This chapter is dedicated to the foundations of quantum theory, or as it's more fashionably called these days, quantum foundations. Here, we will use all of the things we've learned so far to probe some very deep questions:

1. 1. What features of nature are imposed on us by quantum theory?

2. 2. Conversely, what features of a physical theory are imposed on us by (our current understanding of) nature?

3. 3. Which of these features are ‘properly quantum’, in the sense that they have no counterpart in any classical physical theory?

We'll address these questions by looking at one of those most celebrated (and historically controversial) properties of quantum theory: quantum non-locality. First, we will give a precise definition of non-locality and prove that it exists within the theory of quantum processes and in fact already within the comparatively tiny subtheory of causal Clifford maps. Then, we will present a new process theory called spek, which has locality built right in. A remarkable thing is that the two theories of Clifford maps and spek are identical in every respect except one: the phase group of a single system. And (another spoiler alert!) it is indeed this one difference that kills the proof of non-locality that works for quantum theory.

Quantum Non-locality

Quantum non-locality is probably still the least understood of all the new quantum features, in both philosophical and structural terms. Our upbringing in a seemingly ‘classical’ world, and especially our undeniably corrupting ‘classical’ scientific education, tends to make us expect two things from a physical theory:

1. 1. Realism: physical systems have real pre-existing properties, and hence the outcome of ‘measuring’ such a property is fixed in some way prior to the measurement.

2. 2. Locality: it is impossible for one system to affect another distant system instantaneously.