Markov Property – Recent Developments on the Quantum Markov Property

Accardi, Luigi; Fidaleo, Francesco

doi:10.1142/9789812704290_0001

Cited by 5 publications

(9 citation statements)

References 16 publications

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“…There has also been work on Quantum Markov networks within the quantum probability literature [Lei01a,AF03a,AF03b], although Belief Propagation has not been investigated in this literature. This is closer to the spirit of the present work, in the sense that it is based on the generalization of classical probability to a noncommutative, operator-valued probability theory.…”

Section: Related Workmentioning

confidence: 99%

Quantum Graphical Models and Belief Propagation

2008

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Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.

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Section: Related Workmentioning

confidence: 99%

Quantum Graphical Models and Belief Propagation

2008

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“…An immediate application of Proposition 2.2 is that the Markov state ϕ satisfies all the properties listed in Definition 6.1 of [6]. Indeed, it is sufficient to put inside I, α := [n+1, ᾱ = [n. In such a way, α ′ = n], ᾱ′ = n−1] and the projective net of conditional expectations is precisely that formed by the expectations E n] given in (iii) of Proposition 2.2.…”

Section: Markov Statesmentioning

confidence: 93%

Markov States and Chains on the Car Algebra

Accardi

Fidaleo

Mukhamedov

2007

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over Z, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.

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“…Первые попытки построения квантовых аналогов классических марковских полей были рассмотрены в работах [4], [6]- [8]. В этих работах было обобщено понятие квантового марковского состояния, рассмотренное в [9].…”

Section: математические заметкиunclassified

“…{︃cosh4 + sinh 2 cosh = , sinh 2 cosh 2 (1 + cosh ) 2 = 2 . Пусть : ∆ → ∆ -динамическая система, заданная через (4.14).…”

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