Transition effect matrices and quantum Markov chains
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)].
scientific article; zbMATH DE number 5619052
scientific article