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

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.