  Just a few years after the inception of quantum mechanics, there has been a
research program using the nonclassical values of some quasiprobability
distributions to delineate the nonclassical aspects of quantum phenomena. In
particular, in KD (Kirkwood-Dirac) quasiprobability distribution, the
distinctive quantum mechanical feature of noncommutativity which underlies many
nonclassical phenomena, manifests in the nonreal values and/or the negative
values of the real part. Here, we develop a faithful quantifier of quantum
coherence based on the KD nonclassicality which captures simultaneously the
nonreality and the negativity of the KD quasiprobability. The
KD-nonclassicality coherence thus defined, is upper bounded by the uncertainty
of the outcomes of measurement described by a rank-1 orthogonal PVM
(projection-valued measure) corresponding to the incoherent orthonormal basis
which is quantified by the Tsallis $\frac{1}{2}$-entropy. Moreover, they are
identical for pure states so that the KD-nonclassicallity coherence for pure
state admits a simple closed expression in terms of measurement probabilities.
We then use the Maassen-Uffink uncertainty relation for min-entropy and
max-entropy to obtain a lower bound for the KD-nonclassicality coherence of a
pure state in terms of optimal guessing probability in measurement described by
a PVM noncommuting with the incoherent orthonormal basis. We also derive a
trade-off relation for the KD-noncassicality coherences of a pure state
relative to a pair of noncommuting orthonormal bases with a state-independent
lower bound. Finally, we sketch a variational scheme for a direct estimation of
the KD-nonclassicality coherence based on weak value measurement and thereby
discuss its relation with quantum contextuality.
