Absorption in Invariant Domains for Semigroups of Quantum Channels

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.

On a generalized Central Limit Theorem and Large Deviations for Homogeneous Open Quantum Walks

“…ρ n = 0 (lim n→+∞ qρ n q − ρ n = 0). By [8,Theorem 14], we know that q − p V ≤ p T , hence to conclude we only need to show that lim n→+∞ p T ρ n p T = 0. Since p T is superharmonic,…”

Absorption and Fixed Points for Semigroups of Quantum Channels

“…The notion of positive recurrence for semigroups of quantum channels traces back to the 70s and it is a fundamental tool for instance for studying the long-time behaviour of quantum systems; taking inspiration from the theory of classical Markov semigroups, transience and the distinction between positive and null recurrence were further analyzed in [8,17] (in finite dimensional quantum systems, as in the case of Markov chains on a finite state space, null recurrence does not show up and the situation is far less complicated). However, there are still some open issues in the theory of recurrence for semigroups of quantum channels; following on from [3], in this work we present some recent results that improve the understanding of null recurrence in the noncommutative setting.…”

“…Another topic which has drown attention in the last years is the study of the fixed points of the evolution: they are relevant for the asymptotics of the evolution of quantum systems ( [10]) and whether or not they are a W * -algebra has implications, for instance, on the relationship between conserved quantities and symmetries of the semigroup (see [2] about the general problem of when fixed points are an algebra and [12] for a discussion about Noether-type results in the context of quantum channels). Fixed points are well understood in the case of positive recurrent semigroups ([4, 9, 14]), while less is known for general semigroups ( [1,3,11]). In the present work we review and improve some results of [3] which, under mild assumptions, characterize the fixed points sets in terms of absorption operators, which are a noncommutative generalization of absorption probabilities introduced always in [3].…”

“…Fixed points are well understood in the case of positive recurrent semigroups ([4, 9, 14]), while less is known for general semigroups ( [1,3,11]). In the present work we review and improve some results of [3] which, under mild assumptions, characterize the fixed points sets in terms of absorption operators, which are a noncommutative generalization of absorption probabilities introduced always in [3].…”

“…In Section 3 we show that the fixed points set of a recurrent semigroup of quantum channels is a W * -algebra. Moreover, assuming that the recurrent space is absorbing, we provide a description of the fixed points set of the semigroup in terms of absorption operators and we present some relevant consequences (these results have already been proved in [3] under stronger assumptions). We recall that the set of fixed points (sometimes called harmonic operators) of a semigroup of quantum channels is the following set:…”