Incompatibility of rank-one POVMs and quantum uncertainty relations

Abstract

The incompatibility of quantum measurements is a fundamental feature of quantum mechanics with profound implications for uncertainty relations and quantum information processing. In this paper, we extend the notion of {\em $s$-order incompatibility} of measurements, introduced by Xu (inspired by De Bi\`evre's ``complete incompatibility" ), to more general rank-one POVMs, and establish novel uncertainty relations. Furthermore, we investigate the incompatibility of multiple POVMs and its connection to support uncertainty relations. These results may have applications in quantum cryptography, quantum state reconstruction, and quantum compressed sensing.