  Kirkwood-Dirac (KD) quasiprobability is a quantum analog of classical phase
space probability. It offers an informationally complete representation of
quantum state wherein the quantumness associated with quantum noncommutativity
manifests in its nonclassical values, i.e., the nonreal and/or negative values
of the real part. This naturally raises a question: how does such form of
quantumness comply with the uncertainty principle which also arise from quantum
noncommutativity? Here, first, we obtain sufficient conditions for the KD
quasiprobability defined relative to a pair of PVM (projection-valued measure)
bases to have nonclassical values. Using these nonclassical values, we then
introduce two quantities which capture the amount of KD quantumness in a
quantum state relative to a single PVM basis. They are defined respectively as
the nonreality, and the classicality which captures both the nonreality and
negativity, of the associated KD quasiprobability over the PVM basis of
interest, and another PVM basis, and maximized over all possible choices of the
latter. We obtain their lower bounds, and derive trade-off relations
respectively reminiscent of the Robertson and Robertson-Schr\"odinger
uncertainty relations but with lower bounds maximized over the convex sets of
Hermitian operators whose complete sets of eigenprojectors are given by the PVM
bases. We discuss their measurement using weak value measurement and classical
optimization, and suggest information theoretical and operational
interpretations in terms of optimal estimation of the PVM basis and state
disturbance.
