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Communications on Stochastic Analysis

Almost any state of any amplitude Markov chain is recurrent

Luigi Accardi, Hiromichi Ohno

Published Dec 1, 2007• 0 citations
Publisher

Abstract

We introduce "amplitude Markov chains" associated to the matrix elements, in a fixed basis, of a unitary operator (discrete quantum dynamics).We prove the amplitude analogue of the relation between the recurrence probability of a state of a classical Markov chain and its first return probability.This formula is then used to prove the universal property, mentioned in the title, which emphasizes a striking difference between the amplitude Markov chains and their classical analogues.This property is probably the statistical reflex of the reversibility of the quantum evolution.Finally note that, in the finite dimensional case, which is the most important one for the applications to quantum information, the word "almost" in the title can be omitted.

DOI
10.31390/cosa.1.3.07

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We introduce "amplitude Markov chains" associated to the matrix elements, in a fixed basis, of a unitary operator (discrete quantum dynamics).We prove the amplitude analogue of the relation between the recurrence probability of a state of a classical Markov chain and its first return probability.This formula is then used to prove the universal property, mentioned in the title, which emphasizes a striking difference between the amplitude Markov chains and their classical analogues.This property is probably the statistical reflex of the reversibility of the quantum evolution.Finally note that, in the finite dimensional case, which is the most important one for the applications to quantum information, the word "almost" in the title can be omitted.
The asymptotic dynamics of discrete quantum Markov chains generated by the most general physically relevant quantum operations is investigated. It is shown that it is confined to an attractor space in which the resulting quantum Markov chain is diagonalizable. A construction procedure of a basis of this attractor space and its associated dual basis of 1-forms is presented. It is applicable whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation. Moreover, algebraic relations between the attractor space and Kraus operators involved in the definition of a quantum Markov chain are derived. This construction is not only expected to offer significant computational advantages in cases in which the dimension of the Hilbert space is large and the dimension of the attractor space is small, but it also sheds new light onto the relation between the asymptotic dynamics of discrete quantum Markov chains and fixed points of their generating quantum operations. Finally, we show that without any restriction our construction applies to all initial states whose support belongs to the so-called recurrent subspace.
We study the classical probability analogue of the unitary dilations of a quantum dynamical semigroup in quantum probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E is realized by a proper choice of the initial random state of the environment. We also compare these dilations with the unitary dilations of a quantum dynamical semigroup in quantum probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of ∗*∗-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.
A quantum chain is analogous to a classical stationary Markov chain except that the probability measure is replaced by a complex amplitude measure and the transition probability matrix is replaced by a transition amplitude matrix. After considering the general situation, we study a particular example of a quantum chain whose transition amplitude matrix has the form of a Dirichlet matrix. Such matrices generate a discrete analog of the usual continuum Feynman amplitude. We then compute the probability distribution for these quantum chains Une chai←ne de Markov quantique est analogue a une chai←ne de Markov stationnaire classique avec la difference que la mesure de probabilite est remplacee par une mesure d'amplitude complexe, et la matrice des probabilites de transition est remplacee par une matrice d'amplitude de transition. Apres avoir considere le cas general, nous etudions le cas particulier d'une chaine quantique dont la matrice de transition est une matrice de Dirichlet. De telles matrices conduisent a un analogue discret des amplitudes de Feynman continues. Nous calculons ensuite les distributions de probabilite de ces chaines quantiques
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.
We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup in Quantum Probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.