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On Entangled Markov Chains (Non-Commutative Analysis and Micro-Macro Duality)

Luigi Accardi, Takashi Matsuoka, Masanori Ohya

Published Jul 1, 2008• 0 citations
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Abstract

Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of $[1|$ , were $inti^{\vee}oduced$ by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems.In [3], using our entanglement criterion for pure states [16] which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra $||generically^{I\dagger}$ are entangled.On the other hand we showed that this entanglement condition for pure state is sufficient condition for entanglement in the case of mixtures [4].This fact was then applied to prove that EMC with unitarily im- plementable transition operator induce a mixture entangled state on any local algebra.Interestingly it was also shown that all these local states provide a new class of examples (in any dimension) of entangled states which nevertheless satisfy the PPT condition [4].In this article we report the above results.

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Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of [1∣[1|[1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems.In [3], using our entanglement criterion for pure states [16] which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.On the other hand we showed that this entanglement condition for pure state is sufficient condition for entanglement in the case of mixtures [4].This fact was then applied to prove that EMC with unitarily im- plementable transition operator induce a mixture entangled state on any local algebra.Interestingly it was also shown that all these local states provide a new class of examples (in any dimension) of entangled states which nevertheless satisfy the PPT condition [4].In this article we report the above results.
Entangled Markov chains, which can be recognized as quantum Markov chains in the sense of (1∣(1|(1∣ , were inti∨oducedinti^{\vee}oducedinti∨oduced by Accardi and Fidaleo in order to extend the notion of classical random walk to quaritum systems. In (3), using our entanglement criterion for pure states (16) which is based on the notion of the quantum mutual information, we proved that the vector states defining the EMC's on infinite tensor products of matrix algebra ∣∣genericallyI†||generically^{I\dagger}∣∣genericallyI† are entangled.
Entangled Markov chains (EMC) were so baptized on the basis of the conjecture that they provide examples of states, on infinite tensor products of matrix algebras, which are in some sense "entangled".2 In this paper we introduce the notion of multiple (or "many-body") entanglement and extend the two-body criterion of entanglement obtained in Ref. 17 to this case. We then apply this extension to EMC and prove that "generically" they satisfy the entanglement conditions.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state--space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on ∪J⊂Z⊗ˉJFˉC∗\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}∪J⊂Z​⊗ˉ​J​Fˉ​C∗, FFF being any infinite dimensional type III factor, JJJ a finite interval of ZZZ, and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II1II_1II1​ von Neumann factors.
A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.
We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.