Quantized recurrence time in unital iterated open quantum dynamics

Sinkovicz, P.; Kurucz, Z.; Kiss, T.; Asbóth, János K.

doi:10.1103/physreva.91.042108

Cited by 21 publications

(44 citation statements)

References 38 publications

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“…This is not only a generalization of the results of Ref. [7] for unital channels (which includes unitary dynamics), but also of the Kac lemma about Markov chains.…”

Section: Discussionsupporting

confidence: 58%

“…We remark that the operator M[·] does not take us out of the relevant Hilbert space [7], and thus the operatorχ M has all its support in the relevant Hilbert space. We will prove Eq.…”

Section: Appendix: Recurrence Of the Initial Statementioning

confidence: 99%

“…[7] we have shown that in the case of unital dynamics, where all eigenvalues ofχ are equal, the expected return time T can be obtained from the steady stateχ. In that case, we found that the expected return time is an integer, equal to the dimensionality ofχ .…”

Section: First Return Timementioning

confidence: 99%

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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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“…This is not only a generalization of the results of Ref. [7] for unital channels (which includes unitary dynamics), but also of the Kac lemma about Markov chains.…”

Section: Discussionsupporting

confidence: 58%

“…We remark that the operator M[·] does not take us out of the relevant Hilbert space [7], and thus the operatorχ M has all its support in the relevant Hilbert space. We will prove Eq.…”

Section: Appendix: Recurrence Of the Initial Statementioning

confidence: 99%

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Generalized Kac lemma for recurrence time in iterated open quantum systems

2016

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“…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 return to a subspace in Theorem 2.10, following the ideas developed in [9] for UQWs.The structure of the present paper is as follows: the rest of this introduction is devoted to a summary on CP maps, quantum Markov chains and OQWs, which clarifies the relationships among them.…”

mentioning

confidence: 56%

“…In quantum theory there are in principle several nonequivalent notions of recurrence and in this work we follow a recent trend of considering a so-called monitored recurrence for discrete-time quantum systems [1,9,14,18,19,24,26,30,33,34]: the return properties are defined through a process in which the system is monitored after each time step via a projective measurement. This measurement only checks whether the system returns to a required state -or subspace of states-or not, so that quantum coherence is only partially destroyed and non-classical effects arise [19,9].…”

Section: Introductionmentioning

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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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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Quantized recurrence time in unital iterated open quantum dynamics

Cited by 21 publications

References 38 publications

Generalized Kac lemma for recurrence time in iterated open quantum systems

Generalized Kac lemma for recurrence time in iterated open quantum systems

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

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

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