Citation: Pavlovic, Dusko. Quantum measurements without sums. No. RR-06-02. 2006.
Web: https://arxiv.org/abs/quant-ph/0608035
Tags: Philosophical, Monoidal-categories
In this paper, the authors give a foundation for quantum mechanics entirely in terms of the tensor product, without direct sums. Specifically, they do the following:
Since the general algebraic structure has no biproduct, we conclude that the direct sum (and hence the additive aspect of Hilbert spaces) is entirely unnecessary for quantum mechanics.
An interesting point is that one thing you do need direct sums for, however, is measurement. If you are given a (categorically defined) classical object in your category, you can use that to make a quantum measurement scheme. Direct sums are intimately tied to classical objects. In a way, it makes sense that quantum measurements are a classical concept. The output of a quantum measurement is a classical probability distribution over states.
The results of this paper help contextualize string diagrams in modular tensor categories. Why do direct sums correspond to classical probability distributions? Why are direct sums so hard to work into the graphical language? It is because direct sums are a classical notation, and are in a real sense peripheral to quantum mechanics.