Finite-energy capacity equality for unobserved fading

Determine whether entanglement-distribution capacity and secret-key capacity are equal at a fixed finite mean input-photon-number budget for bosonic channels with unobserved, independently varying pure-loss transmissivity.

Background

For fading channels whose transmissivity realization is unobserved by both communicating parties, the paper derives finite-energy upper and lower bounds on the two-way entanglement-distribution and secret-key capacities. The bounds imply that the possible difference between these capacities is at most a term proportional to the mean energy and the binary entropy of the average transmissivity, but they do not establish equality at a fixed finite budget.

The binary on/off fading channel is identified as a concrete test case for improving distillation protocols and comparing achievable rates with the converse. Resolving the equality question would determine whether the distinction between entanglement distribution and secret-key agreement persists under finite-energy constraints when channel-state information is unavailable.