Quantum Markov fields on graphs

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 $C^{\ast }$-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.

23 pages, 1 figure. accepted to "Infinite Dimensional Anal. Quantum Probability & Related Topics"

Cited by in corpus (10)

• On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with XY-model on the Cayley tree of order two
• Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree
• On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three
• On Quantum Markov Chains on Cayley tree III: Ising model
• Quantum Markov States on Cayley trees
• Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
• Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions
• Refinement of quantum Markov states on trees
• Quantum Markov Chains on the Comb graphs: Ising model
• Entropy of Quantum Markov states on Cayley trees