On the completeness of contraction map proof method for holographic entropy inequalities
arXiv:2506.18086 · doi:10.1007/JHEP12(2025)140
The paper proves that for any linear holographic entropy inequality with rational coefficients, the existence of a contraction map is not only sufficient but also necessary, by showing that non‑contraction maps correspond to improper cubical subgraphs that violate the inequality in holographic geometries.
Abstract
The contraction map proof method is the commonly used method to prove holographic entropy inequalities. Existence of a contraction map corresponding to a holographic entropy inequality is a sufficient condition for its validity. But is it also necessary? In this note, we answer that question in affirmative for all linear holographic entropy inequalities with rational coefficients. We show that the pre-image of a non-contraction map is not a hypercube, but a proper cubical subgraph, and show that this manifests as alterations to the geodesic structure in the bulk, which leads to the violation of inequalities by holographic geometries obeying the RT formula.
v1 25 pages, v2 32 pages (journal version)