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Russian Mathematical Surveys

On the ergodic principle for Markov processes associated with quantum quadratic stochastic processes

Farrukh Maksutovich Mukhamedov

Published Dec 31, 2002• 3 citations
Publisher

Abstract

In [1] the non-stationaryMarkov process generated by a quantum quadratic stochastic process (QQSP) was defined, and it was noted that if the QQSP satisfies the ergodic principle, then so does the Markov process. In the present article we use a decomposition of a QQSP (see [2]) to prove that the converse is also true, that is, if the Markov process satisfies the ergodic principle, then so does the corresponding QQSP.

DOI
10.1070/rm2002v057n06abeh000588

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In [1] the non-stationaryMarkov process generated by a quantum quadratic stochastic process (QQSP) was defined, and it was noted that if the QQSP satisfies the ergodic principle, then so does the Markov process. In the present article we use a decomposition of a QQSP (see [2]) to prove that the converse is also true, that is, if the Markov process satisfies the ergodic principle, then so does the corresponding QQSP.
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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.