Unconstrained distillation capacities of a pure-loss bosonic broadcast channel

TL;DR

This paper determines the exact capacity region for entanglement and secret key distillation over a pure-loss bosonic broadcast channel without energy constraints, using state merging and relative entropy bounds.

Contribution

It establishes the unconstrained capacity region for a pure-loss bosonic broadcast channel, matching inner and outer bounds in the infinite-energy limit.

Findings

Inner bound achieved via state merging protocol

Outer bound derived from relative entropy of entanglement

Capacity region matches in the infinite-energy limit

Abstract

Bosonic channels are important in practice as they form a simple model for free-space or fiber-optic communication. Here we consider a single-sender two-receiver pure-loss bosonic broadcast channel and determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they are allowed arbitrary public classical communication. We show how the state merging protocol leads to achievable rates in this setting, giving an inner bound on the capacity region. We also evaluate an outer bound on the region by using the relative entropy of entanglement and a `reduction by teleportation' technique. The outer bounds match the inner bounds in the infinite-energy limit, thereby establishing the unconstrained capacity region for such channels. Our result could provide a useful benchmark for implementing a broadcasting of entanglement and secret key through such channels. An important open question relevant to practice is to determine the capacity region in both this setting and the single-sender single-receiver case when there is an energy constraint on the transmitter.