Phase groups and the origin of non-locality for qubits

TL;DR

This paper introduces a framework to compare quantum-like theories, revealing how the structure of their phase groups influences their non-locality properties, exemplified by stabiliser states and Spekkens's toy theory.

Contribution

It identifies the phase group as a key factor in the non-locality of quantum-like theories and classifies theories based on their phase group structure.

Findings

Stabiliser theory has phase group Z4.

Spekkens's toy theory has phase group Z2 x Z2.

The phase group determines the form of GHZ correlations.

Abstract

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 Spekkens's toy theory. 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 toy theory the group is Z2 x 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 x 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 in Spekkens's toy theory. The results point at a classification of local/non-local behaviours by finite Abelian groups, extending beyond qubits to finitary theories whose observables are all mutually unbiased.