Entanglement-area law for general bosonic harmonic lattice systems (14 pages)

By M. Cramer, J. Eisert, Martin Bodo Plenio, J. Dreissig. 2005

172 citations recorded by the source corpus.

We demonstrate that the entropy of entanglement and the distillable entanglement of regions with respect to the rest of a general harmonic-lattice system in the ground or a thermal state scale at most as the boundary area of the region. This area law is rigorously proven to hold true in noncritical harmonic-lattice systems of arbitrary spatial dimension, for general finite-ranged harmonic interactions, regions of arbitrary shape, and states of nonzero temperature. For nearest-neighbor interactions—corresponding to the Klein-Gordon case— upper and lower bounds to the degree of entanglement can be stated explicitly for arbitrarily shaped regions, generalizing the findings of Phys. Rev. Lett. 94, 060503 2005. These higher-dimensional analogs of the analysis of block entropies in the one-dimensional case show that under general conditions, one can expect an area law for the entanglement in noncritical harmonic many-body systems. The proofs make use of methods from entanglement theory, as well as of results on matrix functions of block-banded matrices. Disordered systems are also considered. We moreover construct a class of examples for which the two-point correlation length diverges, yet still an area law can be proven to hold. We finally consider the scaling of classical correlations in a classical harmonic system and relate it to a quantum lattice system with a modified interaction. We briefly comment on a general relationship between criticality and area laws for the entropy of entanglement.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/db85a85d7159ebd44a2343d8b1aaa845a039d5f7

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Citations in the corpus

The Physics of Communicability in Complex Networks (2011, methodology): “…information theory, where it has been used, for example, to establish area laws for the entanglement entropy; see, e.g., (Cramer and Eisert, 2006; Cramer et al., 2006; Hastings, 2004; Hastings and Koma, 2006; Schuch, 2007; Schuch et al., 2006) and especially (Eisert et al., 2010) for a…”
Computations in quantum tensor networks (2012, background): “On a general scale, this can be made precise in terms of area laws , where the correlations are quantiﬁed by a measure termed ‘entanglement entropy’ of ground states [13, 14, 15], see also the recent review in [16].”
Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions (2021, background): “[75, 92, 93] for a detailed discussion of bosonic systems) according to”
Holographic entanglement entropy in Lovelock gravities (2011, result): “This result, known as the “ area law” for EE, was first found numerically [30, 31] and later derived analytically [32–37].”
DECAY BOUNDS AND ( ) ALGORITHMS FOR APPROXIMATING FUNCTIONS OF SPARSE MATRICES (2007, background): “Here T has entries tij = e −0.1|i−j| and D is a random diagonal matrix with entries in the interval (5, 6).”
Decay properties for functions of matrices over $C^*$-algebras (2013, unknown): “Such properties play an important role in various applications including electronic structure computations in quantum chemistry [6,15], quantum information theory [19,20,29,56], computational harmonic analysis [37,41], high-dimensional statistics [2], random matrix theory [51] and numerical analysis”