Phase Groups and the Origin of Non-locality for Qubits

Abstract

We describe a general framework in which we can precisely compare the\nstructures of quantum-like theories which may initially be formulated in quite\ndifferent mathematical terms. We then use this framework to compare two\ntheories: quantum mechanics restricted to qubit stabiliser states and\noperations, and Spekkens's toy theory. We discover that viewed within our\nframework these theories are very similar, but differ in one key aspect - a\nfour element group we term the phase group which emerges naturally within our\nframework. In the case of the stabiliser theory this group is Z4 while for\nSpekkens's toy theory the group is Z2 x Z2. We further show that the structure\nof this group is intimately involved in a key physical difference between the\ntheories: whether or not they can be modelled by a local hidden variable\ntheory. This is done by establishing a connection between the phase group, and\nan abstract notion of GHZ state correlations. We go on to formulate precisely\nhow the stabiliser theory and toy theory are `similar' by defining a notion of\n`mutually unbiased qubit theory', noting that all such theories have four\nelement phase groups. Since Z4 and Z2 x Z2 are the only such groups we conclude\nthat the GHZ correlations in this type of theory can only take two forms,\nexactly those appearing in the stabiliser theory and in Spekkens's toy theory.\nThe results point at a classification of local/non-local behaviours by finite\nAbelian groups, extending beyond qubits to finitary theories whose observables\nare all mutually unbiased.\n