We explore the uncertainty relation for unitary operators in a new way and
find two uncertainty equalities for unitary operators, which are minimized by
any pure states. Additionally, we derive two sets of uncertainty inequalities
that unveil hierarchical structures within the realm of unitary operator
uncertainty. Furthermore, we examine and compare our method for unitary
uncertainty relations to other prevailing formulations. We provide explicit
examples for better understanding and clarity. Results show that the
hierarchical unitary uncertainty relations establish strong bounds. Moreover,
we investigate the higher-dimensional limit of the unitary uncertainty
equalities.Comment: 15 pages, 4 figure