Discrete Wigner functions and quantum computational speedup

Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions W to represent quantum states in a finite Hilbert space dimension d. I characterize the set Cd of states having non-negative W simultaneously in all definitions of W in this class. For d5 I show Cd is the convex hull of stabilizer states. This supports the conjecture that negativity of W is necessary for exponential speedup in pure-state quantum computation.