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Phase records control the conversion cost of non-Gaussian light

Zixuan He

Published Oct 9, 2026• 0 citations
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Abstract

Detecting non-Gaussian light establishes that a resource is present. Its structure and preparation record determine the cost of turning it into a specified output. This paper gives matching conversion-cost laws for weak optical sources and a general input certificate that can exclude every allowed Gaussian receiver before a circuit is designed. Sources with identical energy, covariance and total photon statistics can have different cost exponents; losing a classical phase bit can make a fixed-source conversion impossible. Source, target and accounting. The Wigner-positive two-mode sources are obtained by half loss on each mode: \[\rho_{t,e,q}=\mathcal L_{1/2}^{\otimes2}\!\left[\frac{|00\rangle\langle00|+t^2\{(1-q)|p_+\rangle\langle p_+|+q|p_-\rangle\langle p_-|\}}{1+t^2}\right],\] \[|p_\pm\rangle=\frac{|22\rangle\pm e(|40\rangle+|04\rangle)/\sqrt2}{\sqrt{1+e^2}},\quad 0\le e\le\tfrac14,\quad 0\le q\le\tfrac12,\quad E=\frac{2t^2}{1+t^2}.\] Here \(e\) selects the source, \(q\) is an unflagged phase-record error probability, and \(E\) is delivered mean photon number. Conversion targets \[\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25,\qquad \tfrac12\|\sigma_{\mathrm{out}}-\tau_*\|_1\le\delta,\quad 0<\delta<10^{-5}.\] Joint processing of independent supplied copies, Gaussian ancillas, squeezing, memory, Gaussian measurements, feedback, discarding and postselection are allowed. Cost is \(C=\mathbb E K/p_s\): every source request counts, including vacuum requests and failed attempts. The task uses ideal Gaussian operations and charges supplied copies. 1. A source-selection law controlled by record accuracy. Optimizing both the source parameter \(e\) and the allowed Gaussian protocol gives, uniformly in \(q\in[0,1/2]\) as \(E\to0\), \[\log C_{\mathrm{best}}(E,q)=\Theta_\delta\!\left(\max\{E^{-1/3},\sqrt{q/E}\}\right).\] Errors of order \(E^{1/3}\) or smaller preserve the fastest logarithmic cost. Above that scale, record quality controls the optimum. Equivalently, for \(q=q_0E^a\), fixed \(0<q_0\le1/2\) and \(a\ge0\), \[\log C_{\mathrm{best}}=\Theta_\delta(E^{-\beta}),\qquad \beta=\max\{1/3,(1-a)/2\}.\] At fixed nonzero error, choosing the phase-insensitive source keeps conversion possible at logarithmic cost of order \(E^{-1/2}\). 2. The same elementary diagnostics, exponentially different costs. The normalized excitation polynomials \[p_A(z)=\frac{z_1^4-z_2^4}{\sqrt{48}},\qquad p_B(z)=\frac{(z_1^2-z_2^2)^2}{8}\] give calibrated half-loss sources with identical energy, covariance and total photon-number statistics, but for the same target and tolerance, \[\log C_A(E)=\Theta_\delta(E^{-1/3}),\qquad \log C_B(E)=\Theta_\delta(E^{-1/2}).\] The distinction is the local structure of their excitation-polynomial zeros. Full Gaussian-protocol lower bounds and matching heterodyne constructions establish both orders. 3. Certification survives errors that obstruct conversion. In the symmetric quartic family, the full first-mode photon distribution is independent of \(q\). In particular, \[P_x(1)=\frac{E(4+e^2)}{16(1+e^2)}\ge E/5,\qquad P_x(N\ge5)=0.\] Every state in the full one-mode Gaussian convex hull instead satisfies \[P(N\ge5)\ge\frac{P(1)^6}{4608000}.\] Photon counting therefore supplies a sufficient \(O(E^{-6})\) request budget at fixed false-positive and missed-detection probabilities against independent repetitions of any one unknown Gaussian-hull state. Yet, in the joint limit \(t\to0\), \(e/t\to\infty\), finite-cost conversion has the threshold \[q_c(t,e;\delta)=\Theta_\delta\!