# Representations¶

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.