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.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.

Random Operators and Stochastic EquationsDec 1, 20040 citations

Infinite dimensional entangled Markov chains

Francesco Fidaleo

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 $\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}$ , $F$ being any infinite dimensional type $I$ factor, $J$ a finite interval of $Z$ , 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.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 $\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}$, $F$ being any infinite dimensional type $I$ factor, $J$ a finite interval of $Z$, 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.

arXiv:math/0411202Nov 9, 20040 citations

ENTANGLED MARKOV CHAINS GENERATED BY SYMMETRIC CHANNELS

Takayuki Miyadera

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.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.

arXiv:quant-ph/0602056Infinite Dimensional Analysis Quantum Probability and Related TopicsSep 1, 20056 citations

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

Luigi Accardi, Takashi Matsuoka, Masanori Ohya

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.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.

Jul 1, 20080 citations

Discrete Quantum Markov Chains

Ulrich Faigle, Alexander Schönhuth

A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.

Nov 4, 20103 citations

On Entangled Markov Chains

Luigi Accardi, Takashi Matsuoka, Masanori Ohya

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.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.

Jan 1, 20081 citation

\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}

F

being any infinite dimensional type

I

factor,

J

a finite interval of

Z

, 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.

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.

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.