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Journal of Stochastic Analysis

Invariant Projections for Covariant Quantum Markov Semigroups

Franco Fagnola, Emanuela Sasso, Veronica Umanità

Published Nov 6, 2020• 1 citation
Publisher

Abstract

In this paper we investigate consequences of covariance of a uniformly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections.We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed.We apply this results in the study circulant Quantum Markov Semigroups.

DOI
10.31390/josa.1.4.03

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In this paper we investigate consequences of covariance of a uniformly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections.We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed.We apply this results in the study circulant Quantum Markov Semigroups.
In this thesis I analyze the structure of decoherence-free subalgebra N(T) of a uniformly continuous covariant Quantum Markov Semigroup (QMS) with respect to a representation ? of a compact group G on a Hilbert space h. In particular, I obtain that, when ? is irreducible, N(T) is isomorphic to (?(k) ? 1m)^d for suitable Hilbert spaces k and m, and an integer d related to the connected components of G. I extend this result when ? is reducible and N(T) is atomic by the decomposition of h due to the Peter?Weyl theorem. The thesis is then concluded by exploring the possibility of extending these results beyond the direct sum by using the concept of direct integral.
We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.
Abstract We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.