# 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}
}
• Published 4 November 2019
• Mathematics
• Journal of Statistical Physics
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…
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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
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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
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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.

## References

SHOWING 1-10 OF 39 REFERENCES

• Mathematics
• 2004
We clarify the meaning of diagonalizability of quantum Markov states. Then, we prove that each non homogeneous quantum Markov state is diagonalizable. Namely, for each Markov state $\phi$ on the spin
• Mathematics
• 2020
We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the $$\ell • Mathematics • 2011 In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By • Mathematics Reviews in Mathematical Physics • 2019 In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC • Mathematics Journal of Statistical Physics • 2019 In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the • Mathematics • 2016 The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not • Mathematics • 2010 In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and • Physics, Mathematics • 2013 In this paper, we consider the classical Ising model on the Cayley tree of order$$k$$k ($$k\ge 2k≥2), and show the existence of the phase transition in the following sense: there exists two
• Mathematics
Annales Henri Poincaré
• 2019
In the present paper, we consider a quantum Markov chain corresponding to the XY-model with competing Ising interactions on the Cayley tree of order two. Earlier, it was proved that this state does
• Mathematics
Quantum Information Processing
• 2020
It is seen that the Gaussian and the mixture of Gaussians in the limit depend on the structure of the invariant states of the intrinsic quantum Markov semigroup whose generator is given by the Kraus operators which generate the open quantum random walks.