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 \(\sigma\) at a beam splitter of signal transmissivity \(\eta\). The requirement is \[\Phi_{\eta,\sigma}(\rho)\le cI\qquad\text{for every normal input state }\rho,\qquad 0<c<1.\] Equivalently, every rank-one output projection has probability at most \(c\) and the receiver's output min-entropy is at least \(-\ln c\). Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let \(d=\lceil 1/c\rceil-1\). Throughout \(0<\eta\le 1/\lceil 1/c\rceil\), the minimum mean photon numbers of the actual bath are \[N_{\mathrm{all}}(c,\eta)=\frac{d-cd(d+1)/2}{1-\eta}.\] \[N_{\mathrm G}(c,\eta)=\frac{c^{-1}-1}{1-\eta}.\] 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 \(j\ge2\) and \(0<\eta\le1/j\), \[N_{\mathrm G}(1/j,\eta)=2N_{\mathrm{all}}(1/j,\eta).\] 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 \[\lambda_*=(\underbrace{c,\ldots,c}_{d\text{ entries}},r).\] It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly \[S_{\min}(\Phi_{\eta,\sigma})=-dc\ln c-r\ln r.\] 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 \[S_{\mathrm{out}}\ge m\ln j\] for every finite number \(m\) of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with \(\eta\) 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, \(\sigma=\lvert1\rangle\langle1\rvert\), at every \(0<\eta<1\). With \[h(\eta)=-\eta\ln\eta-(1-\eta)\ln(1-\eta),\] it proves: Exact \(S_{\mathrm{out}}\ge mh(\eta)\) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by \(2\pi/q\), \(q\ge4\). 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. \(S_{\mathrm{out}}\ge Mh(\eta)-o(M)\) 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 \(\eta\), all outputs remain a uniformly positive trace distance from every state of rank below \(2^m\), 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.
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 (\sigma) at a beam splitter of signal transmissivity (\eta). The requirement is [\Phi_{\eta,\sigma}(\rho)\le cI\qquad\text{for every normal input state }\rho,\qquad 0<c<1.] Equivalently, every rank-one output projection has probability at most (c) and the receiver's output min-entropy is at least (-\ln c). Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let (d=\lceil 1/c\rceil-1). Throughout (0<\eta\le 1/\lceil 1/c\rceil), the minimum mean photon numbers of the actual bath are [N_{\mathrm{all}}(c,\eta)=\frac{d-cd(d+1)/2}{1-\eta}.] [N_{\mathrm G}(c,\eta)=\frac{c^{-1}-1}{1-\eta}.] 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\ge2) and (0<\eta\le1/j), [N_{\mathrm G}(1/j,\eta)=2N_{\mathrm{all}}(1/j,\eta).] 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 [\lambda_*=(\underbrace{c,\ldots,c}{d\text{ entries}},r).] It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly [S{\min}(\Phi_{\eta,\sigma})=-dc\ln c-r\ln r.] 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 [S_{\mathrm{out}}\ge m\ln j] for every finite number (m) of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with (\eta) 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, (\sigma=\lvert1\rangle\langle1\rvert), at every (0<\eta<1). With [h(\eta)=-\eta\ln\eta-(1-\eta)\ln(1-\eta),] it proves: Exact (S_{\mathrm{out}}\ge mh(\eta)) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by (2\pi/q), (q\ge4). 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. (S_{\mathrm{out}}\ge Mh(\eta)-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 (\eta), all outputs remain a uniformly positive trace distance from every state of rank below (2^m), 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.
