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 \(j > m_i\).