We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated Lp norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated Lp norms are not uniformly convex for 1 ≤ p ≤ 2. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.This part is organised as follows: in Section 2.1 we introduce our notations and recall the definitions of quantum Markov semigroups, their decoherence-free algebra and the notion of environment-induced decoherence. Section 2.2 is devoted to the exposition of the weighted L p norms and the L p Dirichlet forms associated to a quantum Markov semigroup. The main results of this article are presented in Section 2.3, namely the equivalence between hypercontractivity and logarithmic Sobolev inequality in the context of amalgamated L p spaces, and the existence of universal constants. In Section 2.4 we apply our framework to the estimation of decoherence rates. Finally, the study of hypercontractivity for the CB-norms is presented in Section 2.5. Quantum Markov semigroups and environment-induced decoherenceLet (H, .|. ) be a finite dimensional Hilbert space of dimension d H . We denote by B(H) the Banach space of bounded operators on H, by B sa (H) the subspace of self-adjoint operators on H, i.e. B sa (H) = {X = B(H); X = X * }, and by B + sa (H) the cone of positive semidefinite operators on H, where the adjoint of an operator Y is written as Y * . The identity operator on H is denoted by I H , dropping the index H when it is unnecessary. In the case when H ≡ C k , we will also use the notation I k ...

The mixing time of Markovian dissipative evolutions of open quantum manybody systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

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