Conditions tighter than noncommutation needed for nonclassicality

TL;DR

This paper investigates the conditions under which the Kirkwood-Dirac distribution exhibits nonclassical behavior, revealing that noncommutation alone is insufficient and providing new criteria for quantum advantages.

Contribution

It establishes that noncommutation is not enough for nonclassicality and introduces sufficient conditions for the Kirkwood-Dirac distribution to be nonclassical, advancing understanding of quantum phenomena.

Findings

Noncommutation does not necessarily imply nonclassicality.

Sufficient conditions for KD nonclassicality are identified.

Quantitative bounds on achievable KD nonclassicality are provided.

Abstract

Kirkwood discovered in 1933, and Dirac discovered in 1945, a representation of quantum states that has undergone a renaissance recently. The Kirkwood-Dirac (KD) distribution has been employed to study nonclassicality across quantum physics, from metrology to chaos to the foundations of quantum theory. The KD distribution is a quasiprobability distribution, a quantum generalization of a probability distribution, which can behave nonclassically by having negative or nonreal elements. Negative KD elements signify quantum information scrambling and potential metrological quantum advantages. Nonreal elements encode measurement disturbance and thermodynamic nonclassicality. KD distributions' nonclassicality has been believed to follow necessarily from noncommutation of operators. We show that noncommutation does not suffice. We prove sufficient conditions for the KD distribution to be nonclassical (equivalently, necessary conditions for it to be classical). We also quantify the KD nonclassicality achievable under various conditions. This work resolves long-standing questions about nonclassicality and may be used to engineer quantum advantages.