Two different spaces can be used to represent joint probability distributions:

  • joint probability distributions of the form \(P(ab...|xy...)\) are contained in the correlation space,
  • local decompositions of the form \(q_{\alpha \beta ...}\) are contained in the space of strategy weights.

The correlation space contains signaling and non-properly normalized joint probability distributions. As such, we defined in FaacetsPaper the no-signaling subspace and its canonical projection. Because the no-signaling subspace is not fully dimensional, several of its parametrizations are used in the literature and have been implemented in Faacets: the Collins-Gisin notation (see CollinsGisin), and the binary Correlators notation (see e.g. Sliwa). We extended the Correlators notation in FaacetsPaper to non-binary outputs. We also extended both the Collins-Gisin and the Correlators notations to include signaling terms, such that there is a bijection between signaling probabilities \(P(ab...|xy...)\), the Signaling Collins-Gisin and Correlators notation. The transformation between the Non-signaling probabilities, Collins-Gisin and Correlators notations is also bijective.

As described in FaacetsPaper , Bell expressions can be projected in the non-signaling subspace by setting the signaling terms to 0 in the Collins-Gisin or Correlators notations.

Local decompositions can be specified using either weights corresponding to deterministic points, or using strategy correlators as specified in the BilocalityPaper.

In summary, we distinguish:

Representation name Shorthand Bijection with To represent Used for
Signaling probabilities SP SC, SG Probability distributions

Permutation group algorithms

Canonical expresions

Non-signaling probabilities NP NC, NG Prob. dist.
Signaling Collins-Gisin SG SP, SC Prob. dist.  
Non-signaling Collins-Gisin NG NP, NC Prob. dist.  
Signaling Correlators SC SP, SG Prob. dist. Product decompositions and projection in non-signaling subspace
Non-signaling Correlators NC NP, NG Prob. dist.
Strategy Correlators T W Local decompositions  
Strategy Weights W T Local dec.  

Probabilites representations

In this representation, we simply write enumerate the coefficients of the joint probability distribution \(P(ab...|xy...)\) (or the coefficients of a Bell expression acting on such distributions) in the following order: we increment first Alice’s outcome $a$, then increment Alice’s setting $x$, then increment Bob’s outcome $b$, then increment Bob’s setting $y$, and so on.

There is no difference in the order of terms between Signaling and Non-signaling Probabilities, except that distributions or expressions in the Non-signaling Probabilities representation are non-signaling or have been projected.

For the CHSH scenario with two parties and binary measurement settings/outcomes, the order of terms is: P(11|11), P(21|11), P(11|21), P(21|21), P(12|11), P(22|11), P(12|21), P(22|21), P(11|12), P(21|12), P(11|22), P(21|22), P(12|12), P(22|12), P(12|22), P(22|22).

Correlators representations

To be described.

Collins-Gisin representations

To be described.

Strategy Correlators representation

To be described.

Strategy Weights representation

To be described.