Symmetries in Bell nonlocality, part 3

Let us now turn our attention to the symmetries of the full scenario. We first recall the definitions of part 1.

addpath([pwd, '/../../../external/replab']);
replab_init('verbose', 0);
outputGroup = replab.S(2);
outputRep = outputGroup.naturalRep;
inputGroup = replab.S(2);
ioGroup = inputGroup.wreathProduct(outputGroup);
ioRep = ioGroup.imprimitiveRep(outputRep);

replab_init: Initialization done.

The same story repeats for relabelings of parties: the scenario involves two homogeneous parties. We thus have two copies of the group relabeling inputs and/or outputs (one for Alice, one for Bob), and a copy of \(S_2\) that permutes the parties. This group desribing all possible relabellings in this scenario is thus given by:

scenarioGroup = outputGroup.wreathProduct(ioGroup);

scenarioGroup.sample

ans =
{
  [1,1] =

     1   2

  [1,2] =
  {
    [1,1] =
    {
      [1,1] =

         2   1

      [1,2] =
      {
        [1,1] =

           1   2

        [1,2] =

           1   2

      }

    }

    [1,2] =
    {
      [1,1] =

         2   1

      [1,2] =
      {
        [1,1] =

           1   2

        [1,2] =

           2   1

      }

    }

  }

}

The representation on the behavior \(P(ab|xy)\) is however a primitive representation, as \(P(a|x; b|y)\) ressembles a tensor. Inside each party, we use the imprimitive representation constructed before.

probRep = scenarioGroup.primitiveRep(ioRep);

Decomposition of the full probability space

We can now compute the decomposition of this representation on \(P(ab|xy)\):

dec = probRep.decomposition

dec =

Orthogonal reducible representation
   dimension: 16
       field: 'R'
       group: Finite group of order 128
   isUnitary: true
      parent: Orthogonal representation
component(1): Isotypic component R(1) (trivial)
component(2): Isotypic component R(1) (nontrivial)
component(3): Isotypic component R(2) (nontrivial)
component(4): Isotypic component R(4) (nontrivial)
component(5): Isotypic component R(4) (nontrivial)
component(6): Isotypic component R(4) (nontrivial)

There are 6 irreducible components

dec.nComponents

ans = 6

These components correspond to the following physical elements ( see arXiv:1610.01833 ) for more details:

• The global normalization of probabilities

• The difference of normalization between \(x=y\) and \(x\ne y\)

• The difference of normalization between \(x=0\) and \(x=1\)

• The signaling between Alice and Bob

• The marginal probabilities of Alice and Bob

• The correlation between Alice and Bob