Standard linear optics requires that the transmittance of an optic be independent of photon number: for a linear element the input–output relation a_out = t·a_in + r·b_in is an operator identity, so single-photon and bright-light transmission are equal as a theorem rather than as an approximation. We are not aware of a dedicated test of that equality with a controlled scatterer at the 10⁻⁴ level. We state a hypothesis in which the equality fails in the single-quantum sector by a fixed dimensionless coupling λ*, and specify the tabletop experiment that would exclude it. The observable is Δ = D_quantum − D_classical, where D is the half-difference of a two-arm log-ratio under an ABBA swap of a weak scatterer, measured once with counting detectors and once with photodiodes through identical geometry. Standard quantum mechanics fixes Δ = 0 exactly; the hypothesis predicts Δ = −λ*·(−p ln p), or −0.046·λ* at a 1% transmission deficit. The loss map is stated as a photon-number-dependent Kraus channel, K = Σ exp[−½λ*·I_R(n)]|n⟩⟨n| with P_R(n) = 1 − (1−p)^n, which is linear, completely positive, local and non-signalling; invariance of a remote marginal is verified explicitly on a two-mode squeezed vacuum to 3×10⁻¹⁷. An earlier thermal formulation of the same idea is excluded by published SNSPD efficiency data at 2 K by a factor of 2×10⁵ and is withdrawn. An attempt to derive the criterion distinguishing which optical elements incur the cost failed under independent adversarial review and is reported as a failure; the surviving readings are converted into five pre-registered experimental arms with their nulls stated in advance. A borrowed teaching bench at 780 nm reaches a total uncertainty of 2.9×10⁻⁴ per operating point at 2×10⁸ counts, giving 5σ sensitivity at λ* ≥ 0.031 and 95% exclusion to 0.013 at a 1% scatterer; an extended scan through the turnover at p = 36.8% reaches 0.0039 and 0.0016 respectively, subject to a costed high-loss error-budget row and seven stated pass criteria. Five signatures distinguish the effect from systematics, including a −p ln p scaling and a maximum located at p = 0.368. The cheapest arm needs no interferometer: the single-photon versus bright transmission of one 50/50 beam splitter at 10⁻³, which bounds one reading of the hypothesis below λ* = 0.003 in an afternoon’s run on a prepared bench. The protocol is pre-registered before any bench is borrowed. The result will be published in either direction; a null is reported as the best bound in this sector.
Standard linear optics requires that the transmittance of an optic be independent of photon number: for a linear element the input–output relation a_out = t·a_in + r·b_in is an operator identity, so single-photon and bright-light transmission are equal as a theorem rather than as an approximation. We are not aware of a dedicated test of that equality with a controlled scatterer at the 10⁻⁴ level. We state a hypothesis in which the equality fails in the single-quantum sector by a fixed dimensionless coupling λ*, and specify the tabletop experiment that would exclude it. The observable is Δ = D_quantum − D_classical, where D is the half-difference of a two-arm log-ratio under an ABBA swap of a weak scatterer, measured once with counting detectors and once with photodiodes through identical geometry. Standard quantum mechanics fixes Δ = 0 exactly; the hypothesis predicts Δ = −λ*·(−p ln p), or −0.046·λ* at a 1% transmission deficit. The loss map is stated as a photon-number-dependent Kraus channel, K = Σ exp[−½λ*·I_R(n)]|n⟩⟨n| with P_R(n) = 1 − (1−p)^n, which is linear, completely positive, local and non-signalling; invariance of a remote marginal is verified explicitly on a two-mode squeezed vacuum to 3×10⁻¹⁷. An earlier thermal formulation of the same idea is excluded by published SNSPD efficiency data at 2 K by a factor of 2×10⁵ and is withdrawn. An attempt to derive the criterion distinguishing which optical elements incur the cost failed under independent adversarial review and is reported as a failure; the surviving readings are converted into five pre-registered experimental arms with their nulls stated in advance. A borrowed teaching bench at 780 nm reaches a total uncertainty of 2.9×10⁻⁴ per operating point at 2×10⁸ counts, giving 5σ sensitivity at λ* ≥ 0.031 and 95% exclusion to 0.013 at a 1% scatterer; an extended scan through the turnover at p = 36.8% reaches 0.0039 and 0.0016 respectively, subject to a costed high-loss error-budget row and seven stated pass criteria. Five signatures distinguish the effect from systematics, including a −p ln p scaling and a maximum located at p = 0.368. The cheapest arm needs no interferometer: the single-photon versus bright transmission of one 50/50 beam splitter at 10⁻³, which bounds one reading of the hypothesis below λ* = 0.003 in an afternoon’s run on a prepared bench. The protocol is pre-registered before any bench is borrowed. The result will be published in either direction; a null is reported as the best bound in this sector.
