Rapid mixing implies exponential decay of correlations

Michael J. Kastoryano; Jens Eisert

doi:10.1063/1.4822481

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arXiv · 1303.6304

Rapid mixing implies exponential decay of correlations

Michael J. Kastoryano·Jens Eisert

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Abstract

We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information - a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped free-fermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrix-product operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.

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Michael J. Kastoryano, Jens Eisert. 2013-03-25. Rapid mixing implies exponential decay of correlations. https://doi.org/10.1063/1.4822481

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