  Wigner functions are broadly used to probe non-classical effects in the
macroscopic world. Here we develop an orbital-free functional framework to
compute the 1-body Wigner quasi-probability for both fermionic and bosonic
systems. Since the key variable is a quasi-density, this theory is particularly
well suited to circumvent the problem of finding the Pauli potential or
approximating the kinetic energy in orbital-free density functional theory. As
proof of principle, we find that the universal functional for the building
block of optical lattices results from a translation, a contraction, and a
rotation of the corresponding functional of the 1-body reduced density matrix,
indicating a strong connection between these functional theories. Furthermore,
we relate the concepts of Wigner negativity and $v$-representability, and find
a manifold of ground states with negative Wigner functions.
