Optimal entanglement witnesses for continuous-variable systems

Abstract

This paper is concerned with all tests for continuous-variable entanglement\nthat arise from linear combinations of second moments or variances of canonical\ncoordinates, as they are commonly used in experiments to detect entanglement.\nAll such tests for bi-partite and multi-partite entanglement correspond to\nhyperplanes in the set of second moments. It is shown that all optimal tests,\nthose that are most robust against imperfections with respect to some figure of\nmerit for a given state, can be constructed from solutions to semi-definite\noptimization problems. Moreover, we show that for each such test, referred to\nas entanglement witness based on second moments, there is a one-to-one\ncorrespondence between the witness and a stronger product criterion, which\namounts to a non-linear witness, based on the same measurements. This\ngeneralizes the known product criteria. The presented tests are all applicable\nalso to non-Gaussian states. To provide a service to the community, we present\nthe documentation of two numerical routines, FULLYWIT and MULTIWIT, which have\nbeen made publicly available.\n