De-linearizing Linearity: Projective Quantum Axiomatics From Strong Compact Closure

Elaborating on our joint work with Abramsky in [2,3] we further unravel the linear structure of Hilbert spaces into several constituents which play different roles in quantum mechanics. Some prove to be very crucial for particular features of the theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases. In particular do we pass from a vector space formalism to a rather projective one as it was intended in the (in)famous Birkhoffvon Neumann paper [7]. The bulk of the linear structure required to reason about quantum mechanics is multiplicative since it 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 most importantly, the passage from a formalism of the vector space kind to a rather projective one in terms of the preparation-state agreement axiom. Additive types for objects including a zero object provide pseudoprojections 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 pseudo-linear and satisfies the diagonal axiom, which brings us to our main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases.