Quantum Markov fields on graphs

TL;DR

This paper extends quantum Markov chains to graph-based $C^*$-algebras, introducing generalized states and chains, and constructs entangled Markov fields on trees and Cayley graphs, broadening the framework's applicability.

Contribution

It introduces generalized quantum Markov states and chains on graphs, expanding the theoretical framework beyond traditional spin systems.

Findings

Construction of entangled Markov fields on tree graphs

Analysis of generalized d-Markov chains on Cayley trees

Extension of quantum Markov chains to graph-based $C^*$-algebras

Abstract

We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on $C^*$-algebras defined by general graphs. As examples of generalized d-Markov chains, we construct the entangled Markov fields on tree graphs. The concrete examples of generalized d-Markov chains on Cayley trees are also investigated.