Two working documents that the deposited paper (Draft 4, v1.0; DOI 10.5281/zenodo.22678044; printed title The Asymmetric-Loss Equal-Born Test: a pre-registered search for record-information-dependent optical loss) cites and undertakes to deposit alongside it. They are deposited here as they stood on September 7, 2026, the day Draft 4 was completed, each with a dated deposit note at its head; nothing beneath the note is altered. Supplement 1 — Pre-Bench Closure (September 4, 2026), with postscript layers of September 6 and 7. The arithmetic companion to the paper. Part A runs the no-signalling check on the programme's surviving statement, finds that the normalised form permits superluminal signalling and is excluded, and records the single repair — a loss channel — that leaves the bench prediction numerically unchanged and converts the hypothesis from a nonlinear modification of the Born rule into a linear one. Part B carries the consequences into the protocol: the turnover of −p ln p at p = 1/e and the extended scan that would see it, a fourth signature, the cost of the high-loss points, the total-rate observable, the absorber control with its prediction stated in advance, and the consolidated protocol amendments. Part C makes the structural repairs: locating the branching event, the J-conservation contradiction, and the cost functional, with both candidates carried and the location of the peak left to decide between them. Part D drafts the pre-registration and scores the open tasks. The September-6 layer, headed "Postscript — September 5–6", records the two independent referee passes (one blind, one adversarial), the photon-number-dependent loss map, the derivation of the record criterion put to a third referee instructed to kill it and killed on twelve counts, the pre-registered arms and the correction of a handover error; the addendum of September 7 records the closing of the high-loss row. 17 pages. Supplement 2 — The high-p row: proposal, adversarial pass, and the row as costed (September 7, 2026). The error-budget row for the extended scan (p = 10%, 36.8%, 60% and 70%) that the paper's §9 named as the single item blocking pre-registration of the extended points, kept in the three steps by which it was reached. Appendix A is the first-pass proposal exactly as written for review, errors included. Appendix B is the report of an independent adversarial referee instructed to kill it, verbatim: eight errors and six systematics the first pass had not carried. Appendix C is the row as costed after every finding was re-derived and accepted, with the seven pass criteria written into the pre-registration. Its findings: the reach at the turnover (5σ at λ* ≥ 0.0039, 95% exclusion to 0.0016) stands only under the seven criteria; the bright-phase thermal family has the sign of the predicted anomaly and is bounded by a power-dependence test stated in advance; the far point of the scan moves from 70% to 60%; and the extended scan is research-group metrology, not teaching-laboratory work. 25 pages with its deposit note. The coloured highlights in Supplement 1 are its dated layers (rose, September 6; periwinkle, September 7; sky, the deposit note of September 16, 2026), each explained at its head. The line "Not for deposit" on the cover of Supplement 2 recorded the authors' working practice on September 7 and is superseded by the deposit note above it; the paper's Draft-4 note undertakes to deposit the document. The reconciliation document the paper also cites is the programme's working record, still being expanded, and is available from the authors.
Why this measurement is worth a bench: see section 2 of the proposal, "Why anyone should want to measure it". Contact: concursion.alebt@gmail.com. Two companion documents to the deposited paper, doi:10.5281/zenodo.22678044, and its pre-registration, osf.io/3h6yf. The question: does a 50/50 beam splitter transmit a single photon in exactly the same proportion as it transmits bright light? Standard linear optics says yes, exactly. The first run compares the two to one part in a thousand, with one source; only a calibrated attenuator changes between the bright phase and the single-photon phase. ALEBT_proposal_paper_v1_oct20.pdf: the proposal written for a host laboratory (the observable, the hypothesis, the error budget, the stages, the equipment and the open problems). ALEBT_equipment_list_v2_oct26.pdf: the first-run equipment list, with answers to the questions raised by groups we have written to (existing measurements, fibre-coupled detectors, differences between detectors).
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 Pe be the input-picture projector for vacuum in every relative output mode of a beam splitter at angle π/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 (Pe+P−e)/2. For any normal two-arm state Ω with total mean photons at most E, and ∣e∣≤θ≤π/4, TrΩ(2Pe+P−e−P0)≤21+3Eθ2. The exact projector curvature on the total k-photon sector is k+3k2−2k. 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 ⨂i=1B(ρi⊗σi): 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(ρ)=infα∥ρ−∣α⟩⟨α∣∥1,DU2=2(B−n)1∑i∈U[D(ρi)2+D(σi)2]. 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 DU2≤min{4,c4[δ+ην+ηn(B−n)Bln(1/βsig)]}. Here δ bounds optical inflation of the passing probability, η>0 is the retained detection-efficiency lower bound, and ν is a separate absolute click-probability decrement. The coefficient is c=1−exp(−4/Cˉ), where Cˉ≥C⋆ comes from the established sharp Wehrl theorem; the displayed examples use Cˉ=4.114205. Total failure probability is at most βcal+βsig. 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.
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.
A quantum-enhanced telescope (interferometer) can have significant advantages in precisely measuring the angular position of remote objects through sending out ancillary quantum entanglement pairs alongside the large baselines of telescopes. In principle, the larger the baseline, the higher the expected measurement resolution. However, the well-known major problem of photon transmission loss would drastically undermine the practical application of all quantum-enhanced telescopes and optical interferometers that transmit photons along the baseline, especially for the large baselines required for high angular resolution. Here, we propose a photon-transmission-loss-independent quantum-enhanced interferometer that is robust to the transmission loss and keeps its advantages unchanged given whatever large baseline. This removes the well-known barrier to the practical application of quantum-enhanced telescopes. Moreover, in contrast to the previous results of quantum-entanglement-enhanced telescopes requesting sophisticated resources—perfect single-photon entanglement, photon-number-resolving detectors, or quantum memories—our method requires only threshold detectors and tunable coherent-state or two-mode squeezed-state sources, which are mature technologies nowadays, and it does not need any quantum memory. We demonstrate the high performance of our method using Fisher information. For the phase difference of weak light from a remote star, the high-quality result of Fisher information remains constant under arbitrarily large transmission loss, and hence all the advantageous results and the performance of our method are actually independent of transmission loss.
Quantum non-Gaussian states of light have fundamental properties that are essential for a multitude of applications in quantum technology. However, many of these features are difficult to detect using standard criteria due to optical losses and detector inefficiency. As the statistics of light are unknown, the loss correction on the data is unreliable, despite the fact that the losses can be precisely measured. To address this issue, we employ a loss-mitigated verification technique utilizing quantum non-Gaussian witnesses, which incorporate the known optical losses and detector inefficiency into their derivation. This approach allows us to address the considerable challenge of experimentally demonstrating unheralded quantum non-Gaussian states of single photons and photon pairs.