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Scenario

[(2 2) (2 2) (2 2)]

Description

Mermin 1990, inequality for 3 parties

Cross references

pubs/Mermin1990/eq11_3 pubs/Sliwa2003/2 solved/2-2/2-2/2-2/local/33

Bell expression and bounds

$$ \left (\begin{array}{rr:rr} \tooltip{-1}{P(111|111)} & \tooltip{1}{P(211|111)} & \tooltip{0}{P(111|211)} & \tooltip{0}{P(211|211)} \\ \tooltip{1}{P(121|111)} & \tooltip{-1}{P(221|111)} & \tooltip{0}{P(121|211)} & \tooltip{0}{P(221|211)} \\ \hdashline \tooltip{0}{P(111|121)} & \tooltip{0}{P(211|121)} & \tooltip{-1}{P(111|221)} & \tooltip{1}{P(211|221)} \\ \tooltip{0}{P(121|121)} & \tooltip{0}{P(221|121)} & \tooltip{1}{P(121|221)} & \tooltip{-1}{P(221|221)}\end{array} \quad \begin{array}{rr:rr} \tooltip{1}{P(112|111)} & \tooltip{-1}{P(212|111)} & \tooltip{0}{P(112|211)} & \tooltip{0}{P(212|211)} \\ \tooltip{-1}{P(122|111)} & \tooltip{1}{P(222|111)} & \tooltip{0}{P(122|211)} & \tooltip{0}{P(222|211)} \\ \hdashline \tooltip{0}{P(112|121)} & \tooltip{0}{P(212|121)} & \tooltip{1}{P(112|221)} & \tooltip{-1}{P(212|221)} \\ \tooltip{0}{P(122|121)} & \tooltip{0}{P(222|121)} & \tooltip{-1}{P(122|221)} & \tooltip{1}{P(222|221)}\end{array} \quad \begin{array}{rr:rr} \tooltip{0}{P(111|112)} & \tooltip{0}{P(211|112)} & \tooltip{-1}{P(111|212)} & \tooltip{1}{P(211|212)} \\ \tooltip{0}{P(121|112)} & \tooltip{0}{P(221|112)} & \tooltip{1}{P(121|212)} & \tooltip{-1}{P(221|212)} \\ \hdashline \tooltip{1}{P(111|122)} & \tooltip{-1}{P(211|122)} & \tooltip{0}{P(111|222)} & \tooltip{0}{P(211|222)} \\ \tooltip{-1}{P(121|122)} & \tooltip{1}{P(221|122)} & \tooltip{0}{P(121|222)} & \tooltip{0}{P(221|222)}\end{array} \quad \begin{array}{rr:rr} \tooltip{0}{P(112|112)} & \tooltip{0}{P(212|112)} & \tooltip{1}{P(112|212)} & \tooltip{-1}{P(212|212)} \\ \tooltip{0}{P(122|112)} & \tooltip{0}{P(222|112)} & \tooltip{-1}{P(122|212)} & \tooltip{1}{P(222|212)} \\ \hdashline \tooltip{-1}{P(112|122)} & \tooltip{1}{P(212|122)} & \tooltip{0}{P(112|222)} & \tooltip{0}{P(212|222)} \\ \tooltip{1}{P(122|122)} & \tooltip{-1}{P(222|122)} & \tooltip{0}{P(122|222)} & \tooltip{0}{P(222|222)}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$
$$ \left (\begin{array}{r|r:r} \tooltip{2}{P_(|)} & \tooltip{0}{P_A(1|1)} & \tooltip{-4}{P_A(1|2)} \\ \hline \tooltip{-4}{P_B(1|1)} & \tooltip{4}{P_AB(1,1|1,1)} & \tooltip{4}{P_AB(1,1|2,1)} \\ \hdashline \tooltip{0}{P_B(1|2)} & \tooltip{-4}{P_AB(1,1|1,2)} & \tooltip{4}{P_AB(1,1|2,2)}\end{array} \quad \begin{array}{r|r:r} \tooltip{-4}{P_C(1|1)} & \tooltip{4}{P_AC(1,1|1,1)} & \tooltip{4}{P_AC(1,1|2,1)} \\ \hline \tooltip{4}{P_BC(1,1|1,1)} & \tooltip{-8}{P_ABC(1,1,1|1,1,1)} & \tooltip{0}{P_ABC(1,1,1|2,1,1)} \\ \hdashline \tooltip{4}{P_BC(1,1|2,1)} & \tooltip{0}{P_ABC(1,1,1|1,2,1)} & \tooltip{-8}{P_ABC(1,1,1|2,2,1)}\end{array} \quad \begin{array}{r|r:r} \tooltip{0}{P_C(1|2)} & \tooltip{-4}{P_AC(1,1|1,2)} & \tooltip{4}{P_AC(1,1|2,2)} \\ \hline \tooltip{4}{P_BC(1,1|1,2)} & \tooltip{0}{P_ABC(1,1,1|1,1,2)} & \tooltip{-8}{P_ABC(1,1,1|2,1,2)} \\ \hdashline \tooltip{-4}{P_BC(1,1|2,2)} & \tooltip{8}{P_ABC(1,1,1|1,2,2)} & \tooltip{0}{P_ABC(1,1,1|2,2,2)}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$
