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Diagonalizability of Quantum Markov States on Trees
Journal of Statistical Physics ( IF 1.6 ) Pub Date : 2021-01-01 , DOI: 10.1007/s10955-020-02674-1
Farrukh Mukhamedov , Abdessatar Souissi

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 existence of a Umegaki conditional expectation $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ E : A V → D V such that $$\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ φ = φ ⌈ D V ∘ E . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $${\mathcal {D}}_V$$ D V .

中文翻译:

树上量子马尔可夫态的对角化

我们在一般树图 $$G= (V, E)$$ G = ( V , E ) 中引入了量子马尔可夫态 (QMS),扩展了凯莱树的情况。我们研究了马尔可夫性质,并结合了所考虑树的更精细结构。本文的主要结果涉及在 UHF 代数 $${\mathcal {A}}_V$$ AV 上的局部忠实 QMS $$\varphi $$ φ 在所考虑的树上通过合适的条件最大阿贝尔子代数的期望。即,我们证明存在 Umegaki 条件期望 $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ E : AV → DV 使得 $$ \varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ φ = φ ⌈ DV ∘ E . 而且,
更新日期:2021-01-01
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