Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility

TL;DR

This paper establishes criteria based on support uncertainty for identifying KD-nonclassical states in quantum systems, especially when bases are completely incompatible, with implications for quantum metrology.

Contribution

It introduces sharp bounds on support uncertainty that guarantee KD-nonclassicality and defines complete incompatibility between bases, advancing understanding of quantum nonclassicality.

Findings

States with support uncertainty greater than minimal are KD-nonclassical in completely incompatible bases.

Complete incompatibility implies minimal support uncertainty states are not KD-nonclassical.

Results apply to mutually unbiased bases and their perturbations.

Abstract

Given two orthonormal bases in a d-dimensional Hilbert space, one associates to each state its Kirkwood-Dirac (KD) quasi-probability distribution. KD-nonclassical states - for which the KD-distribution takes on negative and/or nonreal values - have been shown to provide a quantum advantage in quantum metrology and information, raising the question of their identification. Under suitable conditions of incompatibility between the two bases, we provide sharp lower bounds on the support uncertainty of states that guarantee their KD-nonclassicality. In particular, when the bases are completely incompatible, a notion we introduce, states whose support uncertainty is not equal to its minimal value d+1 are necessarily KD-nonclassical. The implications of these general results for various commonly used bases, including the mutually unbiased ones, and their perturbations, are detailed.