  Sums play a prominent role in the formalisms of quantum mechanics, be it for
mixing and superposing states, or for composing state spaces. Surprisingly, a
conceptual analysis of quantum measurement seems to suggest that quantum
mechanics can be done without direct sums, expressed entirely in terms of the
tensor product. The corresponding axioms define classical spaces as objects
that allow copying and deleting data. Indeed, the information exchange between
the quantum and the classical worlds is essentially determined by their
distinct capabilities to copy and delete data. The sums turn out to be an
implicit implementation of this capabilities. Realizing it through explicit
axioms not only dispenses with the unnecessary structural baggage, but also
allows a simple and intuitive graphical calculus. In category-theoretic terms,
classical data types are dagger-compact Frobenius algebras, and quantum spectra
underlying quantum measurements are Eilenberg-Moore coalgebras induced by these
Frobenius algebras.