\left((E/e^2)^{1/3}\right).\] For fixed \(e>0\), any fixed positive unflagged error thus eventually obstructs this output. Flagged erasures behave differently: independently retaining a correct bit with probability \(a_r>0\) adds at most a factor \(1/a_r\) to a protocol that discards erasures. 4. A transferable exclusion certificate and an explicit receiver. A verified positive phase-space minorant with a physical pure Gaussian curvature reference supplies a source budget \(b_\kappa\). For a requested one-mode output whose trace-distance gap from the Gaussian convex hull is at least \(g\), every allowed Gaussian protocol obeys \[C\ge\frac{g^2-\kappa/(2-\kappa)}{b_\kappa},\qquad \frac{\kappa}{2-\kappa}<g^2.\] The manuscript constructs the certificate and explicit incomplete-gamma budgets for the quartic sources. It also gives a centred-root heterodyne receiver. In its specified joint limit, with scaled error \(\lambda=d_e q(e/t)^{2/3}\), its optimized limiting one-photon population is \[\sup w=\frac1{2[1+4\,3^{-3/4}\lambda^{3/4}]}.\] The paper supplies \(d_e\), the receiver action, and a photon-statistical output witness. These receiver formulas connect the desired output to tolerable record error; the general certificate supplies the protocol-independent cost bound. Together they let source designers compare preparation structure, record quality and total request budget for the same optical task.

DOI
10.5281/zenodo.23266096

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Quantum limits and one-qubit advantage in optical predictive compression

Zixuan He

What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on \(A\) is prepared with its actual purifier \(C\). A normal encoder acts on \(C\) to produce memory \(M\), while a prescribed Gaussian channel maps \(A\) to the relevant optical output \(B\). The source-information cost and useful correlation are \(r=I(A:M)\) and \(v=I(B:M)\), respectively. All logarithms are natural. The paper determines \[ C_{\mathrm Q}(R)=\sup_{\mathcal N:\,I(A:M)\le R}I(B:M). \] For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number \(n\) and a thermal attenuator with transmissivity \(0<\eta<1\), write the output occupation as \(\eta n+b\), where \(b\ge0\) is the environment's added thermal occupation. With \(g(x)=(x+1)\log(x+1)-x\log x\), the exact optimum over all normal classical records is \[ C_{\mathrm{cl}}(n,R)=g(\eta n+b)-g(\eta u_R+b),\qquad u_R=g^{-1}\!\bigl(\max\{g(n)-R,0\}\bigr). \] A legal encoder applies vacuum loss with transmissivity \(\lambda\) to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed \(b>0\) and \(0<\lambda<1/2\), at the common budget \(R_n=g(n)+g(\lambda n)-g((1-\lambda)n)\), its actual relevance satisfies \[ v_2=\frac{\eta\lambda}{1+b}\,n\log(1/n)+O(n),\qquad C_{\mathrm{cl}}(n,R_n)=2\lambda\eta\log(1+1/b)\,n+O\!\left(\frac{n}{\log(1/n)}\right). \] The full optical tail is included in the resource accounting. At \(n=0.01\), \(\eta=b=0.5\), \(\lambda=0.2\) and \(R=0.023875\), the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For \(k\) uses, the retained predictive memory obeys \(\dim K_k\le2^k\) and \(I(A^k:K_k)\le kR\), including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let \(\alpha_k\) be the missed-correlation probability and \(\beta_k\) the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is \[ \alpha_k\le\epsilon\quad\Longrightarrow\quad \beta_k\ge\exp\!\left[-\frac{k c_nR+\log2}{1-\epsilon}\right],\qquad c_n=\eta\frac{\log(1+1/(\eta n+b))}{\log(1+1/n)}. \] At the finite parameters above and \(\epsilon=0.1\), the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.

