Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality

• URL: http://arxiv.org/abs/2207.07451v1
• Date: Fri, 15 Jul 2022 12:55:18 GMT
• Title: Relating incompatibility, noncommutativity, uncertainty and Kirkwood-Dirac nonclassicality
• Authors: Stephan De Bievre
• Abstract summary: We study the notion of completely incompatible observables and its links to the support uncertainty and to the Kirkwood-Dirac nonclassicality of pure quantum states. We show furthermore that complete incompatibility implies several weaker notions of incompatibility, among which features a strong form of noncommutativity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We provide an in-depth study of the recently introduced notion of completely incompatible observables and its links to the support uncertainty and to the Kirkwood-Dirac nonclassicality of pure quantum states. The latter notion has recently been proven central to a number of issues in quantum information theory and quantum metrology. In this last context, it was shown that a quantum advantage requires the use of Kirkwood-Dirac nonclassical states. We establish sharp bounds of very general validity that imply that the support uncertainty is an efficient Kirkwood-Dirac nonclassicality witness. When adapted to completely incompatible observables that are close to mutually unbiased ones, this bound allows us to fully characterize the Kirkwood-Dirac classical states as the eigenvectors of the two observables. We show furthermore that complete incompatibility implies several weaker notions of incompatibility, among which features a strong form of noncommutativity.

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