# Scenarios and parties¶

Bell experiments are performed using several observers, called parties. These parties are often numbered and named alphabetically Alice, Bob, Charlie and so on. A sequence of parties is a Bell scenario.

A party is unambiguously described by a sequence giving the number of outcomes for each measurement settings.

A scenario can be represented in plain text using the following grammar:

• Scenario := [Party Party ... ]
• Party := (Input Input ...)
• Input := number >= 2

Examples:

• the CHSH scenario is written down [(2 2) (2 2)],
• the Sliwa scenario is written down [(2 2) (2 2) (2 2)],
• the I2233 scenario is written down [(3 3) (3 3)],
• the I3322 scenario is written down [(2 2 2) (2 2 2)].

In the FaacetsPaper, we introduced the following notation: $$[(k_{11} k_{12} \ldots k_{1 m_1})~(k_{21} k_{22} \ldots k_{2 m_2}) \ldots (k_{n 1} k_{n 2} \ldots k_{n m_n})]$$, with $$m_i \ge 1$$ is the number of measurement settings for the $$i^\text{th}$$ party and $$k_{i j} \ge 2$$ is the number of measurement outcomes for the $$j^\text{th}$$ measurement setting of the $$i^\text{th}$$ party.

## Canonical parties¶

Notice that parties (3 2) and (2 3) have essentially the same measurement structure up to a reordering to measurement settings. We thus define a party as canonical when the number of outcomes for successive measurement settings is non-increasing.

Thus, the canonical form of (2 3) is (3 2).

## Canonical scenarios¶

Notice also that scenarios [(2 2) (3 3)] and [(3 3) (2 2)] are identical under reordering of parties. We thus prescribe that a scenario is canonical when its parties are themselves canonical and ordered lexicographically: for all successive non-identical parties $$i$$ and $$i+1$$, there is a $$j \ge 0$$ such that $$\forall k < j$$ we have $$k_{i k} = k_{i+1, k}$$ and $$k_{i j} > k_{i+1, j}$$. For these ordering purposes, we define $$k_{i j} = 0$$ for $$j > m_i$$.