An area law for entanglement from exponential decay of correlations

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F. G.S.L. Brandao and M. Horodecki
Nature Physics 9, 721-726 (2013)

Many-particle quantum systems in principle need an exponential number of parameters (relative to the
number of particles) to describe. Intuitively, one expects that physically relevant systems, such as those
appearing in nature, need far fewer parameters to be described, and are thus far simpler than the generic case.
However, given a system at low temperature, one would like to know if it is indeed possible to find such a
simple description of the state. One property that many physical systems have is a so called 'exponential
decay of correlations' - correlations between observables on distant parts of the system should decay
exponentially fast in the distance.

In their work, Brandao and Horodecki show that 1D systems that display an exponential decay of
correlations, also show an 'area law' for entanglement, meaning that the entanglement of the system is
proportional to the size of its boundary, as opposed to its size (i.e. volume), as is the case for generic
quantum systems. They then show that this further implies the existence of an efficient classical description
of such states, in terms of so called 'matrix product states', used extensively in numerical simulations of
quantum systems. In total this thus finally provides a rigorous justification of the physical intuition that many
physical systems are simple to describe.