2012/06/13 by Fernando G. S. L. Brandão, Fernando G. S. L. Brandao, Michał Horodecki +1 · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum many-body systems #cond-mat.stat-mech #cond-mat.str-el #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1007/s00220-014-2213-8
published as Comm. Math. Phys. 333, 761 (2015) · 35 pages, 6 figures; v2 minor corrections; v3 published version
arxiv published 2012/06/13 · openalex publication_date 2014/11/01 · arxiv created 2015/01/07 · arxiv updated 2015/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.