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 \(\sigma_t=(1-t)|0\rangle\langle0|+t|1\rangle\langle1|\), with \(0\le t\le1/2\), 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 \(\sigma=p_0|0\rangle\langle0|+p_1|1\rangle\langle1|+p_2|2\rangle\langle2|\) with \(p_1\le0.1\), \(p_2\le0.001\), and \(p_0=1-p_1-p_2\). A capacity decomposition with a matched thermal benchmark. Let \(E\) be the mean signal photon budget per use, \(\mu\) the fixed mean ancillary photon number, \(k\) the coherent-input response, and \(G_{1+\mu}\) the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies \[C_\sigma(E)=\ln\!\left(1+\frac{E}{1+\mu}\right)+D(k\Vert G_{1+\mu})-\Delta_\sigma(E).\] 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 \(\Delta_\sigma(E)=U_\sigma(E)-C_\sigma(E)\), where \(U_\sigma(E)=1+\ln(E+1+\mu)-h(k)\), 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 \(\widehat k(\xi)=e^{-u}P(u)\), where \(u=|\xi|^2\), \(P(u)=1-\mu u+vu^2\), and \(v=p_2/2\). Then \[\lim_{E\to\infty}\frac{-\ln\Delta_\sigma(E)}{E}=2\min_{P(\zeta)=0}|\zeta|.\] 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 \[0.009742\lt C_{\sigma_{1/2}}(1)-C_{\tau_{1/2}}(1)\lt 0.010469\quad\text{nat/use}.\] 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 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 (\sigma_t=(1-t)|0\rangle\langle0|+t|1\rangle\langle1|), with (0\le t\le1/2), 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 (\sigma=p_0|0\rangle\langle0|+p_1|1\rangle\langle1|+p_2|2\rangle\langle2|) with (p_1\le0.1), (p_2\le0.001), and (p_0=1-p_1-p_2). A capacity decomposition with a matched thermal benchmark. Let (E) be the mean signal photon budget per use, (\mu) the fixed mean ancillary photon number, (k) the coherent-input response, and (G_{1+\mu}) the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies [C_\sigma(E)=\ln!\left(1+\frac{E}{1+\mu}\right)+D(k\Vert G_{1+\mu})-\Delta_\sigma(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 (\Delta_\sigma(E)=U_\sigma(E)-C_\sigma(E)), where (U_\sigma(E)=1+\ln(E+1+\mu)-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 (\widehat k(\xi)=e^{-u}P(u)), where (u=|\xi|^2), (P(u)=1-\mu u+vu^2), and (v=p_2/2). Then [\lim_{E\to\infty}\frac{-\ln\Delta_\sigma(E)}{E}=2\min_{P(\zeta)=0}|\zeta|.] 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\lt C_{\sigma_{1/2}}(1)-C_{\tau_{1/2}}(1)\lt 0.010469\quad\text{nat/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.
This work gives a resource law for engineering optical noise: how many photons must a single environment mode contain to prevent any signal from producing an output that is too spectrally concentrated? It provides an explicit optimal bath preparation, a matched Gaussian benchmark, and separately proved guarantees for correlated repeated use. Exact photon cost of a spectral noise guarantee A signal mixes with an independently prepared bath σ at a beam splitter of signal transmissivity η. The requirement is Φη,σ(ρ)≤cIfor every normal input state ρ,0<c<1. Equivalently, every rank-one output projection has probability at most c and the receiver's output min-entropy is at least −lnc. Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let d=⌈1/c⌉−1. Throughout 0<η≤1/⌈1/c⌉, the minimum mean photon numbers of the actual bath are Nall(c,η)=1−ηd−cd(d+1)/2.NG(c,η)=1−ηc−1−1. The first optimization includes all normal bath states and has a unique finite-support minimizer. The second includes all Gaussian density operators, with arbitrary squeezing and displacement; a centered thermal bath attains it. The coupling and spectral requirement are identical in both comparisons. At c=1/j, for every integer j≥2 and 0<η≤1/j, NG(1/j,η)=2Nall(1/j,η). Between reciprocal budgets the ratio varies. The cost counts mean photons in the physical bath, not laboratory preparation work or the energy of an auxiliary purification. A physical preparation and its complete single-use spectrum Writing r=1−dc, the attaining vacuum-output spectrum is λ∗=(d entriesc,…,c,r). It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly Smin(Φη,σ)=−dclnc−rlnr. The other positive-order Rényi minima are fixed by the same spectrum. Coherent inputs attain these minima. The proof identifies an actual optical dilation whose port antisymmetry enforces the all-input occupation bound. Exact examples also show that baths with identical energy and covariance can have different optimized entropy floors. What survives entangled repeated use For the same resource-optimal flat-output families of ranks j=2,3,4, separate estimates prove Sout≥mlnj for every finite number m of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with η along their specified families. Further results give sharp tensorized purity bounds for the entire three-level diagonal bath simplex and for every Fock environment at balanced coupling. The paper then treats a different, fixed bath, σ=∣1⟩⟨1∣, at every 0<η<1. With h(η)=−ηlnη−(1−η)ln(1−η), it proves: Exact Sout≥mh(η) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by 2π/q, q≥4. Coherence and entanglement within residue sectors and infinite tails remain allowed. The residue-zero restricted channel also has strong minimum-output-entropy additivity with arbitrary normal partner channels. The same exact entropy floor for up to three globally total-phase-invariant input modes, without a mean-energy limit. Sout≥Mh(η)−o(M) for physical-mode permutation-invariant inputs of finite mean energy density, with explicit errors and full infinite-tail control. For every fixed m and interior η, all outputs remain a uniformly positive trace distance from every state of rank below 2m, without any input-energy bound. This covers nonbalanced parameters where the older purity-based distance bound vanishes; the new positive constant is spectrally implicit. Interpretation and scope The input classes above are not nested, and an entropy rate, a finite-block entropy floor and a low-rank distance are distinct guarantees. The fixed single-photon bath is not generally the resource-optimal bath. Complete unrestricted nonbalanced single-photon entropy additivity and exact classical capacity remain open. A spectral noise guarantee constrains output mixing; it does not mean that all input information or reference correlations are erased. The article separates this issue through actual resolved-erasure qubit restrictions, antidegradability of the specified optimal reciprocal families, and a classical capacity bracket under a true signal-energy budget. The integrated 52-page article contains six scientific figures, the complete proofs, and the mathematical data needed to reconstruct its finite certificates. Logarithms and entropy values use natural units unless bits are explicitly stated.
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.
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.
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.
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.
Wigner-positive light can carry more vacuum–one-photon coherence than every Gaussian mixture under the same mean photon-number budget. We determine the exact Gaussian benchmark and the sharp low-energy scales of this separation. Relative to the unconstrained ceiling U(E)=E(1−E), the Gaussian coherence shortfall is U(E)−MG(E)∼E5/2, whereas the optimum over all Wigner-positive states satisfies limE↓0Elog[U(E)−M∞(E)]=−1/2. A hard Fock cutoff changes the cost: with support on levels 0,…,K, the shortfall is (KK!/2)EK+1/2[1+O(K2E)], uniformly for K≥2 as K2E→0. The same comparison gives a direct certification rule and its phase-noise budget. After Gaussian random phase fluctuations of variance Γ, the optimized Wigner-positive source exceeds the full Gaussian coherence benchmark precisely below Γcert(E)=2E2+O(E3). Contact constraints determine the rank and leading mixed population of finite-support optimizers; a complete three-level frontier gives an exact test case. The exact benchmark lets an energy-and-coherence measurement exclude all Gaussian-mixture explanations and sets the phase-noise tolerance of that test.