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Decomposition of quantum Markov chains and its applications
© 2018 Elsevier Inc. Markov chains have been widely employed as a fundamental model in the studies of probabilistic and stochastic communicating and concurrent systems. It is well-understood that decomposition techniques play a key role in reachability analysis and model-checking of Markov chains. (Discrete-time) quantum Markov chains have been introduced as a model of quantum communicating systems [1] and also a semantic model of quantum programs [2]. The BSCC (Bottom Strongly Connected Component) and stationary coherence decompositions of quantum Markov chains were introduced in [3–5]. This paper presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. We further establish a limit theorem for them. As an application, an algorithm to find a maximum dimensional noiseless subsystem of a quantum communicating system is given using decomposition techniques of quantum Markov chains
Fermi Markov states
We investigate the structure of the Markov states on general Fermion
algebras. The situation treated in the present paper covers, beyond the
d--Markov states on the CAR algebra on Z (i.e. when there are d--annihilators
and creators on each site), also the non homogeneous case (i.e. when the
numbers of generators depends on the localization). The present analysis
provides the first necessary step for the study of the general properties, and
the construction of nontrivial examples of Fermi Markov states on the
d--standard lattice, that is the Fermi Markov fields. Natural connections with
the KMS boundary condition and entropy of Fermi Markov states are studied in
detail. Apart from a class of Markov states quite similar to those arising in
the tensor product algebras (called "strongly even" in the sequel), other
interesting examples of Fermi Markov states naturally appear. Contrarily to the
strongly even examples, the latter are highly entangled and it is expected that
they describe interactions which are not "commuting nearest neighbor".
Therefore, the non strongly even Markov states, in addition to the natural
applications to quantum statistical mechanics, might be of interest for the
information theory as well.Comment: 32 pages. Journal of Operator Theory, to appea
Quantum walks can find a marked element on any graph
We solve an open problem by constructing quantum walks that not only detect
but also find marked vertices in a graph. In the case when the marked set
consists of a single vertex, the number of steps of the quantum walk is
quadratically smaller than the classical hitting time of any
reversible random walk on the graph. In the case of multiple marked
elements, the number of steps is given in terms of a related quantity
which we call extended hitting time.
Our approach is new, simpler and more general than previous ones. We
introduce a notion of interpolation between the random walk and the
absorbing walk , whose marked states are absorbing. Then our quantum walk
is simply the quantum analogue of this interpolation. Contrary to previous
approaches, our results remain valid when the random walk is not
state-transitive. We also provide algorithms in the cases when only
approximations or bounds on parameters (the probability of picking a
marked vertex from the stationary distribution) and are
known.Comment: 50 page
Infinite dimensional entangled Markov chains
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 , being any infinite dimensional type
factor, 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 r\^ole 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 von Neumann factors.Comment: 16 page
Ergodic Properties of Quantum Birth and Death Chains
We study a class of quantum Markov processes that, on the one hand, is
inspired by the micromaser experiment in quantum optics and, on the other hand,
by classical birth and death processes. We prove some general geometric
properties and irreducibility for non-degenerated parameters. Furthermore, we
analyze ergodic properties of the corresponding transition operators. For
homogeneous birth and death rates we show how these can be fully determined by
explicit calculation. As for classical birth and death chains we obtain a rich
yet simple class of quantum Markov chains on an infinite space, which allow
only local transitions while having divers ergodic properties.Comment: 26 pages, 7 figure
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