Classification of incompatibility for two orthonormal bases

TL;DR

This paper introduces the concept of s-order incompatibility for pairs of orthonormal bases in a complex Hilbert space, generalizing previous notions and exploring their relations with other quantum properties, with applications to the Fourier transform.

Contribution

It defines s-order incompatibility, relates it to support uncertainty and matrix rank, and applies it to analyze the Fourier transform in finite dimensions.

Findings

s-order incompatibility generalizes complete incompatibility

Relations established between s-order incompatibility, support uncertainty, and matrix rank

Fourier transform's incompatibility order determined for any finite dimension

Abstract

For two orthonormal bases of a $d$-dimensional complex Hilbert space, the notion of complete incompatibility was introduced recently by De Bi\`{e}vre [Phys. Rev. Lett. 127, 190404 (2021)]. In this work, we introduce the notion of $s$-order incompatibility with positive integer $s$ satisfying $2\leq s\leq d+1.$ In particular, $(d+1)$-order incompatibility just coincides with the complete incompatibility. We establish some relations between $s$-order incompatibility, minimal support uncertainty and rank deficiency of the transition matrix. As an example, we determine the incompatibility order of the discrete Fourier transform with any finite dimension.