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arXiv:cond-mat/9504090

Boundary conditions for quantum lattice systems

M. Fannes, Reinhard F. Werner

Published Apr 24, 1995• 11 citations
arXivHTMLPDF

Abstract

For classical lattice systems, the Dobrushin-Lanford-Ruelle theory of boundary conditions states that the restriction of a global equilibrium state to a subsystem can be obtained as an integral over equilibrium states of the subsystem alone. The Hamiltonians for the subsystem are obtained by fixing a configuration for the variables in the complement of the subsystem, or more generally, by evaluating the full interaction Hamiltonian with respect to a state for the complement. We provide examples showing that the quantum mechanical version of this statement is false. It fails even if the subsystem is classical, but embedded into a quantum environment. We suggest an alternative characterization of the local restrictions of global equilibrium states by inequalities involving only local data.

DOI
10.48550/arxiv.cond-mat/9504090

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In the field of quantum information theory, the concept of quantum fidelity is employed to quantify the similarity between two quantum states. It has been observed that the fidelity between two states describing a bipartite quantum system A⊗B is always less than or equal to the quantum fidelity between the states in subsystem A alone. While this fidelity inequality is well understood, determining the conditions under which the inequality becomes an equality remains an open question. In this paper, we present the necessary and sufficient conditions for the equality of fidelities between a bipartite system A⊗B and subsystem A, considering pure quantum states. Moreover, we provide explicit representations of quantum states that satisfy the fidelity equality, based on our derived results.
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We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.
Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error ε\varepsilonε using Npolylog⁡(N/ε)N\operatorname{polylog}(N/\varepsilon)Npolylog(N/ε) gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.
Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosman's complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.