How does the shape of detector noise affect optical communication? Two receivers can have the same noise covariance yet support different communication rates. This work determines both the quantum noise floor and the photon scale for approaching the capacity ceiling of specified non-Gaussian optical measurements. The central distinction is between the size of a high-energy communication advantage and the finite signal energy needed to realize it. Exact quantum noise floor. The receiver is a displacement-covariant measurement with a fixed, unflagged ancillary state. For every vacuum–single-photon seed σt=(1−t)∣0⟩⟨0∣+t∣1⟩⟨1∣, with 0≤t≤1/2, the output Shannon entropy is minimized uniquely by coherent states over all normal quantum inputs, without an input energy or photon-number cutoff. A separate entropy theorem covers three-level seeds σ=p0∣0⟩⟨0∣+p1∣1⟩⟨1∣+p2∣2⟩⟨2∣ with p1≤0.1, p2≤0.001, and p0=1−p1−p2. A capacity decomposition with a matched thermal benchmark. Let E be the mean signal photon budget per use, μ the fixed mean ancillary photon number, k the coherent-input response, and G1+μ the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies Cσ(E)=ln(1+1+μE)+D(k∥G1+μ)−Δσ(E). The first term is the exact capacity of the matched thermal receiver, allowing arbitrary quantum encoders. The relative entropy is the limiting high-energy capacity gain. The deficit Δσ(E)=Uσ(E)−Cσ(E), where Uσ(E)=1+ln(E+1+μ)−h(k), is the finite-energy shortfall from that gain. The converse permits mixed, non-Gaussian and entangled block codewords under an average total signal-photon constraint. Response zeros determine the sharp convergence rate. For each fixed nonvacuum seed in the stated families, write the coherent-response transform as k(ξ)=e−uP(u), where u=∣ξ∣2, P(u)=1−μu+vu2, and v=p2/2. Then limE→∞E−lnΔσ(E)=2minP(ζ)=0∣ζ∣. Positive classical modulation of coherent states attains this exponent with exact signal-energy accounting. In the nonvacuum two-level family, ordinary Gaussian modulation instead leaves an algebraic deficit. Uniform estimates cover compact three-level parameter families through the merger of two real zeros into a complex pair. Thus the response's relative entropy determines the high-energy gain, while its zeros determine the leading photon cost of approaching that gain. Finite-energy guarantees. Explicit bounds compare ordinary Gaussian modulation with a positive variance-mixture preparation and an upper bound valid for every quantum code. For t=1/2 and E=1, the capacity gain over the thermal receiver with the same ancillary mean photon number obeys 0.009742<Cσ1/2(1)−Cτ1/2(1)<0.010469nat/use. The better achievable bound depends on signal energy; no exact finite-energy optimizer is assumed. The matched-covariance noise comparison builds on classical additive-noise results of Ihara and Diggavi–Cover. The contribution here connects an exact all-input quantum noise floor to a sharp capacity remainder and positive, energy-accounted preparations. The results concern the stated receiver families and ideal measurement model; ancillary preparation and implementation overhead are separate resources. The integrated article includes complete proofs and five figures.
How much classical information can a noisy optical channel carry when both signal energy and preshared entanglement are limited? This work gives an exact capacity frontier for this joint resource problem. It allows one to determine the best asymptotic communication rate, price the benefit of additional entanglement, and distinguish the effect of tolerating errors from that of changing how entanglement is supplied. Exact capacity under two resource constraints The setting is any fixed bosonic Gaussian channel Φ with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian HG=21RTGR, with G>0 and ground energy E0; R collects the canonical quadratures. The budgets satisfy E≥E0 and P≥0. For an n-use code, the mean total signal energy is at most nE, while the same initially shared maximally entangled state has Schmidt rank Kn satisfying logKn≤nP. Thus P is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is C(E,P)=maxU≥Vphysical21Tr(GU)≤E,SG(V)≤P[SG(ΦU)+SG(V)−SG(ΦcV)]. Here V is a Gaussian seed covariance, U is the average-input covariance, and U−V is the displacement-modulation covariance. A covariance is physical when V+iΩ/2≥0, where Ω is the canonical commutation matrix and the vacuum covariance is I/2. The notation SG denotes Gaussian von Neumann entropy, and Φc is the full complementary channel, including any environment purification. All logarithms are natural, so rates are in nats per channel use. This formula replaces optimization over arbitrary quantum block codes by a finite-dimensional covariance optimization. It supplies both an achievable rate and a converse against non-Gaussian, correlated encodings. At P=0, it also determines the unassisted capacity of general fixed finite-mode Gaussian channels, including channels without a passive decomposition into independent modes. The reduction specifies the exact capacity value; it does not require