Entanglement of Bipartite Gaussian States: a Simple Criterion and its Geometric Interpretation

TL;DR

This paper introduces a simplified criterion for determining entanglement in bipartite Gaussian states, based on symplectic matrix calculations, and provides a geometric interpretation involving quantum blobs.

Contribution

It presents a more practical entanglement criterion for Gaussian states and offers a geometric perspective, improving upon previous complex conditions.

Findings

Simpler entanglement criterion using symplectic matrices

Geometric interpretation via quantum blobs

Applicable to bipartite Gaussian states

Abstract

Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..