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Stochastic Models

Quantum Markov Semi-Groups of Quasi-Generic Spin Systems

LEOPOLDO PANTALEÓN-MARTÍNEZ, Roberto B. Quezada

Published Nov 5, 2008• 0 citations
Publisher

Abstract

We prove existence of the quantum Markov semi-groups associated with the class of generators deduced by Accardi and Kozyrev[ 1 Accardi , L. ; Kozyrev S. Lectures on quantum interacting particle systems . In Quantum Interacting Particle Systems ; Accardi , L. , Fagnola , F. , Eds.; World Scientific Publishing Co .: River Edge, NJ , 2002 ; 1 – 195 .[Crossref] , [Google Scholar] ] through the stochastic limit of quasi-generic quantum spin systems. We introduce new conditions of polynomial and exponential decay of the coefficients, which replace the more restrictive finite range condition used by the above mentioned authors.

DOI
10.1080/15326340802437827

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WORLD SCIENTIFIC eBooks•Jan 1, 2002•0 citations
We prove existence of the quantum Markov semi-groups associated with the class of generators deduced by Accardi and Kozyrev[ 1 Accardi , L. ; Kozyrev S. Lectures on quantum interacting particle systems . In Quantum Interacting Particle Systems ; Accardi , L. , Fagnola , F. , Eds.; World Scientific Publishing Co .: River Edge, NJ , 2002 ; 1 – 195 .[Crossref] , [Google Scholar] ] through the stochastic limit of quasi-generic quantum spin systems. We introduce new conditions of polynomial and exponential decay of the coefficients, which replace the more restrictive finite range condition used by the above mentioned authors.
We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.
In this paper we study a class of quantum Markov semigroups whose restriction to an abelian sub-algebra coincides, on the configurations with finite support, with the exclusion type semigroups introduced in Liggett's book14 of exchange rates [Formula: see text] not symmetric in the index site r, s. We find a sufficient condition for the existence of infinitely many faithful diagonal (or classical) invariant states for the semigroup, that satisfy a quantum detailed balance condition. This class of semigroups arises naturally in the stochastic limit of quantum interacting particles in the sense of Accardi and Kozyrev.1 We call these semigroups asymmetric exclusion quantum Markov semigroups and the associated processes asymmetric exclusion quantum processes.
We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian HS, acting on h, interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator HS determines a privileged orthonormal basis of h. These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.
We study a special class of generic quantum Markov semigroups, on the algebra of all bounded operators on a Hilbert space [Formula: see text], arising in the stochastic limit of a generic system interacting with a boson–Fock reservoir. This class depends on an orthonormal basis of [Formula: see text]. We obtain a new estimate for the trace distance of a state from a pure state and use this estimate to prove that, under the action of a semigroup of this class, states with finite support with respect to the given basis converge to equilibrium with a speed which is exponential, but with a polynomial correction which makes the convergence increasingly worse as the dimension of the support increases (Theorem 5.1). We interpret the semigroup as an algorithm, its initial state as input and, following Belavkin and Ohya,10 the dimension of the support of a state as a measure of complexity of the input. With this interpretation, the above results mean that the complexity of the input "slows down" the convergence of the algorithm. Even if the convergence is exponential and the slow down the polynomial, the constants involved may be such that the convergence times become unacceptable from a computational standpoint. This suggests that, in the absence of estimates of the constants involved, distinctions such as "exponentially fast" and "polynomially slow" may become meaningless from a constructive point of view. We also show that, for arbitray states, the speed of convergence to equilibrium is controlled by the rate of decoherence and the rate of purification (i.e. of concentration of the probability on a single pure state). We construct examples showing that the order of magnitude of these two decays can be quite different.
The problem of extending ideas and results on the dynamics of infinite classical lattice systems to the quantum domain naturally arises in different branches of physics (nonequilibrium statistical mechanics, quantum optics, solid state, ...) and new momentum from the development of quantum computer and quantum neural networks (which are in fact interacting arrays of binary systems) has been found. The stochastic limit of quantum theory allowed to deduce, as limits of the usual Hamiltonian systems, a new class of quantum stochastic flows which, when restricted to an appropriate Abelian subalgebra, produces precisely those interacting particle systems studied in classical statistical mechanics. Moreover, in many interesting cases, the underlying classical process "drives" the quantum one, at least as far as ergodicity or convergence to equilibrium are concerned. Thus many deep results concerning classical systems can be directly applied to carry information on the corresponding quantum system. The thermodynamic limit itself is obtained thanks to a technique (the four-semigroup method, new even in the classical case) which reduces the infinitesimal structure of a stochastic flow to that of four semigroups canonically associated to it (Chap. 1). Simple and effective methods to analyze qualitatively the ergodic behavior of quantum Markov semigroups are discussed in Chap. 2. Powerful estimates used to control the infinite volume limit, ergodic behavior and the spectral gap (Gaussian, exponential and hypercontractive bounds, classical and quantum logarithmic Sobolev inequalities, ...) are discussed in Chap. 3