Sufficient conditions, lower bounds, and trade-off relations for quantumness in Kirkwood-Dirac quasiprobability

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 arises from quantum noncommutativity? Here, first, we obtain sufficient conditions for the KD quasiprobability defined relative to a pair of projection-valued measure (PVM) 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. We then suggest an information theoretical interpretation of the KD nonreality relative to a PVM basis as a lower bound to the maximum total root-mean-squared error in an optimal estimation of the PVM basis, and thereby obtain a lower bound and a trade-off relation for the root-mean-squared error. Finally, we suggest an interpretation of the KD nonclassicality relative to a PVM basis as a lower bound to the total state disturbance caused by a nonselective projective binary measurement associated with the PVM basis, and derive a lower bound and a trade-off relation for the disturbance.