Quantum Recurrence of a Subspace and Operator-Valued Schur Functions

Bourgain, Jean; Grünbaum, F. Alberto; Velázquez, L.; Wilkening, Jon

doi:10.1007/s00220-014-1929-9

Cited by 73 publications

(124 citation statements)

References 34 publications

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“…Among the different notions of recurrence appearing in the quantum literature, we will refer here to a recent one based on a monitoring process, developed for unitary quantum walks [1,9,19,24], and later extended to open quantum walks [5,14,18,26]. Consider a discrete time evolution given by iterating a quantum channel Φ on a Hilbert space H. Given a subspace H 0 ⊂ H, we will identify I(H 0 ) with the subspace constituted by those operators ρ ∈ I(H) with ran ρ ⊂ H 0 and ker ρ ⊃ H ⊥ 0 .…”

Section: Recurrence For Quantum Markov Chains and Schur Functionsmentioning

confidence: 99%

“…The first link between recurrence and Schur functions appeared in [19], where the FR-functions encoding the return properties of a state subject to a unitary discrete evolution were identified as Schur functions. This result was extended in [9] to the return properties of a subspace, in which case the corresponding FR-functions become matrix valued Schur functions. The identification of the FR-functions for classical Markov chains as Schur functions only came later [18], and required the extension of the previous results on unitaries in Hilbert spaces to operators in Banach spaces defined by stochastic matrices.…”

Section: Introductionmentioning

confidence: 98%

“…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%

“…Apart from their role as a divide and conquer technique, these splitting properties have striking consequences, such as the invariance of the return probability under certain local perturbations [18] (see also the comments in [19,Sect. 6] and [9,Sect. 4]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Recurrence For Quantum Markov Chains and Schur Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Grünbaum

Lardizabal

Velázquez

2019

Ann. Henri Poincaré

Self Cite

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“…At the same time quantum walks and their time-continuous counterpart are relevant in a quantum algorithmical context, for example in search algorithms, to examine the distinctness of elements, in quantum information processing and applications to the graph isomorphism problem [6,11,27,28,48,51]. From a mathematical point of view, quantum walks on a one-dimensional lattice can also be seen as a particular class of CMV matrices giving a link between quantum dynamical systems and orthogonal polynomials on the unit circle, a connection which has proved to be fruitful in both directions [12,15,16,19,24,39]. One can use spectral methods to deduce bounds on spreading [31,32].…”

Section: Introductionmentioning

confidence: 99%

Singular continuous Cantor spectrum for magnetic quantum walks

et al. 2020

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In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux Φ: while for Φ/(2π) rational the spectrum is known to consist of bands, we show that for Φ/(2π) irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.arXiv:1908.09924v1 [quant-ph]

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A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

Cedzich

Grünbaum

Velázquez

et al. 2015

Comm Pure Appl Math

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Abstract. Khrushchev's formula is the cornerstone of the so called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix-valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix-valued setting which we overcome in this paper.It was recently discovered that orthogonal polynomials on the unit circle and their matrix-valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchev's formula has profound and surprising implications.We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchev's formula via 'quantum' diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix-valued measures. Actually, this path counting approach is so fruitful that it provides different matrix generalizations of Khrushchev's formula, some of them new even in the case of scalar measures.Furthermore, the path counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchev's formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchev's formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.

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Quantum Recurrence of a Subspace and Operator-Valued Schur Functions

Cited by 73 publications

References 34 publications

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Singular continuous Cantor spectrum for magnetic quantum walks

A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

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