A coupling method for quantum Markov processes (Q2902614)

The book under review is the PhD thesis of the author and concerns the study of quantum Markov processes, with the special focus on the case of those processes which do not admit stationary states. The main theme of the thesis is developing the quantum coupling construction as a new tool for analysing quantum dynamics.NEWLINENEWLINEThe first two chapters are devoted to recalling the basic operator algebraic facts, followed by introducing the terminology of quantum stochastic processes, quantum Markov processes, Markov dilations, tensor dilations, and uniform mixing, all in the language of von Neumann algebras. The latter concepts are given various equivalent characterizations and interpretations; in contrast to the original works of \textit{B.\ Kümmerer} (see for example [Lect. Notes Math. 1866, 259--330 (2006; Zbl 1114.81057)]), here the quantum dilations and quantum processes are studied not only in the framework of faithful normal states, but rather faithful normal semifinite weights. In the third chapter, the author presents the notion of quantum coupling, first describing the classical case and then proposing a noncommutative generalisation. In the same chapter, a quantum coupling inequality based on the concept of a diagonal projection is proved, and some consequences for the uniform mixing established. Chapter 4 contains the discussion of the relations of quantum couplings with quantum coding [\textit{R. Gohmet} et al., Ergodic Theory Dyn. Syst. 26, No. 5, 1521--1548 (2006; Zbl 1121.81087)] and asymptotic completeness [\textit{B. Kümmerer} and \textit{H. Maassen}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3, No. 1, 161--176 (2000; Zbl 1243.81104)], with the focus on the latter. The fifth chapter concerns some tentative results on semiclassical couplings, related to the dual evolutions on the state space of a von Neumann algebra, barycentric measure decompositions and quantum trajectories. Finally a very short sixth chapter describes certain open problems.NEWLINENEWLINEThe results presented in the book are regularly illustrated with physically motivated examples; the arguments are clearly described and generally self-contained, often providing also the proofs of the results established earlier by other authors (providing appropriate references). Most of the concrete applications and computations concern the case of \(B(H)\), with \(H\) finite or infinite-dimensional, playing the role of the von Neumann algebra in question.