Classicality in discrete Wigner functions

Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions $W$ to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions $W$ can be defined so that the only pure states having non-negative $W$ for all such functions are stabilizer states, as conjectured by Galvão, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of $W$ for all definitions of $W$ in the class form a subgroup of the Clifford group. This means pure states with non-negative $W$ and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

Figure 1

Operators in $\left\{U_c\right\}$ map translations $T(\vec{q},\vec{p})$ into $T(\overrightarrow{q^{\prime }},\overrightarrow{p^{\prime }})$ (up to a sign); representation of the transformation of translations under the action of (a) translation operators; (b) discrete squeezing; (c) Fourier transform.