Transition effect matrices and quantum Markov chains
type: Item
instance of: scholarly article
title: Transition effect matrices and quantum Markov chains
zbMATH Open document ID: 1179.81019
published in: Foundations of Physics
publication date: 2009-10-21T00:00:00Z
review text: The paper under review is concerned with the analysis of the notion of transition effect matrices, a generalisation of stochastic matrices to quantum-probabilistic context. A transition effect matrix is a (possibly infinite) matrix whose entries are positive contractions on a Hilbert space and whose rows sum to \(I\) in the strong operator topology. These act on `vector states' (understood here as possibly infinite sequences of positive trace-class operators whose traces sum to \(1\)) via a combination of the matrix multiplication and the so-called sequential product of positive Hilbert operators: \(A \circ B:= A^{\frac{1}{2}} B A^{\frac{1}{2}}\). By analogy with the standard setup of Markov chains the author studies the properties such as invertibility, bi-stochasticity or unitarity of a given transition effect matrix and invariance or equilibrium of `vector states'. For other approaches to quantum Markov chains see the lectures of \textit{L. Accardi} [Investig. 16, 1--128 (2001; Zbl 1033.81048)] and \textit{N.\,Konno} [Lecture Notes in Mathematics 1954, 309--452 (2008; Zbl 1329.82011)].
Mathematics Subject Classification ID: 81P15
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 81S25
zbMATH DE Number: 5619052
zbMATH Keywords: quantum Markov chains
zbMATH Keywords: quantum effects
zbMATH Keywords: quantum measurement
zbMATH Keywords: sequential product
zbMATH Keywords: equilibrium states
author: Stan Gudder
reviewed by: Adam Skalski
MaRDI profile type: Publication
full work available at URL: s10701-008-9269-2
OpenAlex ID: W2047488223
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DOI: 10.1007/S10701-008-9269-2
description: scientific article; zbMATH DE number 5619052