A Study of Wigner Functions for Discrete-Time Quantum Walks

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

doi:10.1166/jctn.2013.3101

Cited by 6 publications

(9 citation statements)

References 34 publications

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“…As for the other cases in which the phase-space formalism can also account for the dynamics of the system, it would be possible to formulate the evolution of such a system fully in terms of its Wigner function and to use the proposed measure, η(ρ), for classifying quantumness in evolving states. In this sense, a related problem is the application of this formalism to study the discrete time quantum walk, as we have started discussing in [37]. Although the examples presented in this paper are focused on pure states, the same concepts apply also to mixed states.…”

Section: Discussionmentioning

confidence: 99%

Wigner function for a particle in an infinite lattice

2012

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We study the Wigner function for a quantum system with a discrete, infinite-dimensional Hilbert space, such as a spinless particle moving on a onedimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase-space construction, propose a meaningful definition of the Wigner function in this case and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system, which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states and to their superpositions.

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Section: Discussionmentioning

confidence: 99%

Wigner function for a particle in an infinite lattice

2012

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“…We therefore obtain the definition of the Wigner function on the total probability. A different definition of the Wigner function in a lattice system for spinless and spinor particles has recently been proposed [23,24]. However, this alternative definition leads to an ambiguous interpretation of the displayed results, mainly because the marginals do not coincide with the probability distributions and because of the presence of 'ghost' image artifacts [25].…”

Section: Wigner Function and Quantum Walksmentioning

confidence: 99%

Decoherence models for discrete-time quantum walks and their application to neutral atom experiments

et al. 2014

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We discuss decoherence in discrete-time quantum walks in terms of a phenomenological model that distinguishes spin and spatial decoherence. We identify the dominating mechanisms that affect quantum-walk experiments realized with neutral atoms walking in an optical lattice. From the measured spatial distributions, we determine with good precision the amount of decoherence per step, which provides a quantitative indication of the quality of our quantum walks. In particular, we find that spin decoherence is the main mechanism responsible for the loss of coherence in our experiment. We also find that the sole observation of ballistic-instead of diffusive-expansion in position space is not a good indicator of the range of coherent delocalization. We provide further physical insight by distinguishing the effects of short-and long-time spin dephasing mechanisms. We introduce the concept of coherence length in the discrete-time quantum walk, which quantifies the range of spatial coherences. Unexpectedly, we find that quasi-stationary dephasing does not modify the local properties of the quantum walk, but instead affects spatial coherences. For a visual representation of decoherence phenomena in phase space, we have developed a formalism based on a discrete analogue of the Wigner function. We show that the effects of spin and spatial decoherence differ dramatically in momentum space.

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“…is a parameter defining the bias of the coin toss, I is the identity operator in  l , and σ y and σ z are Pauli matrices acting on  s . The QW dynamics can be described entirely in terms of the WM [47], via a recursion formula that relates + W m k t ( , , 1)to other components of this function at time t. Using equation (56), one obtains, after some algebra:…”

Section: Discrete Time 421 Quantum Walkmentioning

confidence: 99%

“…. A complete analysis of the time evolution in phase space with the help of the WF can be found in [47]. Note that a different definition of the WF was used in [48] for the reduced density matrix of the walker (after tracing the coin) to study the evolution and the effects of decoherence for the quantum walk.…”

Section: Discrete Time 421 Quantum Walkmentioning

confidence: 99%

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Wigner formalism for a particle on an infinite lattice: dynamics and spin

2015

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The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.

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A Study of Wigner Functions for Discrete-Time Quantum Walks

Cited by 6 publications

References 34 publications

Wigner function for a particle in an infinite lattice

Wigner function for a particle in an infinite lattice

Decoherence models for discrete-time quantum walks and their application to neutral atom experiments

Wigner formalism for a particle on an infinite lattice: dynamics and spin

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