•Oct 25, 2026•0 citations
Detecting non-Gaussian light establishes that a resource is present. Its structure and preparation record determine the cost of turning it into a specified output. This paper gives matching conversion-cost laws for weak optical sources and a general input certificate that can exclude every allowed Gaussian receiver before a circuit is designed. Sources with identical energy, covariance and total photon statistics can have different cost exponents; losing a classical phase bit can make a fixed-source conversion impossible. Source, target and accounting. The Wigner-positive two-mode sources are obtained by half loss on each mode: [\rho_{t,e,q}=\mathcal L_{1/2}^{\otimes2}!\left[\frac{|00\rangle\langle00|+t^2{(1-q)|p_+\rangle\langle p_+|+q|p_-\rangle\langle p_-|}}{1+t^2}\right],] [|p_\pm\rangle=\frac{|22\rangle\pm e(|40\rangle+|04\rangle)/\sqrt2}{\sqrt{1+e^2}},\quad 0\le e\le\tfrac14,\quad 0\le q\le\tfrac12,\quad E=\frac{2t^2}{1+t^2}.] Here (e) selects the source, (q) is an unflagged phase-record error probability, and (E) is delivered mean photon number. Conversion targets [\tau_=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25,\qquad \tfrac12|\sigma_{\mathrm{out}}-\tau_|1\le\delta,\quad 0<\delta<10^{-5}.] Joint processing of independent supplied copies, Gaussian ancillas, squeezing, memory, Gaussian measurements, feedback, discarding and postselection are allowed. Cost is (C=\mathbb E K/p_s): every source request counts, including vacuum requests and failed attempts. The task uses ideal Gaussian operations and charges supplied copies. 1. A source-selection law controlled by record accuracy. Optimizing both the source parameter (e) and the allowed Gaussian protocol gives, uniformly in (q\in[0,1/2]) as (E\to0), [\log C{\mathrm{best}}(E,q)=\Theta_\delta!\left(\max{E^{-1/3},\sqrt{q/E}}\right).] Errors of order (E^{1/3}) or smaller preserve the fastest logarithmic cost. Above that scale, record quality controls the optimum. Equivalently, for (q=q_0E^a), fixed (0<q_0\le1/2) and (a\ge0), [\log C_{\mathrm{best}}=\Theta_\delta(E^{-\beta}),\qquad \beta=\max{1/3,(1-a)/2}.] At fixed nonzero error, choosing the phase-insensitive source keeps conversion possible at logarithmic cost of order (E^{-1/2}). 2. The same elementary diagnostics, exponentially different costs. The normalized excitation polynomials [p_A(z)=\frac{z_1^4-z_2^4}{\sqrt{48}},\qquad p_B(z)=\frac{(z_1^2-z_2^2)^2}{8}] give calibrated half-loss sources with identical energy, covariance and total photon-number statistics, but for the same target and tolerance, [\log C_A(E)=\Theta_\delta(E^{-1/3}),\qquad \log C_B(E)=\Theta_\delta(E^{-1/2}).] The distinction is the local structure of their excitation-polynomial zeros. Full Gaussian-protocol lower bounds and matching heterodyne constructions establish both orders. 3. Certification survives errors that obstruct conversion. In the symmetric quartic family, the full first-mode photon distribution is independent of (q). In particular, [P_x(1)=\frac{E(4+e^2)}{16(1+e^2)}\ge E/5,\qquad P_x(N\ge5)=0.] Every state in the full one-mode Gaussian convex hull instead satisfies [P(N\ge5)\ge\frac{P(1)^6}{4608000}.] Photon counting therefore supplies a sufficient (O(E^{-6})) request budget at fixed false-positive and missed-detection probabilities against independent repetitions of any one unknown Gaussian-hull state. Yet, in the joint limit (t\to0), (e/t\to\infty), finite-cost conversion has the threshold [q_c(t,e;\delta)=\Theta_\delta!\left((E/e^2)^{1/3}\right).] For fixed (e>0), any fixed positive unflagged error thus eventually obstructs this output. Flagged erasures behave differently: independently retaining a correct bit with probability (a_r>0) adds at most a factor (1/a_r) to a protocol that discards erasures. 4. A transferable exclusion certificate and an explicit receiver. A verified positive phase-space minorant with a physical pure Gaussian curvature reference supplies a source budget (b_\kappa). For a requested one-mode output whose trace-distance gap from the Gaussian convex hull is at least (g), every allowed Gaussian protocol obeys [C\ge\frac{g^2-\kappa/(2-\kappa)}{b_\kappa},\qquad \frac{\kappa}{2-\kappa}<g^2.] The manuscript constructs the certificate and explicit incomplete-gamma budgets for the quartic sources. It also gives a centred-root heterodyne receiver. In its specified joint limit, with scaled error (\lambda=d_e q(e/t)^{2/3}), its optimized limiting one-photon population is [\sup w=\frac1{2[1+4,3^{-3/4}\lambda^{3/4}]}.] The paper supplies (d_e), the receiver action, and a photon-statistical output witness. These receiver formulas connect the desired output to tolerable record error; the general certificate supplies the protocol-independent cost bound. Together they let source designers compare preparation structure, record quality and total request budget for the same optical task.
