  Lieb-Robinson bounds are powerful analytical tools for constraining the
dynamic and static properties of non-relativistic quantum systems. Recently, a
complete picture for closed systems that evolve unitarily in time has been
achieved. In experimental systems, however, interactions with the environment
cannot generally be ignored, and the extension of Lieb-Robinson bounds to
dissipative systems which evolve non-unitarily in time remains an open
challenge. In this work, we prove two Lieb-Robinson bounds that constrain the
dynamics of open quantum systems with long-range interactions that decay as a
power-law in the distance between particles. Using a combination of these
Lieb-Robinson bounds and mixing bounds which arise from "reversibility" --
naturally satisfied for thermal environments -- we prove the clustering of
correlations in the steady states of open quantum systems with long-range
interactions. Our work provides an initial step towards constraining the
steady-state entanglement structure for a broad class of experimental
platforms, and we highlight several open directions regarding the application
of Lieb-Robinson bounds to dissipative systems.
