PHASE GROUPS AND LOCAL HIDDEN VARIABLES

Abstract

This report extends previous work [3] in which quantum mechanics and a quantum-like theory (Spekkens’s toy bit theory [5]) were compared within the framework of symmetric monoidal categories. In this context, each quantum-like theory is naturally associated with an Abelian group, termed its phase group. Quantum mechanics and the toy bit theory exhibit different phase groups, Z4 and Z2 × Z2 respectively, and it was shown that this difference exactly underlies the fact that while the predictions of the toy theory can be modelled by local hidden variables, those of the stabiliser theory cannot. In this report we extend this work to more general groups: given a quantum-like theory with some phase group we derive a group theoretic criterion which determines whether a local hidden variable interpretation is impossible for the theory. This result is essentially a generalisation of Mermin’s famous no-go theorem [4] employing the GHZ state.