Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

The Kirkwood-Dirac (KD) quasiprobability distribution is known for its role in quantum metrology, thermodynamics, as well as quantum foundations. In this work we classify unitary evolutions that preserve KD positivity. We identify conditions under which positivity preservation is equivalent to $l_1$-norm preservation, and exhibit unitaries that preserve positivity on KD-positive distributions while failing to preserve the $l_1$-norm of non-positive ones. We further prove that unitaries inducing stochastic updates of KD quasiprobabilities form a strict subset of the positivity-preserving unitaries. By adapting the classical sampling algorithm of Pashayan et al. [Phys. Rev. Lett. 115, 070501], we obtain efficient simulation methods for all identified classes of positivity-preserving unitaries. Our classification is complete for distributions defined on Fourier-conjugate bases in dimensions $d=p^k$ and $d=pq$, where $p,q$ are distinct primes, as well as for generic randomly chosen bases. As a consequence, no resource theory in the Fourier-conjugate $d=pq$ setting can simultaneously regard KD non-positivity as a monotone and include all efficiently simulable positivity-preserving unitaries among its free operations.