Phase Groups and the Origin of Non-locality for Qubits

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 × 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 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.

Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction

“…The latter sort of theory, which makes use of a classical phase space over a discrete field, has been developed in Refs. [57,82]. The proof that it is operationally equivalent to the stabilizer theory for qutrits proceeds by showing that it reproduces the discrete Wigner representation for odd-dimensional systems that was proposed by Gross [83] (which is positive on stabilizer states).…”

The Search for Structure in Quantum Computation

“…For example, one can ask whether teleportation could be done in Rel; it cannot! Another striking exploration of this kind is the work by Coecke, Edwards and Spekkens [23] on formalizing a certain toy model originally due to Spekkens and showing that there is a group-theoretic reason for the difference between the two models.…”

Complete set of circuit equations for stabilizer quantum mechanics

“…By the Gottesman-Knill theorem [18], stabilizer quantum mechanics can be efficiently simulated by a classical computer. It has been shown [19] [20] that there is a close relationship between the stabilizer formalism and Spekkens' toy theory [21].…”