We present a proof of the central limit theorem for a pair of mutually noncommuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem. KEY WORDS: central limit theorem; KMS states; quantum spin chain
We present a proof of the central limit theorem for a pair of mutually noncommuting operators in mixing quantum spin chains. The operators are not necessarily strictly local but quasi-local. As a corollary we obtain a direct construction of the time evolution of the algebra of normal fluctuation for Gibbs states of finite range interactions on a one-dimensional lattice. We show that the state of the algebra of normal fluctuation satisfies the β-KMS condition if the microscopic state is a β-KMS state. We show that any mixing finitely correlated state satisfies our assumption for the central limit theorem. KEY WORDS: central limit theorem; KMS states; quantum spin chain
Abstract We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.
We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states—pure or mixed—which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.
This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a discrete quantum state space. The algebra of local observables is constructed. The probability functionals, i.e. the states, are defined in terms of reduced density matrices with their compatibility relations. Several examples of shift-invariant states are discussed: product states, classical states, and limiting Gibbs states. The chapter concludes with the construction of nearest-neighbour dynamics and the need for an appropriate algebra of quasi-local observables.
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.
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 using 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.