  For any state in a $D$-dimensional Hilbert space with a choice of basis, one
can define a discrete version of the Wigner function -- a quasi-probability
distribution which represents the state on a discrete phase space. The Wigner
function can, in general, take on negative values, and the amount of negativity
in the Wigner function has an operational meaning as a resource for quantum
computation. In this note, we study the growth of Wigner negativity for a
generic initial state under time evolution with chaotic Hamiltonians. We
introduce the Krylov-Wigner function, i.e., the Wigner function defined with
respect to the Krylov basis (with appropriate phases), and show that this
choice of basis minimizes the early time growth of Wigner negativity in the
large $D$ limit. We take this as evidence that the Krylov basis (with
appropriate phases) is ideally suited for a dual, semi-classical description of
chaotic quantum dynamics at large $D$. We also numerically study the time
evolution of the Krylov-Wigner function and its negativity in random matrix
theory for an initial pure state. We observe that the negativity broadly shows
three phases: it rises gradually for a time of $O(\sqrt{D})$, then hits a sharp
ramp and finally saturates close to its upper bound of $\sqrt{D}$.
