Elaborating on our joint work with Abramsky in [2,3] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [2,3], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in) famous Birkhoff & von Neumann paper [7]. Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [2] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.