a closed-form optimizer for every device. Energy and strict entanglement supply respond differently to errors For a fixed allowed average error 0≤ε<1, the exact capacity under the mean-energy constraint is Cε(E,P)=C(E0+1−εE−E0,P). Allowing failure rescales the mean energy above the ground state, but leaves the strict initial entanglement budget unchanged. The proof uses an exponential strong converse for codes whose inputs obey a hard total-energy cutoff. In a different model, where a readable flag permits random entanglement supply constrained only on average, the work establishes an achievable bound and a strict small-supply advantage for thermal attenuation at positive energy and fixed nonzero error. That comparison identifies the operational importance of the supply rule; it is not a complete capacity characterization of the flagged model. The communication value of entanglement For a thermal attenuator, let 0<η<1 be the transmissivity, b>0 the output mean photon number for vacuum input, and N the input mean photon number. Write its capacity as CN(P), and define g(N)=(N+1)log(N+1)−NlogN. The gain from entanglement obeys the sharp bound 0≤CN(P)−CN(0)≤1+bηP. For every fixed N>0, the coefficient η/(1+b) is attained as the initial right derivative at P=0. In the low-photon limit, the stronger resource law is CN(P)−CN(0)=1+bηmin{P,g(N)}+O(N),N↓0. The remainder is uniform over P≥0 at fixed η and positive b. Noise therefore determines the conversion efficiency, while g(N)∼Nlog(1/N) sets the entanglement scale sufficient for saturation. This uniform limit concerns the entanglement budget, not a simultaneous limit of vanishing thermal noise. Multimode resource allocation and scope A coupled two-mode example demonstrates how the frontier can guide resource allocation. A global resource-price certificate proves that a legal covariance pair exceeds the entire common-passive-frame Gaussian profile hull by at least 10−5 nat per two-mode use at specified budgets. The comparison includes all common passive frames, local squeezing and average-resource mixtures. The mechanism is the freedom to choose different mode allocations for the seed and average input; the benchmark is this precisely defined Gaussian class, rather than all unentangled encodings. The universal scope is the stated family of fixed finite-mode Gaussian channels, positive quadratic signal costs and strict initial maximally entangled Schmidt-rank supply, with unrestricted joint encoding and collective decoding. The model has no feedback or credit for returned entanglement. The capacity frontier and its resource consequences provide a common way to assess optical communication designs under these operational constraints.
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.
Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let Φ=⨂j=1dΦj, let E≥0 be the total average input photon number per block, and let Cj(Ej) be the explicitly attained one-channel rate specified in the paper. Then C(Φ,E)=χΦ(E)=maxEj≥0,∑jEj≤E∑j=1dCj(Ej),χΦ⊗n(nE)=nC(Φ,E)(n≥1). The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by ln2 gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance I/2, each channel acts as V↦ηV+diag(yq,yp), where 0<η<1, yq≥yp>0, and yqyp≥(1−η)2/4. Define ue=21ypyq,Eth=ue−21+2ηyq−yp. Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing g(x)=(x+1)ln(x+1)−xlnx, the high-energy rate is Cj(E)=g(ηE+2η+yq+yp−1)−g(2η+yqyp−21),E≥Eth. Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: Cj′(Ej∗)=λ(Ej∗>0),Cj′(0+)≤λ(Ej∗=0). A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately E=0.436 photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.
Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has m modes and total mean photon number at most E. Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most d. Independently, the entire register survives with probability 1−p or is erased into an orthogonal readable failure state. The physical output therefore has d+1 levels; a total q-qubit register permits d≤2q−1. Every event is retained, with no postselection. For heterodyne measure d2mz/πm and natural logarithms, the exact all-preparation reduction is Ver(m,d,E,p)=minrankρ≤d,Tr(Nρ)≤E{W(ρ)−(1−p)S(ρ)}. Here W is Wehrl entropy, S is von Neumann entropy and N is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define g(x)=(1+x)ln(1+x)−xlnx, S∗=mg(E/m), the ideal unlimited-memory value Bm(E)=m−Eln(1+m/E), and aE=1−[(1+E/m)ln(1+m/E)]−1. For any target excess satisfying 0<ε<aES∗, pcrit=mg(E/m)ε. A finite register reaches Ver≤Bm(E)+ε if and only if p<pcrit. At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let Sd be the largest entropy of any rank-at-most-d optical state within the same photon budget, and Δd=S∗−Sd. For 0≤p<aE, pS∗+(aE−p)Δd≤Ver−Bm(E)≤pS∗+(1−p)Δd. The first contribution survives at unlimited dimension, whereas Δd tends to zero as storage grows. An exact finite Gibbs program computes Sd, including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At 0<E≤10−4, 2≤d≤m+1 and p<1/8, an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.
