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Article overview
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Phase groups and the origin of non-locality for qubits | Bob Coecke
; Bill Edwards
; Robert W. Spekkens
; | Date: |
25 Mar 2010 | 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. | Source: | arXiv, 1003.5005 | Services: | Forum | Review | PDF | Favorites |
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