A coherence monotone from Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases

Abstract: The Kirkwood-Dirac distribution, serving as an informationally complete representation of a quantum state, has recently garnered increasing attention. We investigate the Kirkwood-Dirac classicality with respect to mutually unbiased bases. For prime dimensional Hilbert spaces, {we demonstrate that a quantum state exhibits Kirkwood-Dirac classicality for two distinct sets of mutually unbiased bases $(A,B)$ and $(A,B')$ if and only if it is incoherent with respect to $A$}. We subsequently introduce a coherence monotone based on Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases. Additionally, we establish that this coherence monotone can be expressed through weak values, suggesting that quantum coherence can be utilized to detect anomalous weak values.