$$ \left (\begin{array}{r|r:r} & \tooltip{0}{<A1>} & \tooltip{0}{<A2>} \\ \hline \tooltip{0}{<B1>} & \tooltip{0}{<A1 B1>} & \tooltip{0}{<A2 B1>} \\ \hdashline \tooltip{0}{<B2>} & \tooltip{0}{<A1 B2>} & \tooltip{0}{<A2 B2>}\end{array} \quad \begin{array}{r|r:r} \tooltip{0}{<C1>} & \tooltip{0}{<A1 C1>} & \tooltip{0}{<A2 C1>} \\ \hline \tooltip{0}{<B1 C1>} & \tooltip{-1}{<A1 B1 C1>} & \tooltip{0}{<A2 B1 C1>} \\ \hdashline \tooltip{0}{<B2 C1>} & \tooltip{0}{<A1 B2 C1>} & \tooltip{-1}{<A2 B2 C1>}\end{array} \quad \begin{array}{r|r:r} \tooltip{0}{<C2>} & \tooltip{0}{<A1 C2>} & \tooltip{0}{<A2 C2>} \\ \hline \tooltip{0}{<B1 C2>} & \tooltip{0}{<A1 B1 C2>} & \tooltip{-1}{<A2 B1 C2>} \\ \hdashline \tooltip{0}{<B2 C2>} & \tooltip{1}{<A1 B2 C2>} & \tooltip{0}{<A2 B2 C2>}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$
$$ \left (\begin{array}{c} \text{-} \text{ } 1 \text{ } P(111|111) \text{ } \text{+} \text{ } 1 \text{ } P(211|111) \text{ } \text{+} \text{ } 1 \text{ } P(121|111) \text{ } \text{-} \text{ } 1 \text{ } P(221|111) \\ \text{-} \text{ } 1 \text{ } P(111|221) \text{ } \text{+} \text{ } 1 \text{ } P(211|221) \text{ } \text{+} \text{ } 1 \text{ } P(121|221) \text{ } \text{-} \text{ } 1 \text{ } P(221|221) \\ 1 \text{ } P(112|111) \text{ } \text{-} \text{ } 1 \text{ } P(212|111) \text{ } \text{-} \text{ } 1 \text{ } P(122|111) \text{ } \text{+} \text{ } 1 \text{ } P(222|111) \\ 1 \text{ } P(112|221) \text{ } \text{-} \text{ } 1 \text{ } P(212|221) \text{ } \text{-} \text{ } 1 \text{ } P(122|221) \text{ } \text{+} \text{ } 1 \text{ } P(222|221) \\ \text{-} \text{ } 1 \text{ } P(111|212) \text{ } \text{+} \text{ } 1 \text{ } P(211|212) \text{ } \text{+} \text{ } 1 \text{ } P(121|212) \text{ } \text{-} \text{ } 1 \text{ } P(221|212) \\ 1 \text{ } P(111|122) \text{ } \text{-} \text{ } 1 \text{ } P(211|122) \text{ } \text{-} \text{ } 1 \text{ } P(121|122) \text{ } \text{+} \text{ } 1 \text{ } P(221|122) \\ 1 \text{ } P(112|212) \text{ } \text{-} \text{ } 1 \text{ } P(212|212) \text{ } \text{-} \text{ } 1 \text{ } P(122|212) \text{ } \text{+} \text{ } 1 \text{ } P(222|212) \\ \text{-} \text{ } 1 \text{ } P(112|122) \text{ } \text{+} \text{ } 1 \text{ } P(212|122) \text{ } \text{+} \text{ } 1 \text{ } P(122|122) \text{ } \text{-} \text{ } 1 \text{ } P(222|122)\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$
$$ \left (\begin{array}{c} 2 \text{ } P_(|) \text{ } \text{-} \text{ } 4 \text{ } P_A(1|2) \text{ } \text{-} \text{ } 4 \text{ } P_B(1|1) \text{ } \text{+} \text{ } 4 \text{ } P_AB(1,1|1,1) \\ 4 \text{ } P_AB(1,1|2,1) \text{ } \text{-} \text{ } 4 \text{ } P_AB(1,1|1,2) \text{ } \text{+} \text{ } 4 \text{ } P_AB(1,1|2,2) \text{ } \text{-} \text{ } 4 \text{ } P_C(1|1) \\ 4 \text{ } P_AC(1,1|1,1) \text{ } \text{+} \text{ } 4 \text{ } P_AC(1,1|2,1) \text{ } \text{+} \text{ } 4 \text{ } P_BC(1,1|1,1) \text{ } \text{-} \text{ } 8 \text{ } P_ABC(1,1,1|1,1,1) \\ 4 \text{ } P_BC(1,1|2,1) \text{ } \text{-} \text{ } 8 \text{ } P_ABC(1,1,1|2,2,1) \text{ } \text{-} \text{ } 4 \text{ } P_AC(1,1|1,2) \text{ } \text{+} \text{ } 4 \text{ } P_AC(1,1|2,2) \\ 4 \text{ } P_BC(1,1|1,2) \text{ } \text{-} \text{ } 8 \text{ } P_ABC(1,1,1|2,1,2) \text{ } \text{-} \text{ } 4 \text{ } P_BC(1,1|2,2) \text{ } \text{+} \text{ } 8 \text{ } P_ABC(1,1,1|1,2,2)\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$
$$ \left (\text{-} \text{ } 1 \text{ } <A1 B1 C1> \text{ } \text{-} \text{ } 1 \text{ } <A2 B2 C1> \text{ } \text{-} \text{ } 1 \text{ } <A2 B1 C2> \text{ } \text{+} \text{ } 1 \text{ } <A1 B2 C2>\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)}\end{array} \right. $$

Symmetry group

Symmetry group of order: 192

Generators:

  • outputPerms :
    • A1(1,2) B2(1,2) C1(1,2)
    • A1(1,2) B1(1,2) C2(1,2)
    • A2(1,2) B2(1,2) C2(1,2)
  • outputInputPerms :
    • B2(1,2) A(1,2) C(1,2)
    • C2(1,2) A(1,2) B(1,2)
  • partyPerms :
    • (B,C)
  • rest :
    • B2(1,2) (A,C)

Extras

Sources