On a Class of Quantum Channels, Open Random Walks and Recurrence

Lardizabal, Carlos F.; Souza, Rafael Rigão

doi:10.1007/s10955-015-1217-x

Cited by 29 publications

(72 citation statements)

References 21 publications

(47 reference statements)

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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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“…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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“…[25]). The speciality of this type of dynamics can be understood if we restrict our attention only to the dynamics of the diagonal elements.…”

Section: Discussionmentioning

confidence: 99%

“…The Pólya number for quantum walks characterizing recurrence has been defined in this way [26,27]. There are also alternative ways to define iterative open quantum dynamics, e.g., the "open quantum random walks" [28], for which there are known results on recurrence and return time [25].…”

Section: Discussionmentioning

confidence: 99%

Generalized Kac lemma for recurrence time in iterated open quantum systems

2016

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We consider recurrence to the initial state after repeated actions of a quantum channel. After each iteration a projective measurement is applied to check recurrence. The corresponding return time is known to be an integer for the special case of unital channels, including unitary channels. We prove that for a more general class of quantum channels the expected return time can be given as the inverse of the weight of the initial state in the steady state. This statement is a generalization of the Kac lemma for classical Markov chains.

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“…We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains.…”

mentioning

confidence: 99%

“…walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].…”

mentioning

confidence: 99%

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Grünbaum

Lardizabal

Velázquez

2019

Ann. Henri Poincaré

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In this work we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called splitting rules. These rules also yield an alternative to the folding trick to transform a doubly infinite system into a semi-infinite one which doubles the number of internal degrees of freedom. The generalization of Schur functions -so-called FR-functions-to the general context of closed operators in Banach spaces is the key for the present applications to open quantum systems. An important class of examples included in this setting are the open quantum random walks, as described by S. Attal et al., but we will state results in terms of general completely positive trace preserving maps. We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains. As a consequence of these splitting rules we will see that, similarly to UQWs, the return probability is also invariant under certain local perturbations.Other novel contributions of this paper deal with generalizations of known results on recurrence in classical Markov chains and UQWs. Different quantum generalizations of Kac's lemma for classical Markov chains are commented in Subsect. 2.1. Besides, Theorem 2.8 constitutes the quantum version of a well known result for classical Markov chains, namely, that finiteness and irreducibility imply the positive recurrence of every state, i.e. every state returns to itself with probability one and in a finite expected return time. A similar result holds for finite-dimensional unitary evolutions, which in addition present the striking particularity of exhibiting integer valued expected times for the return to any state [19]. These results has been generalized to the return to a subspace in [9], while [33] proves that they hold for the return to a state in iterated quantum channels whenever they are unital. We extend this property of unital quantum channels to the r...

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On a Class of Quantum Channels, Open Random Walks and Recurrence

Cited by 29 publications

References 21 publications

Microscopic derivation of open quantum walks

Microscopic derivation of open quantum walks

Generalized Kac lemma for recurrence time in iterated open quantum systems

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

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