Interference in discrete Wigner functions

We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. Phys. Rev. A 70 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider “cat” states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.