No eleventh conditional Ingleton inequality
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Submission date: 11. Apr. 2022
Link to arXiv: See the arXiv entry of this preprint.
A rational probability distribution on four binary random variables X,Y,Z,U is constructed which satisﬁes the conditional independence relations [X ⫫ Y ], [X ⫫ Z∣U], [Y ⫫ U∣Z] and [Z ⫫ U∣XY ] but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (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.