Classification of incompatibility for two orthonormal bases

Abstract

For two orthonormal bases of a $d$-dimensional complex Hilbert space, the notion of complete incompatibility was introduced recently by De Bièvre [Phys. Rev. Lett. 127, 190404 (2021)]. In this work, we introduce the notion of $s$-order incompatibility with positive integer $s$ satisfying $2\le s\le d+1.$ 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.

arXiv comment7 pages, 3 figures

References in corpus (15)

• An Invitation to Quantum Incompatibility
• Quantum theory of successive projective measurements
• Quantum Incompatibility Witnesses
• A Complete Resource Theory of Quantum Incompatibility as Quantum Programmability
• Non-disturbing quantum measurements
• Incompatibility robustness of quantum measurements: a unified framework
• Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility
• Quantum random access codes and incompatibility of measurements
• On uncertainty principles in the finite dimensional setting
• Universality of Sequential Quantum Measurements
• Device-independent quantification of measurement incompatibility
• Optimal quantum observables
• Quantum Incompatibility of a Physical Context
• Quantum measurement incompatibility in subspaces
• Incompatibility probability of random quantum measurements

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