Bell expression and bounds

$$ \left (\begin{array}{rr:rr} \tooltip{-1}{P(11|11)} & \tooltip{1}{P(21|11)} & \tooltip{-1}{P(11|21)} & \tooltip{1}{P(21|21)} \\ \tooltip{1}{P(12|11)} & \tooltip{-1}{P(22|11)} & \tooltip{1}{P(12|21)} & \tooltip{-1}{P(22|21)} \\ \hdashline \tooltip{-1}{P(11|12)} & \tooltip{1}{P(21|12)} & \tooltip{1}{P(11|22)} & \tooltip{-1}{P(21|22)} \\ \tooltip{1}{P(12|12)} & \tooltip{-1}{P(22|12)} & \tooltip{-1}{P(12|22)} & \tooltip{1}{P(22|22)}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$
$$ \left (\begin{array}{r|r:r} \tooltip{-2}{P_(|)} & \tooltip{4}{P_A(1|1)} & \tooltip{0}{P_A(1|2)} \\ \hline \tooltip{4}{P_B(1|1)} & \tooltip{-4}{P_AB(1,1|1,1)} & \tooltip{-4}{P_AB(1,1|2,1)} \\ \hdashline \tooltip{0}{P_B(1|2)} & \tooltip{-4}{P_AB(1,1|1,2)} & \tooltip{4}{P_AB(1,1|2,2)}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$
$$ \left (\begin{array}{r|r:r} & \tooltip{0}{<A1>} & \tooltip{0}{<A2>} \\ \hline \tooltip{0}{<B1>} & \tooltip{-1}{<A1 B1>} & \tooltip{-1}{<A2 B1>} \\ \hdashline \tooltip{0}{<B2>} & \tooltip{-1}{<A1 B2>} & \tooltip{1}{<A2 B2>}\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$
$$ \left (\begin{array}{c} \text{-} \text{ } 1 \text{ } P(11|11) \text{ } \text{+} \text{ } 1 \text{ } P(21|11) \text{ } \text{-} \text{ } 1 \text{ } P(11|21) \text{ } \text{+} \text{ } 1 \text{ } P(21|21) \\ 1 \text{ } P(12|11) \text{ } \text{-} \text{ } 1 \text{ } P(22|11) \text{ } \text{+} \text{ } 1 \text{ } P(12|21) \text{ } \text{-} \text{ } 1 \text{ } P(22|21) \\ \text{-} \text{ } 1 \text{ } P(11|12) \text{ } \text{+} \text{ } 1 \text{ } P(21|12) \text{ } \text{+} \text{ } 1 \text{ } P(11|22) \text{ } \text{-} \text{ } 1 \text{ } P(21|22) \\ 1 \text{ } P(12|12) \text{ } \text{-} \text{ } 1 \text{ } P(22|12) \text{ } \text{-} \text{ } 1 \text{ } P(12|22) \text{ } \text{+} \text{ } 1 \text{ } P(22|22)\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$
$$ \left (\begin{array}{c} \text{-} \text{ } 2 \text{ } P_(|) \text{ } \text{+} \text{ } 4 \text{ } P_A(1|1) \text{ } \text{+} \text{ } 4 \text{ } P_B(1|1) \text{ } \text{-} \text{ } 4 \text{ } P_AB(1,1|1,1) \\ \text{-} \text{ } 4 \text{ } P_AB(1,1|2,1) \text{ } \text{-} \text{ } 4 \text{ } P_AB(1,1|1,2) \text{ } \text{+} \text{ } 4 \text{ } P_AB(1,1|2,2)\end{array}\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$
$$ \left (\text{-} \text{ } 1 \text{ } <A1 B1> \text{ } \text{-} \text{ } 1 \text{ } <A2 B1> \text{ } \text{-} \text{ } 1 \text{ } <A1 B2> \text{ } \text{+} \text{ } 1 \text{ } <A2 B2>\right ) \le \left \{ \begin{array}{rl} \text{local:} & 2\text{ (facet)} \\ \text{no-signaling:} & 4 \\ \text{quantum:} & {} [2.828427124746190, 2.828427124746191] \\ \text{ten-percents-nonfree-settings:} & 2.2\end{array} \right. $$

Symmetry group

Symmetry group of order: 16

Generators:

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

Extras

  • detection_loophole: eta_c = 2/3
  • detection_loophole_singlet: eta_c = [0.8284271247461901, 0.8284271247461902]