Beyond noise variance: capacity gains and photon costs in non-Gaussian optical detection

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 \(σ_t=(1-t)|00|+t|11|\), 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 \(σ=p_0|00|+p_1|11|+p_2|22|\) with \(p_1≤0.1\), \(p_2≤0.001\), 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, \(μ\) 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 \[C_σ(E)=ln\!(1+E/1+μ)+D(k G1+μ)-Δ_σ(E).\] 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 \(Δ_σ(E)=U_σ(E)-C_σ(E)\), where \(U_σ(E)=1+ln(E+1+μ)-h(k)\), 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 \(u=|ξ|^2\), \(P(u)=1-μ u+vu^2\), and \(v=p_2/2\). Then \[limE→∞-lnΔ_σ(E)/E=2minP(ζ)=0|ζ|.\] 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 \[0.009742< C_{σ1/2}(1)-C_{τ1/2}(1)< 0.010469/use.\] 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.