Entanglement and area laws in weakly correlated Gaussian states

Abstract

We examine the evaluation of entanglement measures in weakly correlated Gaussian states. It is shown that they can be expressed in terms of the singular values of a particular block of the generalized contraction matrix. This result enables us to obtain in a simple way asymptotic expressions and related area laws for the entanglement entropy of bipartitions in pure states, as well as for the logarithmic negativity associated with bipartitions and also with pairs of arbitrary subsystems. As illustration, we consider different types of contiguous and noncontiguous blocks in two-dimensional lattices. Exact asymptotic expressions are provided for both first-neighbor and full-range couplings, which lead in the first case to area laws depending on the orientation and separation of the blocks. © 2012 American Physical Society.