  The holographic principle states that on a fundamental level the information
content of a region should depend on its surface area rather than on its
volume. This counterintuitive idea which has its roots in the nonextensive
nature of black-hole entropy serves as a guiding principle in the search for
the fundamental laws of Planck-scale physics. In this paper we show that a
similar phenomenon emerges from the established laws of classical and quantum
physics: the information contained in part of a system in thermal equilibrium
obeys an area law. While the maximal information per unit area depends
classically only on the number of microscopic degrees of freedom, it may
diverge as the inverse temperature in quantum systems. A rigorous relation
between area laws and correlations is established and their explicit behavior
is revealed for a large class of quantum many-body states beyond equilibrium
systems.
