We determine the finite-blocklength asymptotics of classical communication over the pure-loss bosonic channel under a maximum-photon-number occupation constraint. Wilde, Renes and Guha previously derived a second-order achievable rate governed by thermal photon-number fluctuations and left its optimality open. We show that fixed-total-photon-number encoding removes input-number fluctuations while preserving the first-order capacity, leaving binomial photon-survival fluctuations as the operational second-order contribution. For fixed transmissivity 0 < η < 1, signal photon number NS > 0, and error probability 0 < ε < 1, we determine the optimal fixed-error expansion through third order. The operational dispersion is Vsh = η(1 − η)NS [log2(1 + 1/(ηNS))]^2, and the optimal code size contains the third-order correction −(1/2)log2 n up to a bounded O(1) remainder. The achievability construction uses fixed-photon-number shell codes, while the converse applies to arbitrary multimode quantum encodings and arbitrary joint decoding measurements within the stated occupation constraint. This Zenodo record contains the preprint, LaTeX source, vector figures, source data, and reproducibility code used for the numerical checks and figure generation.
We determine the finite-blocklength asymptotics of classical communication over the pure-loss bosonic channel under a maximum-photon-number occupation constraint. Wilde, Renes and Guha previously derived a second-order achievable rate governed by thermal photon-number fluctuations and left its optimality open. We show that fixed-total-photon-number encoding removes input-number fluctuations while preserving the first-order capacity, leaving binomial photon-survival fluctuations as the operational second-order contribution. For fixed transmissivity 0 < η < 1, signal photon number NS > 0, and error probability 0 < ε < 1, we determine the optimal fixed-error expansion through third order. The operational dispersion is Vsh = η(1 − η)NS [log2(1 + 1/(ηNS))]^2, and the optimal code size contains the third-order correction −(1/2)log2 n up to a bounded O(1) remainder. The achievability construction uses fixed-photon-number shell codes, while the converse applies to arbitrary multimode quantum encodings and arbitrary joint decoding measurements within the stated occupation constraint. This Zenodo record contains the preprint, LaTeX source, vector figures, source data, and reproducibility code used for the numerical checks and figure generation.
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.
Abstract: We determine the two-way quantum, entanglement-distribution, private, and secret-key capacities of the pure-loss bosonic channel under an unconditional mean transmitted-photon-number constraint. For transmissivity η and mean photon number N, all four capacities coincide and are given by g(N)−g((1−η)N),where g is the entropy of a thermal mode. This resolves the longstanding finite-energy capacity problem by establishing the optimality of the reverse-coherent-information rate introduced in 2009. Our central tool is sector teleportation simulation, which we develop specifically to transfer the input-energy constraint to the shared entanglement resource, thereby preserving the constraint throughout converse arguments for general adaptive protocols. For every N>0, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value −log2(1−η). For parallel protocols with a hard total-photon-number cutoff, the same finite-energy capacity formula instead satisfies an exponential strong converse. Together, these results establish a unified finite-energy framework for adaptive bosonic communication and reveal how energy constraints fundamentally reshape both attainable rates and converse bounds.
This paper reports the proof of a strong converse for the unconstrained quantum capacity of the pure-loss bosonic channel. At every fixed rate above capacity, the entanglement-generation fidelity of every code is bounded by a constant times the reciprocal of the number of channel uses. The bound holds without an energy constraint and for arbitrary encoded states, including states correlated across all input modes, and arbitrary joint decoders. The proof combines quantum Chebyshev and hockey-stick testing inequalities with a uniform relative-entropy-variance bound for the balanced pure-loss channel, corresponding to transmissivity η=1/2. The variance bound follows by expressing the balanced beam splitter in bright and dark modes: the dark modes are exactly in vacuum, and any state orthogonal to that vacuum contains at least one dark photon. For general transmissivity, dilating the degrading attenuator reduces the problem to this balanced-channel setting and bounds the decoder test by precisely the factor that produces the known quantum-capacity threshold. The resulting argument establishes the strong converse at the unconstrained quantum capacity for every pure-loss bosonic channel.
Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for 4≤n≤25 over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of SU(2) and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number nd=⌈(3d2+1)/4⌉ and construct each corresponding code through distance d=10. To our knowledge, the constructed (n,d)=(28,6),(49,8),(76,10) codes give the smallest reported n for their respective distances; all members through d=9 attain the fixed-n LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.
Sending quantum information over long distances is ultimately limited by photon loss. The rate limit of a lossy bosonic channel of transmissivity η is the two-way assisted capacity -log2(1-η) (the PLOB bound), whose achievability rests on unlimited classical feedback: without feedback the quantum capacity vanishes for all η ≤ 1/2, so classical back-communication acts as an on/off switch for quantum communication. Classical bits are priced for entanglement transformations and, for a fixed bipartite state, the one-way distillation trade-off between classical rate and ebit yield has a regularized entropic characterization; but no framework prices the feedback that assists a channel's quantum capacity — feedback that may steer subsequent channel inputs — and no such trade-off has been evaluated for any bosonic channel. Here we show that quantum communication can be metered in classical bits. Writing C(η,B) for the capacity of the pure-loss channel under a feedback budget of B classical bits per channel use, we construct the first achievable-rate frontier C_{ach}(η,B) ≤ C(η,B) and prove the onset law C_{ach}(η,B) ≥ ηlog2(m_B)/m_B, m_B = ⌈ H_2(η)/B⌉, valid for every B>0 with envelope (η/H_2(η)) Blog2(H_2(η)/B) as B → 0 — an exact identity R = η Blog2(1/B) under fixed-length accounting, so the Blog(1/B) order is accounting-convention invariant. The B=0 switch thereby resolves, at the level of achievable rates, into a continuous but non-analytic onset: even for η ≤ 1/2 any positive budget buys a positive quantum rate, each bit unlocking ~log2(1/B) ebits at small budgets and refuting the naive pricing C ≤ Q + B. At large budgets a metered quantum-state-merging family, consuming backward one-way classical communication only, achieves C_{ach}(η,B) ≥ PLOB - [eη^2/(2ln 2)] 2^{-B}(1+o(1)): each backward bit per use halves the achievable distance to PLOB, which exposes the 73.6% plateau of the purification families as a family artifact. Standard entanglement-measure converses are provably blind to B, so what remains open is only strict saturation at finite B, now confined to an exponentially small window. The frontier is also a design tool: on a network of lossy links, the joint allocation of per-link feedback budgets and photon directions under per-site classical budgets is a convex program, and on a 20-site fibre network its optimum doubles the end-to-end entanglement rate relative to a fixed orientation. An appendix extends the frontier to multimode loss, where Grassmann-quantized beamforming controls the PLOB loss at rate 2^{-B/(M-1)}. Preprint (companion paper: Conservation laws for the classical communication cost of one-way entanglement distillation, with entropic converse bounds for the pure-loss channel). Ancillary files: reproduction scripts and data; code maintained at https://github.com/deeptell-inc/fmplob.
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.