  Originally, quantum probability theory was developed to analyze statistical
phenomena in quantum systems, where classical probability theory does not
apply, because the lattice of measurable sets is not necessarily distributive.
On the other hand, it is well known that the lattices of concepts, that arise
in data analysis, are in general also non-distributive, albeit for completely
different reasons. In his recent book, van Rijsbergen argues that many of the
logical tools developed for quantum systems are also suitable for applications
in information retrieval. I explore the mathematical support for this idea on
an abstract vector space model, covering several forms of data analysis
(information retrieval, data mining, collaborative filtering, formal concept
analysis...), and roughly based on an idea from categorical quantum mechanics.
It turns out that quantum (i.e., noncommutative) probability distributions
arise already in this rudimentary mathematical framework. We show that a
Bell-type inequality must be satisfied by the standard similarity measures, if
they are used for preference predictions. The fact that already a very general,
abstract version of the vector space model yields simple counterexamples for
such inequalities seems to be an indicator of a genuine need for quantum
statistics in data analysis.
