We describe a general framework in which we can precisely compare the structures of quantum-like theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and a toy theory proposed by Spekkens. We discover that viewed within our framework these theories are very similar, but differ in one key aspect- a four element group we term the phase group which emerges naturally within our framework. In the case of the stabiliser theory this group is Z4 while for Spekkens’s theory the group is Z2 × Z2. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. This is done by establishing a connection between the phase group, and an abstract notion of GHZ state correlations. We go on to formulate precisely how the stabiliser theory and toy theory are ‘similar ’ by defining a notion of ‘mutually unbiased qubit theory’, noting that all such theories have four element phase groups. Since Z4 and Z2 × Z2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and those appearing in Spekkens’s theory. The results point at a classification of local/non-local behaviours by finite Abelian groups, extending beyond qubits to any finitary theory whose observables are all mutually unbiased. 1

Dedicated to the many bright young theoretical physicists that failed to escape the fate of having to work in institutions like banks. Preface New? In what sense? Surely I am not the only person who, after extensively justifying why certain mathematical structures naturally arise in physics, gets questions like: “this is all nice maths but what’s the physics? ” Meanwhile I figured out what this truly means: “I don’t see any differential equations! ” Okay, this is indeed a bit overstated. Nowadays any mathematical argument involving groups, when these are moreover referred to as ‘symmetry groups’, stands a serious chance of being eligible for carrying the label ‘physics’. But it hasn’t always been like this. John Slater (cf. the Slater determinant in quantum chemistry) referred to the use of group theory in quantum physics by Weyl, Wigner et al. as der Gruppenpest, what translates as the ‘plague of groups’. Even in 1975 he wrote [14]: “As soon as [my] paper became known, it was obvious that a great many other physicists were as ‘disgusted ’ as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as ‘Slater has slain the Gruppenpest’. I believe that no other