Exponential Decay of Correlations Implies Area Law

TL;DR

This paper proves that exponential decay of correlations in 1D quantum states guarantees an area law for entanglement entropy, enabling efficient classical simulation and providing insights into quantum many-body systems.

Contribution

It establishes a rigorous link between exponential decay of correlations and the area law, and shows that such states can be efficiently approximated by matrix product states.

Findings

Exponential decay of correlations implies an area law for entanglement entropy.

States with exponential decay can be efficiently represented as matrix product states.

Quantum states with short-range correlations are classically simulatable.

Abstract

We prove that a finite correlation length, i.e. exponential decay of correlations, implies an area law for the entanglement entropy of quantum states defined on a line. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings proof of an area law for groundstates of 1D gapped Hamiltonians. 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, thus giving an equivalence between injective matrix product states and states with a finite correlation length. The result can be seen as a rigorous justification, in one dimension, of 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 for 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 entanglement distillation protocols achieving the hashing bound, properties of single-shot smooth entropies, and the quantum substate theorem - and also on some newly developed ones. In particular we derive a new bound on correlations established by local random measurements, and we 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 on the phenomenon of data hiding in quantum states.