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 \(\Phi\) with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian \(H_G=\tfrac12 R^{\mathsf T}GR\), with \(G>0\) and ground energy \(E_0\); \(R\) collects the canonical quadratures. The budgets satisfy \(E\ge E_0\) and \(P\ge0\). 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 \(K_n\) satisfying \(\log K_n\le nP\). Thus \(P\) is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is \[ C(E,P)=\max_{\substack{U\ge V\;\mathrm{physical}\\ \frac12\operatorname{Tr}(GU)\le E,\;S_G(V)\le P}}\bigl[S_G(\Phi U)+S_G(V)-S_G(\Phi^c V)\bigr]. \] 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 \(V+i\Omega/2\ge0\), where \(\Omega\) is the canonical commutation matrix and the vacuum covariance is \(I/2\). The notation \(S_G\) denotes Gaussian von Neumann entropy, and \(\Phi^c\) 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 \(0\le\varepsilon<1\), the exact capacity under the mean-energy constraint is \[ C_\varepsilon(E,P)=C\!\left(E_0+\frac{E-E_0}{1-\varepsilon},\,P\right). \] 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 \(0<\eta<1\) be the transmissivity, \(b>0\) the output mean photon number for vacuum input, and \(N\) the input mean photon number. Write its capacity as \(C_N(P)\), and define \(g(N)=(N+1)\log(N+1)-N\log N\). The gain from entanglement obeys the sharp bound \[ 0\le C_N(P)-C_N(0)\le\frac{\eta}{1+b}\,P. \] For every fixed \(N>0\), the coefficient \(\eta/(1+b)\) is attained as the initial right derivative at \(P=0\). In the low-photon limit, the stronger resource law is \[ C_N(P)-C_N(0)=\frac{\eta}{1+b}\min\{P,g(N)\}+O(N),\qquad N\downarrow0. \] The remainder is uniform over \(P\ge0\) at fixed \(\eta\) and positive \(b\). Noise therefore determines the conversion efficiency, while \(g(N)\sim N\log(1/N)\) 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 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 (\Phi) with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian (H_G=\tfrac12 R^{\mathsf T}GR), with (G>0) and ground energy (E_0); (R) collects the canonical quadratures. The budgets satisfy (E\ge E_0) and (P\ge0). 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 (K_n) satisfying (\log K_n\le 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)=\max_{\substack{U\ge V;\mathrm{physical}\ \frac12\operatorname{Tr}(GU)\le E,;S_G(V)\le P}}\bigl[S_G(\Phi U)+S_G(V)-S_G(\Phi^c V)\bigr]. ] 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+i\Omega/2\ge0), where (\Omega) is the canonical commutation matrix and the vacuum covariance is (I/2). The notation (S_G) denotes Gaussian von Neumann entropy, and (\Phi^c) 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\le\varepsilon<1), the exact capacity under the mean-energy constraint is [ C_\varepsilon(E,P)=C!\left(E_0+\frac{E-E_0}{1-\varepsilon},,P\right). ] 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<\eta<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 (C_N(P)), and define (g(N)=(N+1)\log(N+1)-N\log N). The gain from entanglement obeys the sharp bound [ 0\le C_N(P)-C_N(0)\le\frac{\eta}{1+b},P. ] For every fixed (N>0), the coefficient (\eta/(1+b)) is attained as the initial right derivative at (P=0). In the low-photon limit, the stronger resource law is [ C_N(P)-C_N(0)=\frac{\eta}{1+b}\min{P,g(N)}+O(N),\qquad N\downarrow0. ] The remainder is uniform over (P\ge0) at fixed (\eta) and positive (b). Noise therefore determines the conversion efficiency, while (g(N)\sim N\log(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.
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.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
When does Gaussian encoding miss the communication limit? This paper identifies an encoding class that attains the asymptotic capacity of arbitrary quantum block codes, and an exact resource threshold below which individual Gaussian signal states cease to be optimal. The result gives communication theorists a source-selection rule: classical mixtures of Gaussian packets suffice, while the shape of their displacement distribution can be essential. Channel and operating regime. One use is a two-mode additive Gaussian-noise channel with covariance Yn,c=(nI2cZcZnI2), where Z=diag(1,−1) and vacuum covariance is I2/2. For fixed y>0 and resource budget r>0, cn=n2−y2,En≥cn+2n+21,n→∞. The mean photon budget is 2En per pair. The resource is relative entropy to normalized PPT states across the two input streams, averaged over transmitted messages at at most r nats per pair. Arbitrary mixed, non-Gaussian and cross-use-entangled messages and joint decoding are allowed. The coding limit is taken at each fixed channel before the strong-correlation limit. 1. Exact asymptotic capacity reduction. With g(x)=(x+1)ln(x+1)−xlnx and An=2g(En+n), C=Chull=An−lnn−1−H(r)+o(1). Here Chull restricts every transmitted message to the trace-closed Gaussian convex hull. The regularized classical variational function H is the lower boundary, at cost at most r, of the closed convex upper hull of (mD(f∥w),mh(f)−lnπ),w=Wσ≥0,m≥1. The displacement law f is a classical probability density; its reference w is the nonnegative Wigner function of a physical state. The exact dual is H(r)=supα>1α−1−P(α)−αr,P(α)=supm≥1,Wσ≥0m1ln[πm(α−1)∫Wσα]. This converts optimization over quantum block codes into an entropy variational problem with an explicit quantum admissibility condition. 2. A sharp decision boundary for Gaussian signals. Define k−1−lnk=r, 0<k≤1, T(r)=−lnk and V(r)=1−H(r). Then CG=An−lnn−2+T(r)+o(1),rc=ln2−21,V(r)=T(r)(r≥rc),T(r)<V(r)≤1−ln2+2r(0<r<rc). Thus a Gaussian density is asymptotically optimal above the threshold. Below it, non-Gaussian displacement shapes strictly improve the rate, while Gaussian-packet mixtures still attain the all-state optimum. The low-resource offset is characterized by the variational formula. 3. Sub-vacuum Wigner entropy yields a communication advantage. For messages that are exactly PPT, CPPT=An−lnn−1−hW++o(1),limn→∞(CPPT−CG(0))=1−hW+>0, where hW+=infm,σ[h(Wσ)/m−lnπ], over finite-energy physical states with Wσ≥0, satisfies ln2≤hW+<1. The comparator CG(0) allows Gaussian messages with vanishing resource per use. The achieving messages are simultaneously PPT, Wigner-positive and mixtures of Gaussian states. All rates and logarithms use nats.
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.
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.
A gap between achievable rates and converse bounds persists for energy-constrained bosonic communication in the presence of excess noise. We derive a unified adaptive weak-converse bound for thermal attenuation, noisy amplification, and additive Gaussian noise under an unconditional mean transmitted-photon-number constraint. The bound applies to arbitrary adaptive protocols with quantum memories and constrains the two-way quantum, entanglement-distribution, private, and secret-key capacities. It vanishes throughout the entanglement-breaking region and approaches the corresponding PLOB bound at infinite energy. At finite energy, it improves the evaluated Gaussian squashed-entanglement and PLOB bounds in relevant parameter regimes, while comparison with hashing rates quantifies the remaining gap to achievability. The result provides a common energy-dependent benchmark for noisy bosonic Gaussian channels and extends finite-energy converse methods beyond the quantum-limited setting.