Remarks on quantum Markov states
type: Item
instance of: scholarly article
title: Remarks on quantum Markov states
zbMATH Open document ID: 1352.46060
author: Z. I. Bezhaeva
author: Valery Oseledets
publication date: 2016-01-06T00:00:00Z
review text: The definition of a quantum Markov state was given by \textit{L. Accardi} [Funct. Anal. Appl. 9, 1--7 (1975); translation from Funkts. Anal. Prilozh. 9, No. 1, 1--8 (1975; Zbl 0395.60076)]. In the classical case, according to the authors, this definition gives hidden Markov measures, which, generally speaking, are not Markov measures. The authors use a nonnegative transfer matrix to define a Markov measure. They 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 \textit{F. Fidaleo} and \textit{L. Accardi} [Ann. Mat. Pura Appl. (4) 184, No. 3, 327--346 (2005; Zbl 1102.46043)] is a quantum Markov state in their sense. For the case where the transfer matrix has rank 1, they 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 turns out to be bounded.
reviewed by: Franco Fagnola
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 6526697
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: state on \(C^*\)-algebra
zbMATH Keywords: density matrix
zbMATH Keywords: quantum Markov state
zbMATH Keywords: von Neumann entropy
MaRDI profile type: Publication
full work available at URL: s10688-015-0105-0
OpenAlex ID: W2216820396
cites work: The noncommutative Markovian property
cites work: Entangled Markov chains
cites work: Finitely correlated states on quantum spin chains
cites work: Q5718636
cites work: Linear transformations which preserve trace and positive semidefiniteness of operators
cites work: Completely positive linear maps on complex matrices
DOI: 10.1007/S10688-015-0105-0
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 6526697