Relating incompatibility, noncommutativity, uncertainty, and Kirkwood–Dirac nonclassicality

We provide an in-depth study of the recently introduced notion completely incompatible observables and its links to support uncertainty Kirkwood–Dirac nonclassicality pure quantum states. The latter has been proved central a number issues in information theory metrology. In this last context, it was shown that advantage requires use nonclassical establish sharp bounds very general validity imply is efficient witness. When adapted are close mutually unbiased ones, bound allows us fully characterize classical states as eigenvectors two observables. show furthermore complete incompatibility implies several weaker notions incompatibility, among which features strong form noncommutativity.