The finite-energy multimode entropy photon-number inequality in the OpenAI mathematics release supplies the sharp entropy premise for the previously conditional pure-loss bosonic classical/quantum/entanglement capacity region under an average photon constraint, including arbitrary entanglement across channel uses. The one-mode pure-loss channel has transmissivity 1/2 ≤ η ≤ 1, with forward resource accounting without free feedback; the photon-number constraint is averaged over messages, key and encoder randomness, rather than a peak or per-codeword bound. This note gives the precise consequence and audits a private-resource operational defect: the printed catalytic secrecy condition in the cited coding theorem protects consumed key jointly with generated secrets and excludes the one-time pad. A zero-energy converse refutes the displayed private region under that literal condition. The public/private/secret-key formula is proved under an explicitly corrected generated-secret trace-secrecy convention, with corrected converse accounting and direct key-assisted coding. The formulas and conditional reductions are inherited; this is not an independent proof of EPnI or a full solution of the false unchanged private target. The upstream unrefereed EPnI proof was analytically audited; its Lean build was not reproduced. AI tools were used extensively in research, drafting and verification. Automated adversarial reviews are not conventional human peer review; this preprint has not undergone conventional human refereeing.
The finite-energy multimode entropy photon-number inequality in the OpenAI mathematics release supplies the sharp entropy premise for the previously conditional pure-loss bosonic classical/quantum/entanglement capacity region under an average photon constraint, including arbitrary entanglement across channel uses. The one-mode pure-loss channel has transmissivity 1/2 ≤ η ≤ 1, with forward resource accounting without free feedback; the photon-number constraint is averaged over messages, key and encoder randomness, rather than a peak or per-codeword bound. This note gives the precise consequence and audits a private-resource operational defect: the printed catalytic secrecy condition in the cited coding theorem protects consumed key jointly with generated secrets and excludes the one-time pad. A zero-energy converse refutes the displayed private region under that literal condition. The public/private/secret-key formula is proved under an explicitly corrected generated-secret trace-secrecy convention, with corrected converse accounting and direct key-assisted coding. The formulas and conditional reductions are inherited; this is not an independent proof of EPnI or a full solution of the false unchanged private target. The upstream unrefereed EPnI proof was analytically audited; its Lean build was not reproduced. AI tools were used extensively in research, drafting and verification. Automated adversarial reviews are not conventional human peer review; this preprint has not undergone conventional human refereeing.
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 energy-constrained 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 energy-constrained capacity formula instead satisfies an exponential strong converse. As an extension, we determine the energy-constrained two-way capacities of the quantum-limited amplifier, showing that classical assistance does not improve its unassisted quantum and private rates. 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.
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.
We study quantum communication, entanglement distribution, and secret-key generation through multimode bosonic loss and fading channels, assisted by local quantum operations and two-way classical communication. Using sector teleportation, we determine the exact two-way quantum and secret-key capacities of multimode pure-loss channels under a mean photon-number constraint, allowing arbitrary adaptive quantum protocols. We extend these results to continuous frequency-dependent loss under a power constraint and identify the optimal allocation of photons among optical modes. For constant loss over an unlimited bandwidth, we obtain a closed-form capacity proportional to the square root of power. For randomly varying loss, we determine exact finite-energy capacities when the channel realization is publicly revealed, distinguishing knowledge before and after transmission. When fading remains unobserved, we derive capacity bounds and explicit formulas for an on/off channel. We also establish strong-converse bounds, with exact thresholds for the deterministic loss models considered here, observed fading, and unobserved fading with finitely many possible transmissivities. Under mean energy constraints, these thresholds generally exceed the corresponding finite-energy capacities. These results guide energy allocation and provide benchmarks for energy-constrained optical quantum communication with multimode signals and variable loss.
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.
Abstract: Two-way classical assistance does not increase the energy-constrained quantum or private capacity of the quantum-limited bosonic amplifier. We establish this optimality through a matching converse for adaptive protocols with arbitrary separable quantum memories. For amplification gain κ>1 and an unconditional mean input photon number bounded by N, quantum transmission, entanglement distribution, private communication, and secret-key agreement share the capacityg(κN+κ−1)−g((κ−1)(N+1)), where g denotes the entropy of a thermal mode. The proof extends sector teleportation simulation from pure loss to amplification. A correspondence between their conditional resources provides the entanglement estimate needed to retain the transmitted-energy budget throughout the adaptive converse, despite the amplifier's unbounded output photon number. We also determine the strong-converse thresholds under the same averaged energy constraint. For every N>0, these equal log2[κ/(κ−1)], separating the energy-dependent vanishing-error capacities from an energy-independent strong-converse threshold.
We determine the two-way quantum and secret-key capacities of the pure-loss channel under an average input photon-number constraint. For transmissivity η and photon budget N, both capacities equal g(N)−g((1−η)N), where g is the entropy of a thermal state. This coincides with the rate achieved by the known reverse coherent information of a two-mode squeezed vacuum. Our result provides the last missing capacity formula for the celebrated pure-loss channel, completing the picture of bosonic quantum communication under pure loss.