Covariance matrices for finite-dimensional DFT-related position-momentum pairs are fully characterized via unitary invariants, convex geometry, and SDP, yielding extremal states and application bounds.
Optimal entanglement witnesses for continuous-variable systems
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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. Moreover, we show that for each such test, referred to as 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 non-linear 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.
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Numerical evidence from projections and witnesses on specific Gaussian families leads to the conjecture that full inseparability implies genuine multipartite entanglement for all Gaussian states.
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The uncertainty geometry of finite-dimensional position and momentum
Covariance matrices for finite-dimensional DFT-related position-momentum pairs are fully characterized via unitary invariants, convex geometry, and SDP, yielding extremal states and application bounds.
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On the existence of fully inseparable biseparable Gaussian states
Numerical evidence from projections and witnesses on specific Gaussian families leads to the conjecture that full inseparability implies genuine multipartite entanglement for all Gaussian states.