  For pure states of multi-dimensional quantum lattice systems, which in a
convenient computational basis have amplitude and phase structure of
sufficiently rapid decorrelation, we construct high fidelity approximations of
relatively low complexity. These are used for a conditional proof of area-law
bounds for the states' entanglement entropy. The condition is also shown to
imply exponential decay of the state's mutual information between disjoint
regions, and hence exponential clustering of local observables. The
applicability of the general results is demonstrated on the quantum Ising model
in transverse field. Combined with available model-specific information on
spin-spin correlations, we establish an area-law type bound on the entanglement
in the model's subcritical ground states, valid in all dimensions and up to the
model's quantum phase transition.
