica-moment formulas and classical power-sum inversion.
How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on m bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least 1/32 from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most D, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability a, or is erased with a classical flag. The quantity ka(D,m) is the smallest deterministic original-preparation cap achieving worst-case error at most 1/3. Failed preparations, discarded trials and reserved positions are all charged. A retained register of q qubits has D=2q. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is Θ(m). Along arbitrary growing-mode sequences, the precise criterion for improving that order is ka(D,m)=o(m)⟺am→∞andmaD→∞. The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when a=O(1/m), yet a survival probability tending to zero can still permit an advantage. Let A=max{1,m2/D}. At the fixed energy and separation above, constants c,C>0, independent of m,D,a, give cmin{m,A/a}≤ka(D,m)≤Cmin{m,A/a+min(a−2,m2/3)},0<a≤1. At a=0, the sample cost is Θ(m). For ideal memory the complete rate is k1(D,m)=Θ(min{m,max(1,m2/D)}). With loss, the bounds match whenever A≥min{a−1,am2/3}. For example, a=m−1/6 and D=⌈m11/6⌉ give Θ(m1/3) preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for D≥m2, a=m−1/2, the stated bounds are Ω(m1/2) and O(m2/3). Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order m2 labels, only a positive register of dimension m+1 is retained. The three-visit route requires D≥m+1 and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition N=aΨ+(1−a)Rτ, where Ψ is CPTP and Rτ(X)=Tr(X)τ, an exact flagged-erasure simulation gives kN(D,m)≥ka(D,m). The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction 1−q=O(1/m) therefore forces Θ(m) samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.
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 ε away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget Es. 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 ∑t=1TEK∥αt∥2≤TEp under every channel in the testing promise. The probe budget Ep is distinct from the signal budget Es defining the certification guarantee. Optimal mean-energy query laws. For fixed positive Es,Ep, fixed finite mode number m, and both testing errors at most δ, the minimax number of calls satisfies Tmean(1)(ε,δ)=ΘEs,Ep(ε−4logε1logδ1),Tmean(m)(ε,δ)=Θm,Es,Ep(ε−(2m+2)logδ1),m≥2. The bounds hold for sufficiently small ε, uniformly over 0<δ≤1/4. 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 tlogt for one mode and as tm for fixed m≥2. 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 logTG=Θ(ε−2) at fixed confidence. A fixed brightness cap gives logTcap=Θ(ε−2log(1/ε)). 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.
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 for the m-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ), using natural logarithms; H1 is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr(Nmρm)→0,21∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞. Here Fm is the trace-norm-closed convex hull of Gaussian states and 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, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η and any fixed positive total input-energy budget E, the sharp threshold is ηc(α)={1/2,α/2,0<α≤1,1<α<2. With Pm,E={ρ:Wρ≥0,Tr(Nmρ)≤E}, supm≥1supρ∈Pm,EΔα,m(Lη⊗mρ)={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, 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, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−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 infa∥ρ−∣a⟩⟨a∣∥12≤ln264s(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 is guaranteed by s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥16ln2dFm(L1/2⊗mρ)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 liminfm→∞dFm(ρm,ηc)≥cηc3exp{−c[1−(1−ηc)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>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 within half trace distance 10−6. Let Cm be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound 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+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/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3) and differ by 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, trusted half loss and all-mode on/off detection with efficiency at least ξmin, the click probability q gives s(σ)≤AKq+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=4, τK=0, ξ=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, α=1.9, η=0.9505 and m=105, the certified bound Δα<0.06942 rules out a target deficit of 0.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97, negative deficit near η=0.85, and a Gaussian-hull witness above 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 and L=ln(1/δ). For the maximum deficit Dm,E at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ), where Th is the first mode number allowing a fixed positive deficit h. Beyond a sufficiently large fixed multiple of L/δ, 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)+εm with εm>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k denoting the ordering-weighted norm of photon-number-difference block k, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−η1−r. For vacuum–n-photon coherence and 1<α<2, 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.
An unknown optical phase can hide quantum non-Gaussianity in every individual mode while leaving it certifiable in the joint light. This paper shows when Gaussian measurements can exclude the complete Gaussian convex hull without knowing the source's common phase, and gives explicit finite-sample tests. The results guide whether to preserve phase relations between emissions and how to process the resulting measurement records. 1. Certification against the full Gaussian model. Let Gn be the trace-norm-closed convex hull of all physical n-mode Gaussian states. For a fixed known ρ∈/G1, form the common-phase state Ωn=∫02π(UϕρUϕ†)⊗n2πdϕ,Uϕ=e−iϕN. Finite Gaussian measurements with a laboratory phase reference and bounded classical scores achieve arbitrarily small false-positive and missed-detection probabilities. The null includes arbitrary Gaussian entanglement and classical correlations, with unrestricted displacement, squeezing and energy. In particular, infσn∈Gn∥Ωn−σn∥1⟶2. The norm is the full, unhalved trace norm. An unavailable common phase therefore differs operationally from independently randomising the phase of each emission. 2. An exact coherence boundary and a joint-only resource. For the displaced two-Gaussian-packet family, centred by its actual mean before pure loss, the Gaussian-hull boundary is ρη,ν∈G1⟺ν≤νc,νc=exp[−2(1−η)(v−1/2)2v2R]. Here v>1/2 is the packet quadrature variance, R the momentum displacement, 0<τ,η<1 the packet weight and transmissivity, and 0≤ν≤1 the coherence visibility. The same membership boundary holds after common-phase averaging for every n≥3. At (v,R,τ,ν,η)=(20,2/9,9/50,1,1/2), each phase-averaged single mode is a Gaussian mixture, while the joint common-phase state lies outside Gn for n≥3. Independent phase averaging instead gives a Gaussian mixture at every block size. These states are separable and Wigner-positive: their difference is carried by phase relations across emissions. 3. The same two-packet source has an explicit finite certificate. Heterodyne phase training, a finite homodyne score and an eight-direction bounded tail test provide a complete certificate for the preceding parameter choice. With ideal Gaussian readout, the sufficient raw acquisition budget and certified errors are nraw=1015+2(4×1015)=9×1015,PFP<3.86×10−22,PFN<0.002068. This conservative budget establishes finite-sample certification of the joint-only resource; it is not an optimised experimental acquisition requirement. The PDF contains the measurement coefficients and complete proof. 4. A complementary two-million-trial implementation. For a displaced lossy single-photon source with unknown common phase, the target conditional state is ωφ=D(eiφ)2∣0⟩⟨0∣+∣1⟩⟨1∣D†(eiφ). A bounded homodyne score uses the full-Gaussian one-photon benchmark 33/(4e), 128 phase candidates and decision threshold 0.490. Including the specified source and readout tolerances, n=2×106,PFP<1.86×10−4,PFN<7.908×10−3. All vacuum emissions, loss events and measurement outcomes count towards the raw budget. This is a separate source family from the two-packet example. Together, the results distinguish unknown common phase from destructive phase randomisation and connect full-model exclusion to explicit Gaussian-measurement scores, error bounds and acquisition costs. The 15-page PDF includes the manuscript, two figures and complete supplementary proofs.