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.
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\mathcal G_nGn​ be the trace-norm-closed convex hull of all physical nnn-mode Gaussian states. For a fixed known ρ∉G1\rho\notin\mathcal G_1ρ∈/G1​, form the common-phase state Ωn=∫02π(UϕρUϕ†)⊗n dϕ2π,Uϕ=e−iϕN.\Omega_n=\int_0^{2\pi}(U_\phi\rho U_\phi^\dagger)^{\otimes n}\,\frac{d\phi}{2\pi},\qquad U_\phi=e^{-i\phi N}.Ωn​=∫02π​(Uϕ​ρUϕ†​)⊗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, inf⁡σn∈Gn∥Ωn−σn∥1⟶2.\inf_{\sigma_n\in\mathcal G_n}\|\Omega_n-\sigma_n\|_1\longrightarrow2.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⁡ ⁣[−v2R2(1−η)(v−1/2)2].\rho_{\eta,\nu}\in\mathcal G_1\quad\Longleftrightarrow\quad\nu\le\nu_c,\qquad \nu_c=\exp\!\left[-\frac{v^2R}{2(1-\eta)(v-1/2)^2}\right].ρη,ν​∈G1​⟺ν≤νc​,νc​=exp[−2(1−η)(v−1/2)2v2R​]. Here v>1/2v>1/2v>1/2 is the packet quadrature variance, R\sqrt RR​ the momentum displacement, 0<τ,η<10<\tau,\eta<10<τ,η<1 the packet weight and transmissivity, and 0≤ν≤10\le\nu\le10≤ν≤1 the coherence visibility. The same membership boundary holds after common-phase averaging for every n≥3n\ge3n≥3. At (v,R,τ,ν,η)=(20,2/9,9/50,1,1/2),(v,R,\tau,\nu,\eta)=(20,2/9,9/50,1,1/2),(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\mathcal G_nGn​ for n≥3n\ge3n≥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.n_{\rm raw}=10^{15}+2(4\times10^{15})=9\times10^{15},\qquad P_{\rm FP}<3.86\times10^{-22},\qquad P_{\rm FN}<0.002068.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)∣0⟩⟨0∣+∣1⟩⟨1∣2D†(eiφ).\omega_\varphi=D(e^{i\varphi})\frac{|0\rangle\langle0|+|1\rangle\langle1|}{2}D^\dagger(e^{i\varphi}).ωφ​=D(eiφ)2∣0⟩⟨0∣+∣1⟩⟨1∣​D†(eiφ). A bounded homodyne score uses the full-Gaussian one-photon benchmark 33/(4e)3\sqrt3/(4e)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.n=2\times10^6,\qquad P_{\rm FP}<1.86\times10^{-4},\qquad P_{\rm FN}<7.908\times10^{-3}.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.
