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Nagoya Mathematical Journal

Quantum Markov chains and classical random sequences

Yun Gang Lu

Published Sep 1, 1995• 8 citations
Publisher

Abstract

Quantum Markov chain introduced by Accardi (cf. [1,2,3]) is one of natural generalization of classical Markov chain. It has many interesting applications in physics and the most important one is given by the paper of Fannes-Nachtergaele-Werner ([4]), where an application of quantum Markov chain’s technique enables us to understand the Valence bond states well.

DOI
10.1017/s0027763000005353

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Quantum Markov chain introduced by Accardi (cf. [1,2,3]) is one of natural generalization of classical Markov chain. It has many interesting applications in physics and the most important one is given by the paper of Fannes-Nachtergaele-Werner ([4]), where an application of quantum Markov chain’s technique enables us to understand the Valence bond states well.
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of (1∣(1|(1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems. In (3), using our entanglement criterion for pure states (16) which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.
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 this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of [1∣[1|[1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems.In [3], using our entanglement criterion for pure states [16] which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.On the other hand we showed that this entanglement condition for pure state is sufficient condition for entanglement in the case of mixtures [4].This fact was then applied to prove that EMC with unitarily im- plementable transition operator induce a mixture entangled state on any local algebra.Interestingly it was also shown that all these local states provide a new class of examples (in any dimension) of entangled states which nevertheless satisfy the PPT condition [4].In this article we report the above results.
Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.