Chapter 7 The discrete Wigner function

Abstract

Until quite recently, Wigner functions were used mostly for infinite-dimensional Hilbert spaces. However, many quantum systems can be appropriately described in a finite-dimensional Hilbert space. These include, among other, spin systems, multi-level atoms, optical fields with a fixed number of photons, and electrons occupying a finite number of sites. When considering the coherent superpositions of ‘classical’ states, the discrete Wigner function behaves in a way that differs drastically from its continuous counterpart: the interference spreads over all of phase space, affecting even the regions where the original states are localized. As a consequence, the presence of interference may become hard to identify when using this class of Wigner functions. However, given two orthogonal stabilizer ‘classical’ states it is possible to define a Wigner function such that all coherent super-positions of those states have a phase-space representation in which the quantum interference is localized. This chapter discusses the continuous Wigner function, followed by discrete finite space and finite fields. It also discusses the generalized Pauli group, mutually unbiased bases, and the discrete Wigner function. This is followed by a discussion on reconstruction of the density operator from the discrete Wigner function, its applications.