Complete Incompatibility, Support Uncertainty, and Kirkwood-Dirac Nonclassicality.

For quantum systems with a finite dimensional Hilbert space of states, we show that the complete incompatibility of two observables-a notion we introduce-is equivalent to the large support uncertainty of all states. The Kirkwood-Dirac (KD) quasiprobability distribution of a state-which depends on the choice of two observables-has emerged in quantum information theory as a tool for assessing nonclassical features of the state that can serve as a resource in quantum protocols. We apply our result to show that, when the two observables are completely incompatible, only states with minimal support uncertainty can be KD classical, all others being KD nonclassical. We illustrate our findings with examples.