Nesting is not Contracting

TL;DR

This paper examines how the contraction method for proving holographic entropy inequalities is constrained by entanglement wedge nesting (EWN), revealing that contraction proofs often violate EWN and exploring implications for tight inequalities.

Contribution

It demonstrates that contraction proofs necessarily involve candidate RT surfaces violating EWN and analyzes the structure of these violations in the context of holographic entropy inequalities.

Findings

Contraction proofs often violate EWN near boundary conditions.

All aspects of contraction maps are dictated by EWN constraints.

Non-uniqueness of proofs reflects different EWN violation schemes.

Abstract

The default way of proving holographic entropy inequalities is the contraction method. It divides Ryu-Takayanagi (RT) surfaces on the `greater than' side of the inequality into segments, then glues the segments into candidate RT surfaces for terms on the `less than' side. Here we discuss how proofs by contraction are constrained and informed by entanglement wedge nesting (EWN) -- the property that enlarging a boundary region can only enlarge its entanglement wedge. We propose that: (i) all proofs by contraction necessarily involve candidate RT surfaces, which violate EWN; (ii) violations of EWN in contraction proofs of maximally tight inequalities occur commonly and -- where this can be quantified -- with maximal density near boundary conditions; (iii) the non-uniqueness of proofs by contraction reflects inequivalent ways of violating EWN. As evidence and illustration, we study the recently discovered infinite families of holographic entropy inequalities, which are associated with tessellations of the torus and the projective plane. We explain the logic, which underlies their proofs by contraction. We find that all salient aspects of the requisite contraction maps are dictated by EWN while all their variable aspects set the scheme for how to violate EWN. We comment on whether the tension between EWN and contraction maps might help in characterizing maximally tight holographic entropy inequalities.