  Entanglement is a fundamental feature of quantum physics and a key resource
for quantum communication, computing and sensing. Entangled states are fragile
and maintaining coherence is a central challenge in quantum information
processing. Nevertheless, entanglement can be generated and stabilised through
dissipative processes. In fact, entanglement has been shown to exist in the
steady state of certain interacting quantum systems subject solely to
incoherent coupling to thermal baths. This has been demonstrated in a range of
bi- and multipartite settings using systems of finite dimension. Here we focus
on the steady state of infinite-dimensional bosonic systems. Specifically, we
consider any set of bosonic modes undergoing excitation-number-preserving
interactions of arbitrary strength and divided between an arbitrary number of
parties that each couple weakly to thermal baths at different temperatures. We
show that a unique steady state is always separable.
