Optimal fidelity witnesses for gravitational entanglement

Optomechanical systems open new possibilities in fundamental research at the interface between quantum information and gravity. Recently, an ambitious experimental proposal was suggested by Bose et al. [S. Bose, G. Milburn, Phys. Rev. Lett. 119, 240401 (2017)] to measure the entanglement between two optomechanical systems generated by their gravitational interaction. The scheme relied on witnessing entanglement between the two systems. Here we develop a general framework to study the quality of bipartite entanglement witnesses using fidelity witnesses. We then apply this framework to the gravitational entanglement proposal, optimizing for the detection of entanglement. We construct a witness consisting of only five nontrivial spin measurements, which we compare with other proposed witnesses. With postprocessing our witness can detect entanglement for any choice of phases in the setup, up to a set of measure zero, for a closed system. We also explore the effects of a simple dephasing channel on this witness.