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arXiv:quant-ph/0405142 · Physical Review Letters

Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems

Martin Bodo Plenio, Jens Eisert, J. Dreißig, M. Cramer

Published Feb 17, 2005• 392 citations
arXivHTMLPDF

Abstract

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.

DOI
10.1103/physrevlett.94.060503

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We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.
We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians which are the discrete version of real free scalar fields. For the cubic harmonic lattice Hamiltonian which yields the real Klein Gordon field in the continuum limit we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures and for classical correlations in classical harmonic lattice systems. These findings suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.
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.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
It is well-known that entanglement entropy in field theory at its ground state is dominated by an area law term, presenting a similarity to the entropy of black holes. It is interesting to investigate whether this similarity can be extended by showing that gravitational dynamics emerges from the first law of entanglement thermodynamics. Answering this question requires the specification of the modular Hamiltonian. Motivated by the above, we study entanglement in the toy model of harmonic systems lying at any classicalmost state, i.e. any coherent state. We specify explicitly the reduced density matrix and its time-evolution, as well as the modular Hamiltonian. Interestingly, the time evolution is unitary and we specify the effective Hamiltonian which generates it. Our results provide the tools to investigate the similarity between gravity and entanglement in discretized free scalar field theory in the framework of (Phys Rev Lett 71:666, 1993).
The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size NNN with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out N−lN-lN−l oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for NrightarrowinftyN\\rightarrow\\inftyNrightarrowinfty, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large NNN, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by S=frac0.27763lnbigg(fracNpisinfracpilNbigg)+0.6421S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421S=frac0.27763lnbigg(fracNpisinfracpilNbigg)+0.6421 in the discrete case and S=frac0.37443lnl+0.6421S=\\frac{0.3744}{3} \\ln{l} + 0.6421S=frac0.37443lnl+0.6421 in the infinite case.
We discuss the relation between entanglement and criticality in translationally invariant harmonic lattice systems with nonrandom, finite-range interactions. We show that the criticality of the system as well as validity or breakdown of the entanglement area law are solely determined by the analytic properties of the spectral function of the oscillator system, which can easily be computed. In particular, for finite-range couplings we find a one-to-one correspondence between an area-law scaling of the bipartite entanglement and a finite correlation length. This relation is strict in the one-dimensional case and there is strong evidence for the multidimensional case. We also discuss generalizations to couplings with infinite range. Finally, to illustrate our results, a specific 1D example with nearest and next-nearest-neighbor coupling is analyzed.