  Combining gravity with quantum theory is still work in progress. On the one
hand, classical gravity, is the geometry of space-time determined by the
energy-momentum tensor of matter and the resulting nonlinear equations; on the
other hand, the mathematical description of a quantum system, is Hilbert space
with linear equations describing evolution. In this paper, various measures in
Hilbert space will be presented. In general, distance measures in Hilbert space
can be divided into measures determined by energy and measures determined by
entropy. Entropy measures determine quasi-distance because they do not satisfy
all the axioms defining distance. Finding a general rule to determine such a
measure unambiguously seems to be fundamental.
