  In a recent paper [12], we discussed the serious inconsistency present within
the operational and mathematical definition(s) of the notion of pure state.
Continuing this analysis, in this work we attempt to address the role of
'purity' and 'mixtures' within two different categorical approaches to QM,
namely, the topos approach originally presented by Chris Isham and Jeremy
Butterfield [27, 28, 29] and the more recent logos categorical approach
presented by the authors of this article [10, 11, 13]. While the first approach
exposes the difficulties to produce a consistent understanding of pure states
and mixtures, the latter approach presents a new scheme in which their
reference is erased right from the start in favor of an intensive understanding
of projection operators and quantum superpositions. This new account of the
theory, grounded on an intensive interpretation of the Born rule, allows us not
only to avoid the orthodox interpretation of projection operators --either as
referring to definite valued properties or measurement outcomes-- but also to
consider all matrices (of any rank) on equal footing. It is from this latter
standpoint that we conclude that instead of distinguishing between pure and
mixed states it would be recommendable --for a proper understanding of the
theory of quanta-- to return to the original matrix formulation of quantum
mechanics presented by Werner Heisenberg in 1925.
