  Kirkwood discovered in 1933, and Dirac discovered in 1945, a representation
of quantum states that has undergone a renaissance recently. The Kirkwood-Dirac
(KD) distribution has been employed to study nonclassicality across quantum
physics, from metrology to chaos to the foundations of quantum theory. The KD
distribution is a quasiprobability distribution, a quantum generalization of a
probability distribution, which can behave nonclassically by having negative or
nonreal elements. Negative KD elements signify quantum information scrambling
and potential metrological quantum advantages. Nonreal elements encode
measurement disturbance and thermodynamic nonclassicality. KD distributions'
nonclassicality has been believed to follow necessarily from noncommutation of
operators. We show that noncommutation does not suffice. We prove sufficient
conditions for the KD distribution to be nonclassical (equivalently, necessary
conditions for it to be classical). We also quantify the KD nonclassicality
achievable under various conditions. This work resolves long-standing questions
about nonclassicality and may be used to engineer quantum advantages.
