Stability of energy-constrained bosonic capacities under extensions of LOCC

Abstract: The energy-constrained two-way capacities of pure-loss and quantum-limited amplifier channels have been determined in a combined treatment using finite-round local operations and classical communication with finite or countable outcomes. We prove that the same vanishing-error capacities hold when local instruments have standard Borel outcomes and measurable feedback, and when the classical protocol terminates almost surely without a deterministic bound on its number of rounds. We also specify limiting classes for which the converse remains valid. For quantum and private-state targets, normal strong limits of reduced LOCC channels are covered by complete preservation of separability. For operational secret key, the public realization must be retained: we construct local purifying shields for instruments whose faithful Choi densities factor at each public outcome, and prove an averaged replica bound without representing a continuous record as an orthogonal quantum register. These arguments preserve every transmitted input marginal and require no bound on local memory dimension, energy, or entropy. Energy-controlled limits of complete protocols provide a further extension. The results confirm both capacity formulas under these explicitly defined enlargements and identify why a reduced-map closure alone cannot define secret-key security.