Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Carbone, Raffaella; Pautrat, Yan

doi:10.1007/s00023-015-0396-y

Cited by 48 publications

(114 citation statements)

References 22 publications

(79 reference statements)

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“…We note that using the demonstrated example, one can produce a lot of nontrivial examples of quantum nonhomogeneous Markov processes. Moreover, some of them can be applied to quantum random walks [3,12].…”

Section: Examplesmentioning

confidence: 99%

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Mukhamedov

2015

Positivity

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In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin's ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.

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Section: Examplesmentioning

confidence: 99%

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Mukhamedov

2015

Positivity

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“…The diverse dynamical behaviour of OQWs has been extensively studied [10][11][12][16][17][18][19][20][21]. The asymptotic analysis of OQWs leads to a central limit theorem [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Microscopic derivation of open quantum walks

Sinayskiy

Petruccione

2015

Phys. Rev. A

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Open Quantum Walks (OQWs) are exclusively driven by dissipation and are formulated as completely positive trace preserving (CPTP) maps on underlying graphs. The microscopic derivation of discrete and continuous in time OQWs is presented. It is assumed that connected nodes are weakly interacting via a common bath. The resulting reduced master equation of the quantum walker on the lattice is in the generalised master equation form. The time discretisation of the generalised master equation leads to the OQWs formalism. The explicit form of the transition operators establishes a connection between dynamical properties of the OQWs and thermodynamical characteristics of the environment. The derivation is demonstrated for the examples of the OQW on a circle of nodes and on a finite chain of nodes. For both examples a transition between diffusive and ballistic quantum trajectories is observed and found to be related to the temperature of the bath.

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“…Other topics on OQWs which have been examined are the following: reducibility, periodicity, ergodic properties [13]; large deviations [14]; open quantum Brownian motions [44]; site (vertex) recurrence of OQWs [8,15,16,33]. We refer the reader to [45] for a recent survey.…”

Section: Quantum Markov Chains On a Finite Graph Consider The Setmentioning

confidence: 99%

“…Then, we say that a finite QMC is ergodic if it is irreducible and aperiodic. It is well-known that the iterates of an ergodic QMC acting on any initial density converge to π [13]. We remark that in [13] the term ergodic refers to a distinct notion than the one employed here.…”

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confidence: 98%

Mean hitting times of quantum Markov chains in terms of generalized inverses

Lardizabal

2019

Quantum Inf Process

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We study quantum Markov chains on graphs, described by completely positive maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008) and which includes the dynamics given by open quantum random walks as defined by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such structures we examine a quantum notion of mean time of first visit to a chosen vertex. However, instead of making direct use of the definition as it is usually done, we focus on expressions for such quantity in terms of generalized inverses associated with the walk and most particularly the socalled fundamental matrix. Such objects are in close analogy with the theory of Markov chains and the methods described here allow us to calculate examples that illustrate similarities and differences between the quantum and classical settings. arXiv:1907.01313v2 [math-ph] 9 Jul 2019 Appendix: proofsThroughout the proofs we recall the notation |e I = |e I n k as n, k are clear from context.Proof of Proposition 6.3. The proof is inspired by [24]. We recall that if M ij denotes the (i, j)-th minor of a matrix A, we define the adjugate matrix as adj(A) := ((−1) i+j M ji ) 1≤i,j≤n , and it holds that A adj(A) = adj(A)A = det(A)I By Sylvester's determinant theorem, we have det(I m + AB) = det(I n + BA), from which we obtain as a consequence that det(X + |c r|) = det(X) + r| adj(X)|c

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Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Cited by 48 publications

References 22 publications

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Microscopic derivation of open quantum walks

Mean hitting times of quantum Markov chains in terms of generalized inverses

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