Holographic Entropy Inequalities, Contraction Maps and Partial Cubes

We propose a triality between holographic entropy inequality, contraction maps and graph maps between partial cubes, connecting the disciplines of physics, computer science and mathematics. We use the triality to find all holographic entropy inequalities. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method.