Quantum Uncertainty Equalities and Inequalities for Unitary Operators

We explore the uncertainty relation for unitary operators in a new way andfind two uncertainty equalities for unitary operators, which are minimized byany pure states. Additionally, we derive two sets of uncertainty inequalitiesthat unveil hierarchical structures within the realm of unitary operatoruncertainty. Furthermore, we examine and compare our method for unitaryuncertainty relations to other prevailing formulations. We provide explicitexamples for better understanding and clarity. Results show that thehierarchical unitary uncertainty relations establish strong bounds. Moreover,we investigate the higher-dimensional limit of the unitary uncertaintyequalities.