Convex roofs witnessing Kirkwood-Dirac nonpositivity

TL;DR

This paper develops new convex roof-based witnesses for detecting Kirkwood-Dirac nonpositivity in quantum states, extending previous uncertainty-based criteria to mixed states and providing tools to identify nonclassical features.

Contribution

It introduces two convex roof witnesses for KD nonpositivity applicable to mixed states, enhancing detection methods for quantum nonclassicality.

Findings

Convex roof of support uncertainty extends pure state relation to mixed states.

Convex roof of total KD nonpositivity provides a faithful witness.

New criteria improve identification of nonclassical quantum states.

Abstract

Given two observables $A$ and $B$, one can associate to every quantum state a Kirkwood-Dirac (KD) quasiprobability distribution. KD distributions are like joint classical probabilities except that they can have negative or nonreal values, which are associated to nonclassical features of the state. In the last decade, KD distributions have come to the forefront as a versatile tool to investigate and construct quantum advantages and nonclassical phenomena. KD distributions are also used to determine quantum-classical boundaries. To do so, one must have witnesses for when a state is KD nonpositive. Previous works have established a relation between the uncertainty of a pure state with respect to the eigenbases of $A$ and $B$ and KD positivity. If this $\textit{support uncertainty}$ is large, the state cannot be KD positive. Here, we construct two witnesses for KD nonpositivity for general mixed states. Our first witness is the convex roof of the support uncertainty; it is not faithful, but it extends to the convex hull of pure KD-positive states the relation between KD positivity and small support uncertainty. Our other witness is the convex roof of the total KD nonpositivity, which provides a faithful witness for the convex hull of the pure KD-positive states. This implies that the convex roof of the total nonpositivity captures the nonpositive nature of the KD distribution at the underlying pure state level.