Reply to Comment on ‘Wigner function for a particle in an infinite lattice’

Hinarejos, M.; Perez, A.; Bañuls, Mari Carmen

doi:10.1088/1367-2630/15/6/068002

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…Note that any Wigner function should have the properties described by (3.6)-(3.8) [3,7,16]. It is worthwhile noticing that the discrete Wigner functions have been also extensively studied from the general point of view [29][30][31][32][33][34]. Moreover, a class of discrete-time Wigner functions has been introduced in signal processing [35], where Wigner distributions have been used for time-frequency analysis [36].…”

Section: Wigner Functionmentioning

confidence: 99%

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

Przanowski¹,

Brzykcy²,

Tosiek³

2014

Annals of Physics

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a b s t r a c tA generalized Weyl quantization formalism for a particle on the circle is shown to supply an effective method for defining the number-phase Wigner function in quantum optics. A Wigner function for the stateρ and the kernel K for a particle on the circle is defined and its properties are analysed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner functions are considered.

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Section: Wigner Functionmentioning

confidence: 99%

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

Przanowski¹,

Brzykcy²,

Tosiek³

2014

Annals of Physics

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“…The results (39) and (40) indicate that the decay of the trapping probability with distance from the origin is not purely exponential like for the three-state Wigner walk (23). Nevertheless, the correction to the exponential decay becomes negligible for large x.…”

Section: B Trapping Probabilitymentioning

confidence: 92%

“…This is represented by a stationary peak in the probability distribution located at the origin, which does not vanish with the increasing number of steps but decays exponentially with the distance from the starting point. The Grover walk and its extensions were intensively studied, either for line [30][31][32][33][34][35][36], plane [37][38][39] or higher dimensional lattices [40,41].…”

Section: Introductionmentioning

confidence: 99%

Suitable bases for quantum walks with Wigner coins

2015

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The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by $2j+1$-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in T. Miyazaki et al., Phys. Rev. A 76, 012332 (2007). First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic $2j+1$ to a single one. Moreover, the models with integer $j$ lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to the dimension five

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“…The later method allows for straightforward extensions to quantum walks with larger internal degrees of freedom [14,15] (i.e. larger coin space) and higher-dimensional lattices [16][17][18][19]. For quantum walks without translational invariance it is significantly more difficult to derive the explicit shape of the limit distribution, however, analytical treatment is still tractable in some cases.…”

Section: Introductionmentioning

confidence: 99%

Conditional limit measure of a one-dimensional quantum walk with an absorbing sink

2018

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We consider a two-state quantum walk on a line where after the first step an absorbing sink is placed at the origin. The probability of finding the walker at position j, conditioned on that it has not returned to the origin, is investigated in the asymptotic limit. We prove a limit theorem for the conditional probability distribution and show that it is given by the Konno's density function modified by a pre-factor ensuring that the distribution vanishes at the origin. In addition, we discuss the relation to the problem of recurrence of a quantum walk and determine the Pólya number. Our approach is based on path counting and stationary phase approximation.

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Reply to Comment on ‘Wigner function for a particle in an infinite lattice’

Abstract: In a recent paper (2012 New J. Phys. 14 103009), we proposed a definition of the Wigner function for a particle on an infinite lattice. Here we argue that the criticism to our work raised by Bizarro is not substantial and does not invalidate our proposal.

Cited by 3 publications

References 15 publications

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

Suitable bases for quantum walks with Wigner coins

Conditional limit measure of a one-dimensional quantum walk with an absorbing sink

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