Bell expression symmetriesΒΆ

Information about the symmetry group of an expression can be optionally written down in the YAML file under the optional symmetries key. The Faacets command line tool can also compute this information from scratch.

When symmetries is provided, the following properties are required:

numberOfRepresentatives
An integer giving the number of representative of the Bell expression under relabelings. The order of the symmetry group of the Bell expression can be then computed using Lemma 1 of arXiv.
remarkableGenerators

This section list a set of generators for the symmetry group of the Bell expression. Generators are grouped according to the remarkable subgroups they are part of, according to the following sequence of subgroups:

  • liftings: relabelings involving outcomes of a single setting of a single party
  • outputPermsPerParty: relabelings involving outcomes of a single party
  • outputPerms: general outcomes relabelings
  • inputPermsPerParty: relabelings involving settings of a single party
  • outputInputPermsPerParty: relabelings involving settings and outcomes of a single party
  • outputInputPerms: relabelings involving settings and outcomes
  • partyPerms: relabelings involving parties only
  • rest: additional generators

As an example, here is the symmetry information for the CHSH expression.

symmetries:
  numberOfRepresentatives: 8
  remarkableGenerators:
    outputPerms: ['A1(1,2) A2(1,2) B1(1,2) B2(1,2)']
    outputInputPerms: ['A2(1,2) B(1,2)', 'B1(1,2) A(1,2)']
    rest: ['A(1,2) B(1,2) (A,B)']

Todo

Add link to our paper, to the Faacets command line tool documentation