Absorption in invariant domains for semigroups of quantum channels
type: Item
instance of: scholarly article
title: Absorption in invariant domains for semigroups of quantum channels
zbMATH Open document ID: 1477.46074
author: Raffaella Carbone
author: Federico Girotti
published in: Annales Henri Poincaré
publication date: 2021-07-29T00:00:00Z
review text: The present article concerns the notion of an absorption operator for a quantum Markov evolution, viewed as a noncommutative counterpart of classical absorption probabilities. Suppose that \(H\) is a Hilbert space and \(\Phi:B(H) \to B(H)\) is a quantum channel, i.e., a normal unital completely positive map. A projection \(p\in B(H)\) is said to be reducing for \(\Phi\) if \(\Phi(p)\geq p\); for such a \(p\) (with the space \(pH\) called also an enclosure) we define the associated absorption operator as the strong limit of the bounded increasing sequence \((\Phi^n(p))_{n=1}^\infty\); the authors include also a natural variant of this definition for a continuous evolution. It turns out that the operators \(A(p)\) encode many ergodic/recurrence/decomposition properties of the evolution. The most satisfactory results are established in the situation where \(A(p_{R^+})=1_H\), where \(R_+\) denotes the space generated by supports of all invariant normal states for the evolution; this assumption turns out to be a natural generalization of the one requiring the existence of a faithful invariant normal state. Thus, for example, if \(A(p_{R^+})=1_H\), the fixed point space of the evolution is a von Neumann algebra, coinciding with the weak\(^*\)-closed linear span of the absorption operators. Apart from developing a general theory, the authors also discuss several concrete examples, illustrating the necessity of the assumptions in the main results.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 81S22
Mathematics Subject Classification ID: 81P47
zbMATH DE Number: 7376258
zbMATH Keywords: quantum channel
zbMATH Keywords: quantum Markov semigroup
zbMATH Keywords: absorption probabilities
zbMATH Keywords: ergodic theory
zbMATH Keywords: quantum recurrence
zbMATH Keywords: fixed points
reviewed by: Adam Skalski
MaRDI profile type: Publication
full work available at URL: s00023-021-01016-5
OpenAlex ID: W3127226964
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DOI: 10.1007/S00023-021-01016-5
description: scientific article; zbMATH DE number 7376258