How much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with [Q,P]=i[Q,P]=i[Q,P]=i, mean photon number at most E>0E>0E>0, and a known parity centre. The full class WE\mathcal W_EWE​ consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass GE\mathcal G_EGE​ consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let PLP_LPL​ project onto photon numbers at most LLL, and define the cutoff error by eL(ρ)=12∥ρ−PLρPL∥1.e_L(\rho)=\tfrac12\|\rho-P_L\rho P_L\|_1.eL​(ρ)=21​∥ρ−PL​ρPL​∥1​. This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard OE(L−1/2)O_E(L^{-1/2})OE​(L−1/2) guarantee. At the exact Wigner endpoint, the article proves sup⁡ρ∈WEeL(ρ)≤CE(L+1)−3/4,sup⁡ρ∈GEeL(ρ)≤4EL+1,L≥2.\sup_{\rho\in\mathcal W_E}e_L(\rho)\le C_E(L+1)^{-3/4},\qquad \sup_{\rho\in\mathcal G_E}e_L(\rho)\le \frac{4E}{L+1},\qquad L\ge2.supρ∈WE​​eL​(ρ)≤CE​(L+1)−3/4,supρ∈GE​​eL​(ρ)≤L+14E​,L≥2. The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family τd∈WE\tau_d\in\mathcal W_Eτd​∈WE​ has exact energy EEE, covariance (E+1/2)I(E+1/2)I(E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3)L_d=\Theta(d^3)Ld​=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞.e_{L_d}(\tau_d)\ge c_Ed^{-5/2},\qquad (L_d+1)e_{L_d}(\tau_d)\longrightarrow\infty.eLd​​(τd​)≥cE​d−5/2,(Ld​+1)eLd​​(τd​)⟶∞. Thus the Gaussian-mixture L−1L^{-1}L−1 law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let nX(δ)n_X(\delta)nX​(δ) be the minimum worst-case number of copies for trace-distance accuracy δ\deltaδ over source class XXX. As δ↓0\delta\downarrow0δ↓0, the bounds are log⁡nGE(δ)=Θ(δ−1),cδ−6/5≤log⁡nWE(δ)≤Cδ−4/3.\log n_{\mathcal G_E}(\delta)=\Theta(\delta^{-1}),\qquad c\delta^{-6/5}\le\log n_{\mathcal W_E}(\delta)\le C\delta^{-4/3}.lognGE​​(δ)=Θ(δ−1),cδ−6/5≤lognWE​​(δ)≤Cδ−4/3. These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.
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 mmm 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 1/321/321/32 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 DDD, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability aaa, or is erased with a classical flag. The quantity ka(D,m)k_a(D,m)ka​(D,m) is the smallest deterministic original-preparation cap achieving worst-case error at most 1/31/31/3. Failed preparations, discarded trials and reserved positions are all charged. A retained register of qqq qubits has D=2qD=2^qD=2q. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is Θ(m)\Theta(m)Θ(m). Along arbitrary growing-mode sequences, the precise criterion for improving that order is ka(D,m)=o(m)⟺am→∞ and aDm→∞. k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty. ka​(D,m)=o(m)⟺am→∞ and maD​→∞. The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when a=O(1/m)a=O(1/m)a=O(1/m), yet a survival probability tending to zero can still permit an advantage. Let A=max⁡{1,m2/D}A=\max\{1,m^2/D\}A=max{1,m2/D}. At the fixed energy and separation above, constants c,C>0c,C>0c,C>0, independent of m,D,am,D,am,D,a, give cmin⁡{m,A/a}≤ka(D,m)≤Cmin⁡{m,A/a+min⁡(a−2,m2/3)},0<a≤1. c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1. cmin{m,A/a}≤ka​(D,m)≤Cmin{m,A/a+min(a−2,m2/3)},0<a≤1. At a=0a=0a=0, the sample cost is Θ(m)\Theta(m)Θ(m). For ideal memory the complete rate is k1(D,m)=Θ ⁣(min⁡{m,max⁡(1,m2/D)}). k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right). k1​(D,m)=Θ(min{m,max(1,m2/D)}). With loss, the bounds match whenever A≥min⁡{a−1,am2/3}A\ge\min\{a^{-1},am^{2/3}\}A≥min{a−1,am2/3}. For example, a=m−1/6a=m^{-1/6}a=m−1/6 and D=⌈m11/6⌉D=\lceil m^{11/6}\rceilD=⌈m11/6⌉ give Θ(m1/3)\Theta(m^{1/3})Θ(m1/3) preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for D≥m2D\ge m^2D≥m2, a=m−1/2a=m^{-1/2}a=m−1/2, the stated bounds are Ω(m1/2)\Omega(m^{1/2})Ω(m1/2) and O(m2/3)O(m^{2/3})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 m2m^2m2 labels, only a positive register of dimension m+1m+1m+1 is retained. The three-visit route requires D≥m+1D\ge m+1D≥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\mathcal N=a\Psi+(1-a)\mathcal R_\tauN=aΨ+(1−a)Rτ​, where Ψ\PsiΨ is CPTP and Rτ(X)=Tr⁡(X)τ\mathcal R_\tau(X)=\operatorname{Tr}(X)\tauRτ​(X)=Tr(X)τ, an exact flagged-erasure simulation gives kN(D,m)≥ka(D,m)k_{\mathcal N}(D,m)\ge k_a(D,m)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)1-q=O(1/m)1−q=O(1/m) therefore forces Θ(m)\Theta(m)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.
