Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

Abstract

Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast to the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total non-positivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are non-stochastic yet preserve total non-positivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension $d=pq$, type III gates, which preserve KD positivity but can change the total non-positivity of non-real distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions $d=p^k$ and $d=pq$, with $p, q$ distinct primes. Adapting the sampling algorithm of Pashayan et al. [PRL 115, 070501], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and non-real inputs, whose sampling overhead can grow exponentially. Consequently, for $d=pq$ no resource theory can both treat the KD total non-positivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.