  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 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
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.

Quantum Physics;Statistical Mechanics