Combinatorial properties of holographic entropy inequalities

Abstract:A holographic entropy inequality (HEI) is a linear inequality obeyed by Ryu-Takayanagi holographic entanglement entropies, or equivalently by the minimum cut function on weighted graphs. We establish a new combinatorial framework for studying HEIs, and use it to prove several properties they share, including two majorization-related properties as well as a necessary and sufficient condition for an inequality to be an HEI. We thereby resolve all the conjectures presented in [arXiv:2508.21823], proving two of them and disproving the other two. In particular, we show that the null reduction of any superbalanced HEI passes the majorization test defined in [arXiv:2508.21823], thereby providing strong new evidence that all HEIs are obeyed in time-dependent holographic states.

35 pages, 1 appendix, 2 figures. Watch a 5-minute video abstract here: this https URL v2: clarified that our results apply to contraction-provable HEIs; new appendix on changing the purifier; other minor corrections, updates, and improvements to presentation

High Energy Physics - Theory (hep-th); Combinatorics (math.CO); Quantum Physics (quant-ph)

arXiv:2601.09987 [hep-th]

(or arXiv:2601.09987v2 [hep-th] for this version)