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Journal of Stochastic Analysis

ABOUT FIXED POINTS OF QUANTUM CHANNELS

Raffaella Carbone

Published Mar 1, 2025• 0 citations
Publisher

Abstract

We discuss the structure of the fixed points subspace for a quantum Markov map, reviewing some old and recent results. Our discussion includes various examples that illustrate a range of different scenarios, particularly highlighting the situation when the fixed points form a Von Neumann algebra. Relations with transience and recurrence properties and with absorption problems will also be examined. All this theory is applicable to both discrete and continuous time quantum Markov semigroups

DOI
10.31390/josa.6.1.04

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In this paper, we study some relevant properties of generic quantum Markov semigroups, in particular related to their asymptotic behavior. We can describe the structure of the set of fixed points and of the invariant states in terms of the Hamiltonian’s spectrum and of the communication classes of the classical Markov process associated with the diagonal part of the semigroup. Moreover we study the decoherence-free algebra and we complete the characterization of environmental decoherence for a generic quantum Markov semigroup.
Abstract We introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.