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arXiv:2610.18731

Finite-energy two-way assisted capacities: Exact formulas for multimode loss and fading

Stefano Pirandola, Gaetana Spedalieri

quant-phSubmitted Oct 13, 2026• Announced Oct 14, 2026• 0 citations
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Abstract

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.

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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.
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.
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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.
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 exact quantum communication and entanglement-distribution capacities of three fundamental bosonic channels, namely thermal loss, thermal amplification and additive Gaussian noise, assisted by operations that completely preserve positive partial transpose (PPT). These operations are more powerful than local operations and classical communication (LOCC), but cannot generate distillable entanglement on their own. For thermal loss, the unconstrained capacity coincides with the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound throughout the non-entanglement-breaking regime. The corresponding upper bounds are also attained for thermal amplification and additive noise. We extend these results to energy-constrained communication, establishing exact capacities for all three channels at every prescribed mean input energy. Our results solve a class of bosonic capacity problems under PPT assistance and establish exact benchmarks for determining whether the same communication rates can be achieved via LOCC alone.