An area law for entanglement from exponential decay of correlations

Journal name:
Nature Physics
Volume:
9,
Pages:
721–726
Year published:
DOI:
doi:10.1038/nphys2747
Received
Accepted
Published online

Abstract

Area laws for entanglement in quantum many-body systems give useful information about their low-temperature behaviour and are tightly connected to the possibility of good numerical simulations. An intuition from quantum many-body physics suggests that an area law should hold whenever there is exponential decay of correlations in the system, a property found, for instance, in non-critical phases of matter. However, the existence of quantum data-hiding states—that is, states having very small correlations, yet a volume scaling of entanglement—was believed to be a serious obstruction to such an implication. Here we prove that notwithstanding the phenomenon of data hiding, one-dimensional quantum many-body states satisfying exponential decay of correlations always fulfil an area law. To obtain this result we combine several recent advances in quantum information theory, thus showing the usefulness of the field for addressing problems in other areas of physics.

At a glance

Figures

  1. EDC intuitively suggests an area law.
    Figure 1: EDC intuitively suggests an area law.

    a, The intuition is exemplified in a simple manner by a state consisting of entangled pairs of neighbouring particles. There the correlations are of fixed length 2, as only neighbours are correlated. The particles connected by an edge are in the pure state ψ=(1/2)(|00+|11), and so only the pairs crossing the boundary (dark blue) contribute to the entropy of the region inside the boundary (shaded square). b, For 1D states an area law implies that the entropy of an interval is constant. Again for a system of entangled pairs, only one pair cut the boundary. c,d, A general intuitive argument is the following: if the distance of two parts A and C is larger than the correlation length, the reduced state ρAC should be close to a product state: , then suggesting that the system B can be divided into subsytems B1 and B2 such that the total pure state ψABC is close to the product state . However, for pure bipartite states, the entropy cannot exceed the size of any of subsystems. Therefore, S(A)≤S(B1)O(ξ), and we would obtain that entropy of any interval is constant and proportional to the correlation length ξ.

  2. Data hiding as an obstruction.
    Figure 2: Data hiding as an obstruction.

    a, In quantum information theory there is the phenomenon of data hiding: one can find bipartite states ρAC that can be very well discriminated for example from the maximally mixed state if one has access to both subsystems A and C, but are indistinguishable from the maximally mixed state by parties that can make only local measurements on the subsystems. Such states can be obtained by picking a random pure tripartite state |ψright fenceABC of n qubits, with B having a small fraction of the total number of qubits. These states have decaying correlations (for the above range of sizes of B), but their subsystems A and C are almost maximally mixed, hence following a volume law. This shows that the intuitive argument of Fig. 1 is flawed: the state ρAC of Fig. 1 can be far away from any product state (in a trace norm or fidelity, as needed to carry over the argument outlined in Fig. 1), as is the case for data-hiding states. b, However, random states have strong correlations in a different partition. If we divide it into subsystems ABC′ such that the number of qubits of AB′ is smaller than that of C′, then ρAB is very close to the maximally mixed state in trace distance, and by the decoupling argument42, 43 C′ can be divided into C1C2 such that ψAC1 is maximally entangled. Hence, correlations between A′ and C′ are of the order of 1.

  3. Revealing correlations: main steps of the proof.
    Figure 3: Revealing correlations: main steps of the proof.

    a, Saturation lemma: the mutual information between regions BC and BLBR satisfies I(BC:BLBR)≤εl. b, Using state merging to infer subvolume law from saturation of mutual information. c, Using state merging to infer area law from subvolume law. The pink region plays the role of the party who makes the random measurement in the state merging protocol, the blue region plays the role of the party who obtains the other half of the maximally entangled state, and the white region plays the role of the reference party who does not actively participate in the protocol.

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Affiliations

  1. Department of Computer Science, University College London, London WC1E 6BT, UK

    • Fernando G. S. L. Brandão
  2. National Quantum Information Center of Gdansk, 81824 Sopot, Poland

    • Fernando G. S. L. Brandão
  3. Institute for Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland

    • Michał Horodecki

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