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

Journal of Physics A Mathematical and Theoretical

Decomposable semigroups on C*-algebras and D-divisible dynamical maps

Krzysztof Szczygielski

Published Jun 5, 2026• 0 citations
Publisher

Abstract

Abstract We analyze semigroups of decomposable maps on CC*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including B ( H ) for H a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators and conclude with some simple examples. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras ( J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad ( Commun. Math. Phys. 48 119–130) on completely positive semigroups.

DOI
10.1088/1751-8121/ae785b

Related papers

Decomposable semigroups on C*-algebras and D-divisible dynamical maps

Szczygielski, Krzysztof

We analyze semigroups of decomposable maps on C*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including $B(\mathcal{H})$ for $\mathcal{H}$ a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras (J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad (Commun. Math. Phys. 48 119-130) on completely positive semigroups.

arXiv:2512.08525•Dec 9, 2025•0 citations

D-divisible quantum evolution families

Krzysztof Szczygielski

Abstract We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.

arXiv:2303.04260•Journal of Physics A Mathematical and Theoretical•Oct 27, 2023•5 citations

Characterizing the dynamical semigroups that do not decrease a quantum entropy

Paolo Aniello, Dariusz Chruściński

Abstract In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Rényi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences—in particular, a description of the associated infinitesimal generators—are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.

arXiv:1601.07808•Journal of Physics A Mathematical and Theoretical•Jul 14, 2016•14 citations

Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups

Frederik vom Ende

We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product.

arXiv:2310.04037•Open Systems & Information Dynamics•Jun 1, 2024•6 citations

Divisibility of qubit channels and dynamical maps

David Davalos, Mário Ziman, Carlos Pineda

The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.

arXiv:1812.11437•Quantum•May 20, 2019•38 citations

Divisibility of quantum dynamical maps and collision models

Sergey N. Filippov, Jyrki Piilo, Sabrina Maniscalco +1 more

The divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under infinitesimal perturbations, and characterize Pauli dynamical semigroups exhibiting such a property. We construct collision models with factorized environment particles, which realize additivity and multiplicativity of generators of CP divisible maps. A mixture of dynamical maps is obtained with the help of correlated environment. The mixture of ultimate CP divisible processes is shown to result in a class of eternal CP indivisible evolutions. We explicitly find collision models leading to weakly and essentially non-Markovian Pauli dynamical maps.

arXiv:1708.04994•Physical Review A•Sep 14, 2017•84 citations
Abstract We analyze semigroups of decomposable maps on CC*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including B ( H ) for H a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators and conclude with some simple examples. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras ( J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad ( Commun. Math. Phys. 48 119–130) on completely positive semigroups.
We analyze semigroups of decomposable maps on C*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including B(H)B(\mathcal{H})B(H) for H\mathcal{H}H a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras (J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad (Commun. Math. Phys. 48 119-130) on completely positive semigroups.
Abstract We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.
Abstract In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Rényi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences—in particular, a description of the associated infinitesimal generators—are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.
We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product.
The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.
The divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under infinitesimal perturbations, and characterize Pauli dynamical semigroups exhibiting such a property. We construct collision models with factorized environment particles, which realize additivity and multiplicativity of generators of CP divisible maps. A mixture of dynamical maps is obtained with the help of correlated environment. The mixture of ultimate CP divisible processes is shown to result in a class of eternal CP indivisible evolutions. We explicitly find collision models leading to weakly and essentially non-Markovian Pauli dynamical maps.