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Characterizations of Synchronizability - in terms of road-colored graphs, Markov chains and quantum Markov processes

Knof, Albrun (2024)
Characterizations of Synchronizability - in terms of road-colored graphs, Markov chains and quantum Markov processes.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00026533
Ph.D. Thesis, Primary publication, Publisher's Version

Abstract

The thesis is dedicated to the investigation of the property of synchronizability of road-colored directed graphs, which can also be understood as deterministic finite automata. In this context, Markov chains, a special class of stochastic processes, are considered. They have a canonical representation as directed graphs and, conversely, a directed graph with a probability distribution on the outgoing edges of its vertices induces a Markov chain. Such an identification is also possible between Markov chains and road-colored directed graphs. Especially with regard to the property of synchronizability, this connection has already led to interesting results in both areas, the deepening and expansion of which is one of the main goals of this work. Moreover, it has already been shown in the field of quantum probability theory that the notion of asymptotic completeness of certain quantum Markov processes, motivated by scattering theory, is closely related to the classical idea of synchronizability. Indeed, when considering commutative systems, the synchronizability of a road-colored graph provides an equivalent characterization of the asymptotic completeness of the associated quantum Markov process. In this framework, the present thesis deals with possible generalizations of the classical concept of synchronizability in a quantum mechanical context.

Item Type: Ph.D. Thesis
Erschienen: 2024
Creators: Knof, Albrun
Type of entry: Primary publication
Title: Characterizations of Synchronizability - in terms of road-colored graphs, Markov chains and quantum Markov processes
Language: English
Referees: Kümmerer, Prof. Dr. Burkhard ; Gohm, Dr. Rolf
Date: 22 March 2024
Place of Publication: Darmstadt
Collation: vi, 162 Seiten
Refereed: 20 December 2023
DOI: 10.26083/tuprints-00026533
URL / URN: https://tuprints.ulb.tu-darmstadt.de/26533
Abstract:

The thesis is dedicated to the investigation of the property of synchronizability of road-colored directed graphs, which can also be understood as deterministic finite automata. In this context, Markov chains, a special class of stochastic processes, are considered. They have a canonical representation as directed graphs and, conversely, a directed graph with a probability distribution on the outgoing edges of its vertices induces a Markov chain. Such an identification is also possible between Markov chains and road-colored directed graphs. Especially with regard to the property of synchronizability, this connection has already led to interesting results in both areas, the deepening and expansion of which is one of the main goals of this work. Moreover, it has already been shown in the field of quantum probability theory that the notion of asymptotic completeness of certain quantum Markov processes, motivated by scattering theory, is closely related to the classical idea of synchronizability. Indeed, when considering commutative systems, the synchronizability of a road-colored graph provides an equivalent characterization of the asymptotic completeness of the associated quantum Markov process. In this framework, the present thesis deals with possible generalizations of the classical concept of synchronizability in a quantum mechanical context.

Alternative Abstract:
Alternative abstract Language

Diese Thesis widmet sich der Untersuchung der Eigenschaft der Synchronisierbarkeit straßengefärbter gerichteter Graphen, die auch als deterministische endliche Automaten verstanden werden können. In diesem Rahmen werden Markovketten, eine spezielle Klasse stochastischer Prozesse, betrachtet. Diese besitzen eine kanonische Darstellung als gerichtete Graphen und auch umgekehrt induziert ein gerichteter Graph, versehen mit einer Wahrscheinlichkeitsverteilung auf den ausgehenden Kanten der einzelnen Knoten, eine Markovkette. Eine solche Identifikation ist auch zwischen Markovketten und straßengefärbten gerichteten Graphen möglich. Insbesondere hinsichtlich der Eigenschaft der Synchronisierbarkeit, hat dieser Zusammenhang bereits zu interessanten Erkenntnissen in beiden Gebieten geführt, deren Vertiefung und Ausbau eines der Hauptziele dieser Arbeit ist. Darüber hinaus wurde im Bereich der Quantenwahrscheinlichkeitstheorie bereits gezeigt, dass der aus der Streutheorie motivierte Begriff der asymptotischen Vollständigkeit bestimmter Quantenmarkovprozesse eng mit der klassischen Idee der Synchronisierbarkeit verknüpft ist. Tatsächlich liefert die Synchronisierbarkeit eines straßengefärbten Graphen bei der Betrachtung kommutativer Systeme eine äquivalente Charakterisierung der asymptotischen Vollständigkeit des zugehörigen Quantenmarkovprozesses. In diesem Rahmen befasst sich die vorliegende Thesis mit möglichen Verallgemeinerungen des klassischen Konzepts der Synchronisierbarkeit in einem quantenmechanischen Kontext.

German
Uncontrolled Keywords: Markovketten, Quantenmechanik, Quantenwahrscheinlichkeitstheorie, fast gleichmäßige Konvergenz, Synchronisierbarkeit, synchronisierendes Wort, asymptotische Vollständigkeit, straßengefärbte Graphen, Quantenmarkovprozess, Markov chain, quantum mechanics, quantum probability theory, almost uniform convergence, synchronizability, synchronizing word, asymptotic completeness, road-colores graph, road-coloring, quantum markov process, von Neumann algebra, dual extended tranistion operator, Moller operator, Coupling from the past
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-265335
Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Didactics and Pedagogy of Mathematics
Date Deposited: 22 Mar 2024 13:10
Last Modified: 28 Mar 2024 08:44
PPN:
Referees: Kümmerer, Prof. Dr. Burkhard ; Gohm, Dr. Rolf
Refereed / Verteidigung / mdl. Prüfung: 20 December 2023
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