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arXiv:2512.11517

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

Franco Fagnola, Federico Girotti

quant-phmath-phmath.PRSubmitted Dec 12, 2025• Announced Dec 15, 2025• 0 citations
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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.

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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.
We study regularisation and irreducibility properties of Gaussian quantum Markov semigroups (GQMSs) acting on continuous-variable quantum systems. We first identify a natural notion of regularity for operators in this setting, which allows us to formulate and characterise the smoothing effects of GQMSs in terms of algebraic conditions involving the drift and quantum diffusion matrices. These conditions establish a connection with the controllability theory of quantum linear systems and with the structure of decoherence-free subsystems. We then characterise irreducibility through several equivalent algebraic criteria: one formulated in terms of the drift and quantum diffusion matrices, one in terms of the operators appearing in the generalised GKLS representation of the generator, and a third given by a quantum analogue of Hörmander's condition. A central and somewhat surprising consequence is that, in contrast with the classical case, irreducibility is strictly stronger than conditions ensuring regularisation. Our results provide an algebraic framework for analysing these properties and lay the groundwork for a broader study of reducible Gaussian quantum Markov semigroups and, more generally, more general relevant quantum Markov semigroups on continuous-variable systems.
We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.
Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad \cite{lindblad}, Stinespring \cite{stinespring}, and Kraus \cite{kraus} imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗L(A)=\sum_{n=1}^{\infty}V_n^*AV_n +GA+AG^*L(A)=∑n=1∞​Vn∗​AVn​+GA+AG∗ where VnV_nVn​ and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.
Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to the works of Lindblad [Commun. Math. Phys. 48, 119-130 (1976)], Stinespring [Proc. Am. Math. Soc. 6, 211-216 (1955)], and Kraus [Ann. Phys. 64, 311-335 (1970)] imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗, where Vn and G are elements of the underlying operator algebra. In the present paper, we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus’ result. Finally, we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.