  A new formalism to express and operate on diversity measures of qualitative
variables, built in a Hilbert space, is presented. The abstract character of
the Hilbert space naturally incorporates the equivalence between qualitative
variables and is utilized here to (i) represent the binary character of answers
to categories and (ii) introduce a new criterium for choosing between different
measures of diversity, namely, robustness against uncertainty. The full
potential of the formulation on a Hilbert space comes to play when addressing
the reduction of categories problem, a common problem in data analysis. The
present formalism solves the problem by incorporating strategies inspired by
mathematical and physical statistics, specifically, it makes use of the concept
of partial trace. In solving this problem, it is shown that properly
normalizing the diversity measures is instrumental to provide a sensible
interpretation of the results when the reduction of categories is performed.
Finally, the approach presented here also allows for straightforwardly
measuring diversity and performing category reduction in situations when
simultaneous categories could be chosen.
