Optimal entanglement witnesses for continuous-variable systems

Abstract

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. We moreover characterize all product criteria in the bi- and multi-partite setting, generalizing known product criteria. To provide a service to the community, we have also provided documentation to two numerical routines, FULLYWIT and MULTIWIT, which have been made publicly available.