We establish the fundamental limit of communication capacity within Gaussian schemes under phase-insensitive Gaussian channels, which employ multimode Gaussian states for encoding and collective Gaussian operations and measurements for decoding. We prove that this Gaussian capacity is additive, i.e., its upper bound occurs with separable encoding and separable receivers so that a single-mode communication suffices to achieve the largest capacity under Gaussian schemes. This rigorously characterizes the gap between the ultimate Holevo capacity and the capacity within Gaussian communication, showing that Gaussian regime is not sufficient to achieve the Holevo bound particularly in the low-photon regime. Furthermore, the Gaussian benchmark established here can be used to critically assess the performance of non-Gaussian protocols for optical communication. We move on to identify non-Gaussian schemes to beat the Gaussian capacity and show that a non-Gaussian receiver recently implemented by Becerra et al. [F. E. Becerra et al., Nat. Photon. 7, 147 (2013)] can achieve this aim with an appropriately chosen encoding strategy.
We establish the fundamental limit of communication capacity within Gaussian schemes under phase-insensitive Gaussian channels, which employ multimode Gaussian states for encoding and collective Gaussian operations and measurements for decoding. We prove that this Gaussian capacity is additive, i.e., its upper bound occurs with separable encoding and separable receivers so that a single-mode communication suffices to achieve the largest capacity under Gaussian schemes. This rigorously characterizes the gap between the ultimate Holevo capacity and the capacity within Gaussian communication, showing that Gaussian regime is not sufficient to achieve the Holevo bound particularly in the low-photon regime. Furthermore, the Gaussian benchmark established here can be used to critically assess the performance of non-Gaussian protocols for optical communication. We move on to identify non-Gaussian schemes to beat the Gaussian capacity and show that a non-Gaussian receiver recently implemented by Becerra et al. [F. E. Becerra et al., Nat. Photon. 7, 147 (2013)] can achieve this aim with an appropriately chosen encoding strategy.
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.
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to encode and decode information under a single-channel use. We prove that the optimal capacity of single-channel Gaussian communication is achieved by one of two well-known protocols, i.e., coherent-state communication or squeezed-state communication, depending on the energy (number of photons) as well as the characteristics of the channel. Our result suggests that the coherent-state scheme known to achieve the ultimate information-theoretic capacity is not a practically optimal scheme for the case of using a finite number of channels. We find that overall the squeezed-state communication is optimal in a small-photon-number regime whereas the coherent-state communication performs better in a large-photon-number regime.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated code words with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent-state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations [passive linear optics: beam splitters and phase shifters; second-order nonlinear optics (or active linear optics): squeezers, along with homodyne or heterodyne detection measurements] and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communication, one has to incorporate non-Gaussian, i.e., third- or higher-order nonlinear operations in the receiver. Finally we compare our Gaussian receiver limit with experimentally feasible non-Gaussian receivers and show that in the regime of low received photon flux, it is possible to overcome the Gaussian receiver limit by relatively simple non-Gaussian receivers based on photon counting.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated codewords with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations (passive linear optics: beamsplitters and phase-shifters, second order nonlinear process (squeezers), along with homodyne or heterodyne detection measurements) and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communications, one has to incorporate non-Gaussian (i.e. higher order nonlinear) operations in the receiver.
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.
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.