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

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 $\eta$ and mean photon number $N$, all four capacities coincide and are given by $g(N)-g((1-\eta)N)$,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 $-\log_2(1-\eta)$. 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.