  We provide a unified and strengthened framework for the product form and the
sum form variance-based uncertainty relations by constructing a unified
uncertainty relation. In the unified framework, we deduce that the
uncertainties of the incompatible observables are bounded by not only the
commutator of themselves, but also the quantities related with the other
operator. This operator can provide information so that we can capture the
uncertainty of the measurement result more accurately, and thus is named as the
information operator. The introduction of the information operator can fix the
deficiencies in both the product form and the sum form uncertainty relations,
and provides a more accurate description of the quantum uncertainty relation.
The unified framework also proposes a new interpretation of the uncertainty
relation for non-Hermitian operators; i.e., the "observable" second-order
origin moments of the non-Hermitian operators cannot be arbitrarily small at
the same time when they are generalized-incompatible on the new definition of
the generalized commutator.
