  We introduce the concept of quantum weight as a fundamental property of
insulating states of matter that is encoded in the ground-state static
structure and measures quantum fluctuation in electrons' center of mass. We
find a sum rule that directly relates quantum weight -- a ground state property
-- with the negative-first moment of the optical conductivity above the gap
frequency. Building on this connection to optical absorption, we derive both an
upper bound and a lower bound on quantum weight in terms of electron density,
dielectric constant, and energy gap. Therefore, quantum weight constitutes a
key material parameter that can be experimentally determined from X-ray
scattering.
