We present a detailed analysis of all the algebraic conditions an arbitrary $4\times 4$ 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.

• Received 17 February 2009

DOI:https://doi.org/10.1103/PhysRevA.79.052327