Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Structure of covariant uniformly continuous Quantum Markov Semigroups

Nicolò Ginatta

Published May 28, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.15167/ginatta-nicolo-_phd2021-05-28

Related papers

The general structure of the decoherence-free subalgebra for uniformly continuous quantum Markov semigroups

Emanuela Sasso, Veronica Umanità

By using the decomposition of the decoherence-free subalgebra N(T) in direct integrals of factors, we obtain a structure theorem for every uniformly continuous quantum Markov semigroup. Moreover, we prove that when there exists a faithful normal invariant state, N(T) has to be atomic and decoherence takes place.

arXiv:2101.05121•Journal of Mathematical Physics•Apr 1, 2023•4 citations

Structure of uniformly continuous quantum Markov semigroups

Julien Deschamps, Franco Fagnola, Emanuela Sasso +1 more

The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterization of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.

arXiv:1412.3239•Reviews in Mathematical Physics•Feb 1, 2016•43 citations

Structure of Uniformly Continuous Quantum Markov Semigroups with Atomic Decoherence-free Subalgebra

Franco Fagnola, Emanuela Sasso, Veronica Umanità

We establish the structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra and apply the result to derive a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. We deduce the structure of invariant states and a method for finding decoherence-free subsystems and subspaces. The relationship between environment induced decoherence and the decoherence-free subalgebra is also discussed.

Open Systems & Information Dynamics•Sep 1, 2017•12 citations

The Decoherence-Free Subalgebra of Gaussian Quantum Markov Semigroups

Julián Agredo, Franco Fagnola, Damiano Poletti

Abstract We demonstrate a method for finding the decoherence-free subalgebra $${\mathcal {N}}({\mathcal {T}})$$ N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra $${\mathcal {B}}(\Gamma (\mathbb {C}^d))$$ B ( Γ ( C d ) ) of all bounded operator on the Fock space $$\Gamma (\mathbb {C}^d)$$ Γ ( C d ) on $$\mathbb {C}^d$$ C d . We show that $${\mathcal {N}}({\mathcal {T}})$$ N ( T ) is a type I von Neumann algebra $$L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))$$ L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers $$d_c,d_f\le d$$ d c , d f ≤ d . This result is illustrated by some applications and examples.

arXiv:2112.13781•Milan Journal of Mathematics•Jun 1, 2022•13 citations

Invariant Projections for Covariant Quantum Markov Semigroups

Franco Fagnola, Emanuela Sasso, Veronica Umanità

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.

Journal of Stochastic Analysis•Nov 6, 2020•1 citation

Irreducibility of Quantum Markov Semigroups, uniqueness of invariant states and related properties

Franco Fagnola, Federico Girotti

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.

quant-pharXiv:2512.11517•Dec 15, 2025•0 citations
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 establish the structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra and apply the result to derive a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. We deduce the structure of invariant states and a method for finding decoherence-free subsystems and subspaces. The relationship between environment induced decoherence and the decoherence-free subalgebra is also discussed.
Abstract We demonstrate a method for finding the decoherence-free subalgebra N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B(Γ(Cd)){\mathcal {B}}(\Gamma (\mathbb {C}^d))B(Γ(Cd)) B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ(Cd)\Gamma (\mathbb {C}^d)Γ(Cd) Γ ( C d ) on Cd\mathbb {C}^dCd C d . We show that N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) is a type I von Neumann algebra L∞(Rdc;C)⊗‾B(Γ(Cdf))L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))L∞(Rdc​;C)⊗​B(Γ(Cdf​)) L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers dc,df≤dd_c,d_f\le ddc​,df​≤d d c , d f ≤ d . This result is illustrated by some applications and examples.
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.