No eleventh conditional Ingleton inequality

Abstract

A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studen\'y (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.