Discrete phase space based on finite fields

Abstract

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