  Many applications of quantum information processing (QIP) require
distribution of quantum states in networks, both within and between distant
nodes. Optical quantum states are uniquely suited for this purpose, as they
propagate with ultralow attenuation and are resilient to ubiquitous thermal
noise. Mechanical systems are then envisioned as versatile interfaces between
photons and a variety of solid-state QIP platforms. Here, we demonstrate a key
step towards this vision, and generate entanglement between two propagating
optical modes, by coupling them to the same, cryogenic mechanical system. The
entanglement persists at room temperature, where we verify the inseparability
of the bipartite state and fully characterize its logarithmic negativity by
homodyne tomography. We detect, without any corrections, correlations
corresponding to a logarithmic negativity of $E_\mathrm{N}=0.35$. Combined with
quantum interfaces between mechanical systems and solid-state qubit processors
already available or under development, this paves the way for mechanical
systems enabling long-distance quantum information networking over optical
fiber networks.
