  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.