For a fixed optical receiver, optimizing the classical modulation of Gaussian signals need not reach the best communication rate. This work gives a rigorous way to certify the advantage of a non-Gaussian signal constellation against every one-mode Gaussian alphabet, and to quantify how much of that advantage survives specified receiver and preparation errors. What is compared The receiver is the displacement-covariant phase-space measurement generated by a single-photon ancillary state. It is not a binary photon-click detector. The cost is the actual mean transmitted photon number E; the receiver is fixed, and ancillary preparation and local-oscillator overhead are not included in that signal budget. Let C(E) be the unrestricted capacity and CG(E) the optimum over Gaussian letters with arbitrary Borel modulation, displacement, squeezing and mixedness. For the ideal receiver, CG(E) < C(E), for every 0 < E < ∞. The comparison is global: it does not restrict the Gaussian competitor to a finite constellation or a locally optimized ansatz. A finite-energy certificate with a calibration interval For the actual unflagged receiver seed σv = (1 − v)|1⟩⟨1| + v|0⟩⟨0|, the paper constructs four equiprobable phase-rotated signals supported on photon numbers 0–3. Each signal has exactly 0.1 mean photons. Write I4,v for their mutual information and CG(0.1,v) for the Gaussian optimum of that same receiver. At v = 0.01, in nats per use, I4,0.01 > 0.0839244, CG(0.1,0.01) < 0.0839162,so the certified excess is greater than 8.2 × 10−6. The same constellation obeys I4,v − CG(0.1,v) > 5 × 10−6 for every 0.009 ≤ v ≤ 0.011. This is a continuum guarantee, obtained from a supporting line for the same-source information and a convex upper chord for the Gaussian optimum. It is not interpolation between sampled advantages. The proof retains the full photon tail of Gaussian competitors. Phase tolerance at unchanged energy Suppose all four signals share static photon-number phases θn while retaining their exact amplitudes. After removing irrelevant constant and linear phases, the two residual errors are ε2 = θ2 − 2θ1 + θ0 and ε3 = θ3 − 3θ1 + 2θ0, modulo 2π. If both residual magnitudes are at most δ, their information loss is at most Kδ², with K < 0.132762313267. Thus δ = 0.004 radians (about 0.229°) leaves a certified excess above 2.8 × 10−6 nat/use throughout the receiver interval, with photon cost still exactly 0.1. This phase guarantee concerns a common static source error. A separate trace-distance preparation bound requires an independent check of actual mean energy. Independent phase jitter and arbitrary detector noise are different error models. Collective encodings and interpretation For the ideal receiver, every fixed finite entangled Gaussian block retains a total deficit, and uniformly bounded common blocks retain a per-use deficit. Exact correlation and covariance-repair estimates identify necessary extensive resources for a Gaussian sequence to approach capacity. These results do not settle arbitrary growing-block Gaussian encodings. The quantitative advantage is small—about 0.01% of the rate at the certified point. Its significance is a robust, globally certified separation with explicit resource and error accounting, rather than a large throughput improvement or an experimental demonstration. The article contains the complete proofs, figures, exact reference data and error bounds.
