Mean hitting time formula for positive maps

Lardizabal, C.F. ; Velázquez, L. (Universidad de Zaragoza)

Resumen: In the classical theory of Markov chains, one may study the mean time to reach some chosen state, and it is well-known that in the irreducible, finite case, such quantity can be calculated in terms of the fundamental matrix of the walk, as stated by the mean hitting time formula. In this work, we present an analogous construction for the setting of irreducible, positive, trace preserving maps. The reasoning on positive maps generalizes recent results given for quantum Markov chains, a class of completely positive maps acting on graphs, presented by S.Gudder. The tools employed in this work are based on a proper choice of block matrices of operators, inspired in part by recent work on Schur functions for closed operators on Banach spaces, due to F.A.Grünbaum and one of the authors. The problem at hand is motivated by questions on quantum information theory, most particularly the study of quantum walks, and provides a basic context on which statistical aspects of quantum evolutions on finite graphs can be expressed in terms of the fundamental matrix, which turns out to be an useful generalized inverse associated with the dynamics. As a consequence of the wide generality of the mean hitting time formula found in this paper, we have obtained extensions of the classical version, either by assuming only the knowledge of the probabilistic distribution for the initial state, or by enlarging the arrival state to a subset of states.

Idioma: Inglés

DOI: 10.1016/j.laa.2022.06.011

Año: 2022

Publicado en: LINEAR ALGEBRA AND ITS APPLICATIONS 650 (2022), 169-189
ISSN: 0024-3795

Factor impacto JCR: 1.1 (2022)

Categ. JCR: MATHEMATICS rank: 101 / 329 = 0.307 (2022) - Q2 - T1
Categ. JCR: MATHEMATICS, APPLIED rank: 161 / 267 = 0.603 (2022) - Q3 - T2

Factor impacto CITESCORE: 2.2 - Mathematics (Q2)

Factor impacto SCIMAGO: 0.851 - Algebra and Number Theory (Q1) - Numerical Analysis (Q1) - Discrete Mathematics and Combinatorics (Q1) - Geometry and Topology (Q2)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-20R
Financiación: info:eu-repo/grantAgreement/ES/MCIN/AEI/10.13039/501100011033

Tipo y forma: Article (PostPrint)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)