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AIP conference proceedings

Wigner distributions for qudits

S. Chaturvedi

Published Jan 1, 2006• 0 citations
Publisher

Abstract

Two new approaches to the problem of setting up Wigner distributions for finite level quantum systems are proposed. Both arise by looking at the structure of the familiar Wigner distribution for Cartesian quantum mechanics from different perspectives. The two approaches have one common feature—each involves a ‘square root’ operation though of very different kinds.

DOI
10.1063/1.2400885

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Two new approaches to the problem of setting up Wigner distributions for finite level quantum systems are proposed. Both arise by looking at the structure of the familiar Wigner distribution for Cartesian quantum mechanics from different perspectives. The two approaches have one common feature—each involves a ‘square root’ operation though of very different kinds.
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A complete solution to the problem of setting up Wigner distribution for N-level quantum systems is presented. The scheme makes use of some of the ideas introduced by Dirac in the course of defining functions of noncommuting observables and works uniformly for all N. Further, the construction developed here has the virtue of being essentially input-free in that it merely requires finding a square root of a certain N^2 x N^2 complex symmetric matrix, a task which, as is shown, can always be accomplished analytically. As an illustration, the case of a single qubit is considered in some detail and it is shown that one recovers the result of Feynman and Wootters for this case without recourse to any auxiliary constructs.
We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n-dimensional Euclidean space Rn to a Lie group G. The notion of canonical momentum is carefully analyzed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact G an explicit definition of the Wigner distribution is proposed, possessing all the required properties. Geodesic curves in G which help introduce a notion of the midpoint of two group elements play a central role in the construction.