Irreducibility in Gaussian Quantum Markov Semigroups

par Federico Girotti (Politecnico di Milano)

jeudi 25 juin 2026, 11:00 → 12:00 Europe/Paris

Huron (106) (Bât. 1R1)

Gaussian Quantum Markov Semigroups (GQMSs) constitute the class of all Markovian evolutions of bosonic systems that map quantum Gaussian states into quantum Gaussian states. They first appeared in the physics literature in the study of reduced evolutions of bosonic systems governed by quadratic Hamiltonians and interacting linearly with bosonic fields. Such models efficiently describe a wide range of experiments in quantum optics and correspond to operations that can be realistically implemented in the laboratory.

Besides their physical relevance, there are strong mathematical motivations for the study of GQMSs. Indeed, these semigroups possess a remarkably rich mathematical structure, allowing for a detailed understanding of many aspects of their dynamics, as in the case of their classical counterparts, namely Ornstein–Uhlenbeck semigroups. In the case of finitely many modes, the evolution is described by finitely many parameters, and several problems can be reformulated in linear-algebraic terms.

Despite the ubiquity of quantum Gaussian evolutions in the literature, their rigorous mathematical understanding remains less developed than that of their classical counterparts. Even in the finite-dimensional case, the quantum framework presents several additional challenges arising from its richer structure. Nevertheless, there has been growing interest in the rigorous analysis of GQMSs in recent years, leading to a number of significant advances.

In this talk, we will focus on the study of irreducibility for GQMSs. When a normal invariant state exists, the structure of the semigroup turns out to be remarkably rigid, and the reducibility problem can be successfully addressed through the analysis of the set of normal invariant states. By contrast, the general case requires different techniques. We will show how the study of regularisation properties of the semigroup leads to a characterisation of irreducibility in terms of simple conditions on the parameters defining the evolution.