Correlation matrices of two-mode bosonic systems

Abstract

We present a detailed analysis of all the algebraic conditions an arbitrary $4\ifmmode\times\else\texttimes\fi{}4$ symmetric matrix must satisfy in order to represent the correlation matrix of a two-mode bosonic system. Then, we completely clarify when this arbitrary matrix can represent the correlation matrix of a separable or entangled Gaussian state. In this analysis, we introduce alternative sets of conditions, which are expressed in terms of local symplectic invariants.