A universal energy–entanglement capacity frontier for Gaussian optical channels

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 \(H_G=12 RTGR\), with \(G>0\) and ground energy \(E_0\); \(R\) collects the canonical quadratures. The budgets satisfy \(E≥ E_0\) 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 \(K_n\) satisfying \(log K_n≤ nP\). Thus \(P\) is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is \[ C(E,P)=max_{{U≥ V\;physical\\ 12Tr(GU)≤ E,\;S_G(V)≤ P}}[S_G(Φ U)+S_G(V)-S_G(Φ^c V)]. \] 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 \(S_G\) denotes Gaussian von Neumann entropy, and \(Φ^c\) 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 \(0≤ε<1\), the exact capacity under the mean-energy constraint is \[ C_ε(E,P)=C\!(E_0+E-E_0/1-ε,\,P). \] 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 \(C_N(P)\), and define \(g(N)=(N+1)log(N+1)-Nlog N\). The gain from entanglement obeys the sharp bound \[ 0≤ C_N(P)-C_N(0)≤η/1+b\,P. \] 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 \[ C_N(P)-C_N(0)=η/1+bmin\{P,g(N)\}+O(N), N↓0. \] 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.