Friday, June 15, 2012

1206.2947 (Fernando G. S. L. Brandao et al.)

Exponential Decay of Correlations Implies Area Law    [PDF]

Fernando G. S. L. Brandao, Michal Horodecki
We prove that exponential decay of correlations implies an area law for the entanglement entropy of quantum states defined on a line, despite several previous results suggesting otherwise. As a consequence, we show that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings' area law for groundstates of 1D gapped Hamiltonians. The result can be seen as a rigorous justification, in one dimension, to the intuition that states with exponential decay of correlations, usually associated with non-critical phases of matter, are simple to describe. It also has implications to quantum computing: It shows that unless a pure state quantum computation involves states with long-range correlations, decaying at most algebraically with the distance, it can be efficiently simulated classically. The proof relies on several previous tools from quantum information theory - including the quantum state merging protocol, properties of single-shot smooth entropies, and the quantum substate theorem - and also on developing some new ones. In particular we derive a new bound on correlations established by local random measurements, and give a generalization to the max-entropy of a result of Hastings concerning the saturation of mutual information in multiparticle systems. The proof can also be interpreted as providing a limitation to the phenomenon of data hiding in quantum states.
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