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Brief communications

Remarks on Quantum Markov States

Z. I. Bezhaeva^a, V. I. Oseledets<sup>bc

a</sup> Moscow State Institute of Electronics and Mathematics — Higher School of Economics
^b Financial University under the Government of the Russian Federation, Moscow
^c Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The definition of a quantum Markov state was given by Accardi in 1975. For the classical case, this definition gives hidden Markov measures, which, generally speaking, are not Markov measures. We can use a nonnegative transfer matrix to define a Markov measure. We use a positive semidefinite transfer matrix and select a class of quantum Markov states (in the sense of Accardi) on the inductive limit of the $C^*$-algebras $M_{d^n}$. An entangled quantum Markov state in the sense of Accardi and Fidaleo is a quantum Markov state in our sense. For the case where the transfer matrix has rank $1$, we calculate the eigenvalues and the eigenvectors of the density matrices determining the quantum Markov state. The sequence of von Neumann entropies of the density matrices of this state is bounded.

Keywords: $C^*$-algebra, state on $C^*$-algebra, density matrix, quantum Markov state, von Neumann entropy.

UDC: 519.2

Received: 30.11.2014

DOI: 10.4213/faa3202

 English version: Functional Analysis and Its Applications, 2015, 49:3, 205–209

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