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arXiv:1205.3925 · New Journal of Physics

Wigner function for a particle in an infinite lattice

Margarida Hinarejos, Armando Pérez, Mari Carmen Bañuls

Published Oct 3, 2012• 33 citations
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Abstract

We study the Wigner function for a quantum system with a discrete, infinite-dimensional Hilbert space, such as a spinless particle moving on a one-dimensional 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.

DOI
10.1088/1367-2630/14/10/103009

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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 one-dimensional 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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As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
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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.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counter-example. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.