Entropy rate of message sources driven by quantum walks

Kollár, B.; Koniorczyk, Mátyás

doi:10.1103/physreva.89.022338

Cited by 11 publications

(14 citation statements)

References 34 publications

(52 reference statements)

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“…[1,Sec. 4.2] and [31]. The idea has been recently rediscovered and analysed by Crutchfield and Wiesner under the name of quantum entropy rate [18], [54].…”

Section: Introductionmentioning

confidence: 99%

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Słomczyński

Szczepanek

2017

IEEE Trans. Inform. Theory

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We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM-and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.

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“…[1,Sec. 4.2] and [31]. The idea has been recently rediscovered and analysed by Crutchfield and Wiesner under the name of quantum entropy rate [18], [54].…”

Section: Introductionmentioning

confidence: 99%

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Słomczyński

Szczepanek

2017

IEEE Trans. Inform. Theory

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“…PACS numbers: 03.67.Ac, 05.40.FbQuantum walks are analogs of classical random walks, gaining considerable attention since their introduction in the nineties [1,2]. They did prove themselves as interesting constructs which find their applications gradually [3][4][5][6][7][8][9][10][11][12][13] (for review see [14]). Recent experiments demonstrated their basic properties [15], and stimulated further theoretical studies.…”

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confidence: 99%

Strongly trapped two-dimensional quantum walks

2015

Self Cite

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Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a broad scope of applications. In particular, they can be used as computational primitives, and they are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due to their quantumness. In some cases, however, they can remain localized at their initial state (trapping). The trapping and other fundamental properties of DTQWs are determined by the choice of the coin operator. We introduce and analyze an up to now uncharted type of walks driven by a coin class leading to strong trapping, complementing the known list of walks. This class of walks exhibit a number of exciting properties with the possible applications ranging from light pulse trapping in a medium to topological effects and quantum search.Comment: 5 pages, 4 figures, Accepted for publication in Physical Review

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“…Suppose {W p } p∈N , W p ∈ R(T) is a Cauchy sequence. By (14) there exists W = lim p→∞ W p , where the limit is taken with respect to the norm • DT .…”

Section: Closeness With Respect To • Dtmentioning

confidence: 99%

“…In [1] several mutual relations between these approaches are given. Some works (for example, [2,21,19,4,14]) deal with the finite-dimensional case (#X < ∞). In [8,9] a construction for the measure entropy is proposed for doubly stochastic (bistochastic) operators on various spaces of functions on a measure space.…”

Section: Introductionmentioning

confidence: 99%

$$\mu $$-Norm and Regularity

Treschev

2021

J Dyn Diff Equat

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In [22] we introduce the concept of a µ-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from [22] and present further results on the µ-norm. More precisely, we specify three classes of unitary operators for which the µ-norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.

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Entropy rate of message sources driven by quantum walks

Cited by 11 publications

References 34 publications

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Strongly trapped two-dimensional quantum walks

$$\mu $$-Norm and Regularity

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