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There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility it might belong to mathoverflow as it involves non-trivial inequalities.

Bell inequality violation in two party setting has survived experimental verification.

Instead of two players Alice and Bob let us include two additional players Chihiro and Denzel.

In the original game Alice and Bob are independent players and let us denote the situation $1\times 1$.

If Alice, Bob, Chihiro and Denzel interact independently let us denote the situation $1\times1\times1\times1$.

If Alice and Bob collude let us denote the situation $2\times1\times1$.

If Alice and Bob collude and Chihiro and Denzel collude let us denote the situation $2\times2$.

If Alice, Bob and Chihiro collude let us denote the situation $3\times1$.

  1. Are the Bell inequality violations for these cases quantified somewhere?
  1. Is there experimental results somewhere (it would be out of scope for MO)?
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  • $\begingroup$ Are experiments available for the above cases? $\endgroup$
    – Turbo
    Commented Dec 18, 2020 at 11:36
  • $\begingroup$ Not sure if this is relevant, but there is a nice Coleman lecture with a simple explanation of a Bell-type setting with 1 emitter and 3 observers arxiv.org/abs/2011.12671 (starting page 5). $\endgroup$
    – Sylvain Ribault
    Commented Dec 18, 2020 at 12:45
  • $\begingroup$ @1.. I see that you have created (quantum-games) tag. It might be useful to create also tag-wiki or at least tag-excerpt. It might help other users to use the tag correctly. (This is probably not a problem here, since the tag name seems to be descriptive enough.) Another reason is that the tags used on only one question are automatically deleted after six months unless they have tag-wiki. $\endgroup$
    – Martin Sleziak
    Commented Apr 15, 2021 at 7:49

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One way to classify multiparty Bell inequalities is via the triple $\chi=(n, m, d)$ where $n$ parties choose from among $m$ measurements each obtaining one of $d$ outcomes. I presume you wish to stick to binary observables, so $d=2$. The number $n$ refers to independent parties, "colluding" parties are accounted for by increasing the number $m$. The usual Bell-CHSH inequality (the $1\times 1$ case) refers to $\chi=(2,2,2)$, the $2\times 2$ case is $\chi=(2,4,2)$, and the $1\times 1\times 1\times 1$ case is $\chi=(4,2,2)$. The other cases in the OP, $3\times 1$ and $2\times 1\times 1$ would require a further generalization where the number of measurements can vary from one party to the other.

The full set of Bell inequalities for $\chi=(n,2,2)$ can be found in Trade-offs in multi-party Bell inequality violations in qubit networks. The more general $\chi=(n,m,2)$ case was treated in On tight multiparty Bell inequalities for many settings.

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    $\begingroup$ Is there reference for 2.? Experiments? $\endgroup$
    – Turbo
    Commented Dec 18, 2020 at 11:28
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    $\begingroup$ not that I know of, no multi-party Bell tests have been carried out in the laboratory. $\endgroup$
    – Carlo Beenakker
    Commented Dec 18, 2020 at 11:51
  • $\begingroup$ Strange I think $4$-party should be tested (may be nature behaves differently). $\endgroup$
    – Turbo
    Commented Dec 18, 2020 at 11:53

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