Optimal entanglement witnesses for continuous-variable systems

This paper is concerned with all tests for continuous-variable entanglement that arise from linear combinations of second moments or variances of canonical coordinates, as they are commonly used in experiments to detect entanglement. All such tests for bi-partite and multi-partite entanglement correspond to hyperplanes in the set of second moments. It is shown that all optimal tests, those that are most robust against imperfections with respect to some figure of merit for a given state, can be constructed from solutions to semi-definite optimization problems. Moreover, we show that for each such test, referred to as an entanglement witness based on second moments, there is a one-to-one correspondence between the witness and a stronger product criterion, which amounts to a nonlinear witness, based on the same measurements. This generalizes the known product criteria. The presented tests are all applicable also to non-Gaussian states. To provide a service to the community, we present the documentation of two numerical routines, FullyWit and MultiWit, which have been made publicly available.