Discrete phase space based on finite fields

Kathleen S. Gibbons, Matthew J. Hoffman, and William K. Wootters
Phys. Rev. A 70, 062101 – Published 3 December 2004

Abstract

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2N×2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our N×N phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.

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  • Received 25 May 2004

DOI:https://doi.org/10.1103/PhysRevA.70.062101

©2004 American Physical Society

Authors & Affiliations

Kathleen S. Gibbons1,2, Matthew J. Hoffman1,3, and William K. Wootters1

  • 1Department of Physics, Williams College, Williamstown, Massachusetts 01267, USA
  • 2Department of Theology, University of Notre Dame, Notre Dame, Indiana 46617, USA
  • 3Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA

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Issue

Vol. 70, Iss. 6 — December 2004

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