A graphical framework for proving holographic entanglement entropy inequalities in multipartite systems

TL;DR

The paper develops a graphical framework for proving holographic entanglement entropy inequalities (HEIs) in multipartite systems by introducing a simplex/I-basis formalism and four key theorems (Compatibility, Gapless, Cut, Configuration). It provides a joint-form strategy that reduces CCC-configuration proofs to cross-inequalities and extends results to non-CCC configurations via cuts, enabling systematic verification of HEIs up to 7 parties. Explicit examples for 3–7-partite systems are worked out, including new 5-, 6-, and 7-partite inequalities and their reductions. This work offers a structured pathway to classify and construct HEIs in holography and paves the way for higher-partite generalizations.

Abstract

We present a graphical method for proving holographic entanglement entropy inequalities (HEIs) in general multipartite systems. By introducing a geometric representation of the entanglement structure, we develop a systematic approach that enables one to visualize and verify the validity of HEIs for any number of subsystems $n$. Several theorems are established to formalize this method, and explicit examples are provided for systems with $n = 4$ to $7$ entangled regions.

Figures (11)

• Figure 1: The definition of simplex basis. The six curves are the RT surfaces connecting the ends of entangling regions $A_1$ and $A_2$.
• Figure 2: (a) The graphical proof of SSA. (b) Circular diagram of the SSA.
• Figure 3: Proof of the MMI through the "clean-gap procedure", shown step by step from (a) to (c) using cross inequalities.
• Figure 4: Proving eq.(\ref{['13-24']}) by clean-gap procedure.
• Figure 5: Proof of Compatible Theorem: The sum of the red lines is larger than the sum of the blue lines by the clean-gap procedure, which is in contradiction to eq.(\ref{['ip-qj']}).