| Summary: | Learning physical properties of a quantum system is essential for the
developments of quantum technologies. However, Heisenberg's uncertainty
principle constrains the potential knowledge one can simultaneously have about
a system in quantum theory. Aside from its fundamental significance, the
mathematical characterization of this restriction, known as `uncertainty
relation', plays important roles in a wide range of applications, stimulating
the formation of tighter uncertainty relations. In this work, we investigate
the fundamental limitations of variance-based uncertainty relations, and
introduce several `near optimal' bounds for incompatible observables. Our
results consist of two morphologically distinct phases: lower bounds that
illustrate the uncertainties about measurement outcomes, and the upper bound
that indicates the potential knowledge we can gain. Combining them together
leads to an \emph{uncertainty interval}, which captures the essence of
uncertainties in quantum theory. Finally, we have detailed how to formulate
lower bounds for product-form variance-based uncertainty relations by employing
entropic uncertainty relations, and hence built a link between different forms
of uncertainty relations.
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