Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

TL;DR

This paper classifies which unitary evolutions preserve the positivity of Kirkwood-Dirac quasiprobability distributions and explores their classical simulability, with implications for quantum resource theories.

Contribution

It provides a complete classification of positivity-preserving unitaries for KD distributions in certain dimensions and links positivity preservation to classical simulation methods.

Findings

Identified conditions for KD positivity preservation under unitaries.

Showed that some unitaries preserve positivity without preserving the $l_1$-norm.

Established that stochastic update unitaries form a strict subset of positivity-preserving unitaries.

Abstract

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.