# Diagonalizability of Quantum Markov States on Trees

@article{Mukhamedov2019DiagonalizabilityOQ, title={Diagonalizability of Quantum Markov States on Trees}, author={Farrukh Mukhamedov and Abdessatar Souissi}, journal={Journal of Statistical Physics}, year={2019}, volume={182}, pages={1-15} }

We introduce quantum Markov states (QMS) in a general tree graph $$G= (V, E)$$ G = ( V , E ) , extending the Cayley tree’s case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS $$\varphi $$ φ on a UHF-algebra $${\mathcal {A}}_V$$ A V over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the…

## 6 Citations

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This work takes a huge leap forward at tackling one of the most important open problems in quantum probability, which concerns the calculations of mean entropies of quantum Markov fields, and opens up a new perspective for the generalization of many interesting results related to the one-dimensional QMSs and quantumMarkov chains to multi-dimensional cases.

### Tree-Homogeneous Quantum Markov Chains

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We clarify the structure of tree-homogeneous quantum Markov chains (THQMC) as a multi-dimensional quantum extension of homogeneous Markov chains. We provide a construction of a class of quantum…

### Block Markov Chains on Trees

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We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in…

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In the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and…

### Refinement of quantum Markov states on trees

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It turns out that localized QMS has the mentioned property which is called sub-Markov states, this allows us to characterize translation invariant QMS on regular trees.

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