Fig. 5: Inflation technique for the triangle network.

a The second order inflation graph of the triangle network. Such an inflation doubles the number of latent variables relative to the triangle scenario, having six latent variables \(\{{\lambda }_{AB}^{(1)},{\lambda }_{AB}^{(2)},{\lambda }_{BC}^{(1)},{\lambda }_{BC}^{(2)},{\lambda }_{AC}^{(1)},{\lambda }_{AC}^{(2)}\}\). The inflation quadruples the number of observable random variables of the triangle scenario, having twelve observable random variables {a⁽¹⁾, b⁽¹⁾, c⁽¹⁾, a⁽²⁾, b⁽²⁾, c⁽²⁾, a⁽³⁾, b⁽³⁾, c⁽³⁾, a⁽⁴⁾, b⁽⁴⁾, c⁽⁴⁾}. Distributions compatible with this inflated structure satisfy symmetry properties, and have marginals corresponding to products of triangle-compatible distribution. This can be exploited to derive suitable causal compatibility inequalities that are violated by the experimental data. b This plot depicts the 64 × 64 coefficients \({y}_{{a}_{1}{b}_{1}{c}_{1}{a}_{2}{b}_{2}{c}_{3}}\) for a quadratic inequality of the form of Eq. (7) such that the left-hand side is nonnegative on all distributions compatible with the classical triangle scenario, but which evaluates to the negative number V_exp = − 0.02436 ± 0.00016 on our experimental data. The x-axis ranges over the values of (a₁, b₁, c₁) while the y-axis ranges over the values of (a₂, b₂, c₂), and the color at a given point denotes the value of \({y}_{{a}_{1}{b}_{1}{c}_{1}{a}_{2}{b}_{2}{c}_{3}}\) according to the mapping set out in the legend.