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Just a few years after the inception of quantum mechanics, there has been aresearch program using the nonclassical values of some quasiprobabilitydistributions to delineate the nonclassical aspects of quantum phenomena. Inparticular, in KD (Kirkwood-Dirac) quasiprobability distribution, thedistinctive quantum mechanical feature of noncommutativity which underlies manynonclassical phenomena, manifests in the nonreal values and/or the negativevalues of the real part. Here, we develop a faithful quantifier of quantumcoherence based on the KD nonclassicality which captures simultaneously thenonreality and the negativity of the KD quasiprobability. TheKD-nonclassicality coherence thus defined, is upper bounded by the uncertaintyof the outcomes of measurement described by a rank-1 orthogonal PVM(projection-valued measure) corresponding to the incoherent orthonormal basiswhich is quantified by the Tsallis $\frac{1}{2}$-entropy. Moreover, they areidentical for pure states so that the KD-nonclassicallity coherence for purestate admits a simple closed expression in terms of measurement probabilities.We then use the Maassen-Uffink uncertainty relation for min-entropy andmax-entropy to obtain a lower bound for the KD-nonclassicality coherence of apure state in terms of optimal guessing probability in measurement described bya PVM noncommuting with the incoherent orthonormal basis. We also derive atrade-off relation for the KD-noncassicality coherences of a pure staterelative to a pair of noncommuting orthonormal bases with a state-independentlower bound. Finally, we sketch a variational scheme for a direct estimation ofthe KD-nonclassicality coherence based on weak value measurement and therebydiscuss its relation with quantum contextuality.

Quantum coherence from Kirkwood-Dirac nonclassicality, some bounds, and operational interpretation