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arXiv:2610.06832

Energy-constrained two-way capacities of pure-loss and quantum-limited amplifier channels

Stefano Pirandola

quant-phmath-phphysics.opticsSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Oct 5, 2026v2 · Oct 6, 2026v3 · Oct 14, 2026
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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 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 $-\log_2(1-η)$. 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.

Comments
Submitted version which includes both pure loss and quantum limited amplification. Comments and feedback are welcome

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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 ηηη and mean photon number NNN, all four capacities coincide and are given by g(N)−g((1−η)N)g(N)-g((1-η)N)g(N)−g((1−η)N), where ggg 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>0N>0N>0, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value −log⁡2(1−η)-\log_2(1-η)−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\etaη and mean photon number NNN, all four capacities coincide and are given by g(N)−g((1−η)N)g(N)-g((1-\eta)N)g(N)−g((1−η)N),where ggg 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>0N>0N>0, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value −log⁡2(1−η)-\log_2(1-\eta)−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.
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\kappa>1κ>1 and an unconditional mean input photon number bounded by NNN, quantum transmission, entanglement distribution, private communication, and secret-key agreement share the capacityg(κN+κ−1)−g((κ−1)(N+1))g(\kappa N+\kappa-1)-g((\kappa-1)(N+1))g(κN+κ−1)−g((κ−1)(N+1)), where ggg 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>0N>0N>0, these equal log⁡2[κ/(κ−1)]\log_2[\kappa/(\kappa-1)]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 ηηη and photon budget NNN, both capacities equal g(N)−g((1−η)N)g(N)-g((1-η)N)g(N)−g((1−η)N), where ggg 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.
A gap between achievable rates and converse bounds persists for energy-constrained bosonic communication in the presence of excess noise. We derive a unified adaptive weak-converse bound for thermal attenuation, noisy amplification, and additive Gaussian noise under an unconditional mean transmitted-photon-number constraint. The bound applies to arbitrary adaptive protocols with quantum memories and constrains the two-way quantum, entanglement-distribution, private, and secret-key capacities. It vanishes throughout the entanglement-breaking region and approaches the corresponding PLOB bound at infinite energy. At finite energy, it improves the evaluated Gaussian squashed-entanglement and PLOB bounds in relevant parameter regimes, while comparison with hashing rates quantifies the remaining gap to achievability. The result provides a common energy-dependent benchmark for noisy bosonic Gaussian channels and extends finite-energy converse methods beyond the quantum-limited setting.
We prove that four capacities of single-mode bosonic dephasing channels coincide under an average photon-number constraint: the unassisted quantum capacity, the unassisted private capacity, the two-way classically assisted quantum capacity, and the two-way public-classically assisted private capacity. The assistance may be interactive: the parties can exchange classical messages between channel uses and choose subsequent inputs based on earlier outcomes. For every finite value of the constraint, their common value is the coherent information optimized over Fock-diagonal input states satisfying the constraint. The proof uses a relative entropy comparison with the completely dephasing channel to derive chain rules for the Rains relative entropy and the relative entropy of entanglement. These chain rules give an upper bound on the rates achievable by adaptive protocols whose mean input photon number is constrained on average over all channel uses. We also show that both chain rules are attained by canonical purifications of finite-energy Fock-diagonal states.