  We derive several uncertainty relations for two arbitrary unitary operators
acting on physical states of a Hilbert space. We show that our bounds are
tighter in various cases than the ones existing in the current literature.
Using the uncertainty relation for the unitary operators, we obtain the tight
state-independent lower bound for the uncertainty of two Pauli observables and
anticommuting observables in higher dimensions. With regard to the
minimum-uncertainty states, we derive the minimum-uncertainty state equation by
the analytic method and relate this to the ground-state problem of the Harper
Hamiltonian. Furthermore, the higher-dimensional limit of the uncertainty
relations and minimum-uncertainty states are explored. From an operational
point of view, we show that the uncertainty in the unitary operator is directly
related to the visibility of quantum interference in an interferometer where
one arm of the interferometer is affected by a unitary operator. This shows a
principle of preparation uncertainty, i.e., for any quantum system, the amount
of visibility for two general noncommuting unitary operators is nontrivially
upper bounded.
