Open Quantum Random Walks: Ergodicity, Hitting Times, Gambler’s Ruin and Potential Theory

Lardizabal, Carlos F.; Souza, Rafael Rigão

doi:10.1007/s10955-016-1578-9

Cited by 18 publications

(28 citation statements)

References 45 publications

(101 reference statements)

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“…For instance, the sequence of positions (X n ) n∈N alone is not a Markov chain, since one also needs to carry the information on the internal degree of freedom at all times in order to calculate probabilities in a Markovian way. This has implications, for instance, on the problem of site recurrence for OQWs (see [8,15]) and also on the problem of first visit to a vertex [34,35], which is studied in this work.…”

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

confidence: 99%

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

confidence: 99%

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

Lardizabal

2019

Quantum Inf Process

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“…hence the leading matrix coefficients satisfy κ n+1 a n = κ n . When J is self-adjoint -as an operator on 2 with maximal domain-it represents, in the basis given by the columns of the orthonormal polynomials, the self-adjoint multiplication operator T µ defined in (48), where L 2 µ is now the Hilbert space of square-summable d-vector valued functions with inner product ⟨f g⟩ = ∫ f † dµ g. Hence, J is unitarily equivalent to T µ . This unitary equivalence identifies the columns {p (0) n , p (1) n , .…”

Section: Applications To Orthogonal Polynomials On the Real Line: Oprmentioning

confidence: 99%

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Grünbaum

Velázquez

2018

Advances in Mathematics

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Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces.In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized 'Schur functions' meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FRfunctions satisfy splitting properties which extend useful factorizations of Schur functions.When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refers to as 'the measure of the Nevanlina function'. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple 'Schur algorithm' for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the 'Schur algorithm on the real line'.The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions -as first return generating functions-and their splittings -which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.

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“…A series of results investigating recurrence of open quantum walks can be found in [11,27,28]. In particular, in [11,27,28]…”

Section: Notions Of Recurrence For Open Quantum Walksmentioning

confidence: 99%

“…being equal to 1 or not. Corollary 3.10 shows that, for a semifinite irreducible open quantum walk, inf ρ∈S(hi) P i,ρ (t i < ∞) = 1 for some i if and only if it is true for all i (a fact which is not proved in [11,27,28]), and also that this is equivalent with recurrence in the sense of Definition 3.13. Therefore, an irreducible semifinite OQW is LS-site-recurrent if and only if it is recurrent in our sense.…”

Section: Notions Of Recurrence For Open Quantum Walksmentioning

confidence: 99%

“…They have therefore given rise to various theoretical studies, investigating e.g. ergodic properties, central limit theorems and large deviations properties (see [4,[9][10][11]28]). The approach of [10] was to give analogues for open quantum walks of notions usually associated with Markov chains, such as irreducibility and period, and to investigate their consequences.…”

Section: Introductionmentioning

confidence: 99%

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Passage Times, Exit Times and Dirichlet Problems for Open Quantum Walks

2017

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We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.

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Open Quantum Random Walks: Ergodicity, Hitting Times, Gambler’s Ruin and Potential Theory

Cited by 18 publications

References 45 publications

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

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

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Passage Times, Exit Times and Dirichlet Problems for Open Quantum Walks

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