What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on AAA is prepared with its actual purifier CCC. A normal encoder acts on CCC to produce memory MMM, while a prescribed Gaussian channel maps AAA to the relevant optical output BBB. The source-information cost and useful correlation are r=I(A:M)r=I(A:M)r=I(A:M) and v=I(B:M)v=I(B:M)v=I(B:M), respectively. All logarithms are natural. The paper determines CQ(R)=sup⁡N: I(A:M)≤RI(B:M). C_{\mathrm Q}(R)=\sup_{\mathcal N:\,I(A:M)\le R}I(B:M). CQ​(R)=supN:I(A:M)≤R​I(B:M). For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number nnn and a thermal attenuator with transmissivity 0<η<10<\eta<10<η<1, write the output occupation as ηn+b\eta n+bηn+b, where b≥0b\ge0b≥0 is the environment's added thermal occupation. With g(x)=(x+1)log⁡(x+1)−xlog⁡xg(x)=(x+1)\log(x+1)-x\log xg(x)=(x+1)log(x+1)−xlogx, the exact optimum over all normal classical records is Ccl(n,R)=g(ηn+b)−g(ηuR+b),uR=g−1 ⁣(max⁡{g(n)−R,0}). C_{\mathrm{cl}}(n,R)=g(\eta n+b)-g(\eta u_R+b),\qquad u_R=g^{-1}\!\bigl(\max\{g(n)-R,0\}\bigr). Ccl​(n,R)=g(ηn+b)−g(ηuR​+b),uR​=g−1(max{g(n)−R,0}). A legal encoder applies vacuum loss with transmissivity λ\lambdaλ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed b>0b>0b>0 and 0<λ<1/20<\lambda<1/20<λ<1/2, at the common budget Rn=g(n)+g(λn)−g((1−λ)n)R_n=g(n)+g(\lambda n)-g((1-\lambda)n)Rn​=g(n)+g(λn)−g((1−λ)n), its actual relevance satisfies v2=ηλ1+b nlog⁡(1/n)+O(n),Ccl(n,Rn)=2ληlog⁡(1+1/b) n+O ⁣(nlog⁡(1/n)). v_2=\frac{\eta\lambda}{1+b}\,n\log(1/n)+O(n),\qquad C_{\mathrm{cl}}(n,R_n)=2\lambda\eta\log(1+1/b)\,n+O\!\left(\frac{n}{\log(1/n)}\right). v2​=1+bηλ​nlog(1/n)+O(n),Ccl​(n,Rn​)=2ληlog(1+1/b)n+O(log(1/n)n​). The full optical tail is included in the resource accounting. At n=0.01n=0.01n=0.01, η=b=0.5\eta=b=0.5η=b=0.5, λ=0.2\lambda=0.2λ=0.2 and R=0.023875R=0.023875R=0.023875, the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For kkk uses, the retained predictive memory obeys dim⁡Kk≤2k\dim K_k\le2^kdimKk​≤2k and I(Ak:Kk)≤kRI(A^k:K_k)\le kRI(Ak:Kk​)≤kR, including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let αk\alpha_kαk​ be the missed-correlation probability and βk\beta_kβk​ the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is αk≤ϵ⟹βk≥exp⁡ ⁣[−kcnR+log⁡21−ϵ],cn=ηlog⁡(1+1/(ηn+b))log⁡(1+1/n). \alpha_k\le\epsilon\quad\Longrightarrow\quad \beta_k\ge\exp\!\left[-\frac{k c_nR+\log2}{1-\epsilon}\right],\qquad c_n=\eta\frac{\log(1+1/(\eta n+b))}{\log(1+1/n)}. αk​≤ϵ⟹βk​≥exp[−1−ϵkcn​R+log2​],cn​=ηlog(1+1/n)log(1+1/(ηn+b))​. At the finite parameters above and ϵ=0.1\epsilon=0.1ϵ=0.1, the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.