What can be transmitted with a fixed photon supply? An optical source that supplies exactly K photons across n modes poses a concrete communication problem: how many messages can it transmit at a chosen error probability, and which input states make the best use of those photons? This paper gives directly evaluable coding bounds for pure loss and identifies the exact optimal input ensembles for identical noisy Gaussian channels. The results allow researchers to assess finite optical blocks, quantify the information cost of fixing the photon supply, and determine when entanglement between separately constrained blocks can be dispensed with. Every message is supported on the same exact K-photon sector. Within that sector, encoding may use arbitrary mixed states and entanglement among modes, and the receiver may perform a joint measurement. Communication is unassisted. All logarithms below are natural; information is measured in nats. Finite-message bounds and the optimal third-order rate For n ≥ 2 identical pure-loss channels of transmissivity 0 < η < 1 and K ≥ 1, the number L of surviving photons has the same binomial distribution for every encoded state. Define pℓ=(ℓK)ηℓ(1−η)K−ℓ,dn,ℓ=(ℓn+ℓ−1). If ps*(M;n,K) is the best average success probability for M equiprobable messages, then ∑ℓ=0Kpℓ[1−dn,ℓM−1]+≤ps∗(M;n,K)≤∑ℓ=0Kpℓmin{1,Mdn,ℓ}. Here [x]+ = max{x,0}. The lower bound provides an achievable guarantee; the upper bound limits every admissible code. Evaluating both sums brackets the performance at a specified photon supply and block length. Their proof has a simple physical interpretation: surviving photons determine the available output dimension, while one collective-mode codebook works across all possible survivor counts. For fixed E > 0, fixed error tolerance 0 < ε < 1, Kn = ⌊nE⌋ and en = Kn/n, the bounds determine the largest message count through third order: logM∗(n,Kn,ε)=ng(ηen)+nη(1−η)enh(ηen)ΦN−1(ε)−21logn+O(1). Here g(x) = (x+1)log(x+1) − x log x, h(x) = log(1+1/x), and ΦN is the standard normal distribution function. This fixes both the fluctuation scale and the optimal −½ log n correction. The finite bounds also yield the exact decay exponent of success above capacity and a matching error exponent on a specified interval below capacity. Which inputs are optimal under Gaussian noise? For n identical nonidentity gauge-covariant Gaussian channels, with thermal photon means transformed as r ↦ ar+b, a > 0 and b ≥ max{0,a−1}, the paper determines the exact fixed-photon Holevo optimum and every ensemble attaining it. Each optimal letter places all K photons in one collective mode: ∣K,u⟩=K!(∑j=1nujaj†)K∣0⟩,∑j∣uj∣2=1. An ensemble is optimal precisely when almost all its letters have this form and its average is uniform on the K-photon sector. A finite weighted optimal ensemble exists. These collective modes may themselves extend across, and be entangled among, physical optical modes. Thus the characterization identifies the structure to optimize without imposing a Gaussian-input restriction. The central proof step combines different loss histories that end in the same output photon sector. A common ordering of the comparison eigenstates allows the separate majorization bounds to be combined. For pure loss, a quantitative rigidity estimate further converts a small Holevo deficit into closeness to collective-mode number states. Resource cost and communication across blocks At fixed positive photon density and fixed nonidentity Gaussian noise, requiring an exact photon supply incurs a Holevo penalty of ½ log n plus an explicitly determined constant, relative to the usual mean-energy value. The paper separates this cost into photon-count uncertainty and the mismatch from a thermal output distribution. For separately constrained blocks of quantum-limited loss, quantum-limited amplification or entanglement-breaking Gaussian noise, the one-block Holevo values add exactly. Every optimal letter is a product of blockwise collective-mode number states, and the repeated-block unassisted classical capacity equals the one-block Holevo value. The low-rate reliability interval outside the matching range and repeated-block additivity for intermediate Gaussian noise remain open.