On the completeness of contraction map proof method for holographic entropy inequalities

Technical Summary: On the Completeness of Contraction Map Proof Method for Holographic Entropy Inequalities

Problem Statement
The holographic entropy cone (HEC) characterizes the constraints on entanglement entropies for quantum states admitting a semi-classical bulk dual (holographic states) via the Ryu-Takayanagi (RT) formula. The standard method for proving that a candidate linear inequality is a valid Holographic Entropy Inequality (HEI) is the "contraction map proof method." It is established that the existence of a contraction map satisfying specific homology conditions is a sufficient condition for an inequality's validity. However, a critical open question remained: Is this condition also necessary? Specifically, does the validity of an HEI imply the existence of a corresponding contraction map? While all known facet HEIs admit such maps, a formal proof of this completeness was previously elusive.

Methodology
The authors employ a rigorous framework combining graph theory, information theory, and holographic geometry to address the necessity of contraction maps. The methodology proceeds through the following logical steps:

1. Geometry-Graph Duality: The paper utilizes the established duality between holographic geometries and partial cubes. An RT arrangement in a bulk manifold partitions the space into subregions, which can be represented as a graph where vertices are bulk subregions and edges represent adjacency across RT surfaces. This graph is a partial cube, isometrically embeddable into a hypercube $H_J$.
2. Bitstring Representation: Vertices in these partial cubes are labeled by bitstrings representing the inclusion/exclusion of regions relative to RT surfaces. The graph distance $d_G$ corresponds to the Hamming distance $d_H$ of these bitstrings.
3. Contraction vs. Non-Contraction Maps: A contraction map $f: \{0,1\}^M \to \{0,1\}^N$ is defined as a map that does not increase Hamming distance ($d_H(x, x') \geq d_H(f(x), f(x'))$). The authors analyze the properties of non-contraction maps (where distance increases for some pairs).
4. Graph-Theoretic Analysis of Non-Contractions: The authors investigate the preimage of the image graph under a non-contraction map. They demonstrate that if a map is not a contraction, the preimage of the image graph (excluding the kernel) is a "proper cubical subgraph" of the original hypercube, rather than the full hypercube. Crucially, this subgraph fails to satisfy the adjacency condition ($d_H^{adj} = d_G$), meaning that vertices with bitstrings differing by a single bit (Hamming distance 1) are not connected by an edge in the induced graph.
5. Geometric Interpretation of Failure: The failure of the adjacency condition is interpreted physically as a breakdown in the smoothness of the bulk geometry. Specifically, the absence of an edge between adjacent bitstrings implies the absence of a geodesic connecting the corresponding bulk subregions. The authors introduce "virtual edges" with infinite weight to represent these missing connections.
6. Proof by Contradiction: The core argument assumes the existence of a valid HEI that does not possess a contraction map. The authors construct a specific holographic geometry (via the induced subgraph with virtual edges) where the "missing" geodesics result in infinite entanglement entropy contributions. This construction leads to a violation of the candidate inequality, thereby proving that the inequality cannot be valid for all holographic geometries.

Key Contributions and Results

• Main Theorem (Theorem 4.1): The paper proves that for all linear holographic entropy inequalities with rational coefficients, a candidate inequality is a valid HEI if and only if a contraction map exists satisfying the boundary homology conditions. This establishes the completeness of the contraction map proof method.
• Characterization of Non-Contractions: The authors show that non-contraction maps are intrinsically linked to alterations in the geodesic structure of the bulk. A non-contraction map induces a geometry where the smooth manifold structure is broken (specifically, the RT surfaces cannot be smoothly deformed from the Left-Hand Side configuration to the Right-Hand Side configuration).
• Necessity of Adjacency: The proof highlights that the image graph being a partial cube is a necessary but insufficient condition for validity. The map must also preserve the adjacency condition ($d_H^{adj} = d_G$). The failure to preserve adjacency is the precise mechanism by which non-contraction maps fail to represent valid holographic geometries.
• Examples: The paper provides concrete examples (including a deformed Monogamy of Mutual Information and a five-party example) where false inequalities lead to non-contraction maps. In these cases, the associated graph mappings fail to preserve adjacency, and the authors explicitly construct geometries that violate the proposed inequalities.

Significance and Claims
The paper claims to resolve a fundamental question regarding the sufficiency and necessity of the contraction map method in the study of holographic entropy inequalities. By proving that the existence of a contraction map is a necessary condition for the validity of linear HEIs with rational coefficients, the authors:

• Validate the algorithmic approach proposed in previous work (e.g., [15]) for generating all HEIs, confirming that no valid HEI is missed by this method.
• Clarify the physical interpretation of the mathematical constraints: the rigidity of contraction maps is not merely a combinatorial artifact but a reflection of the underlying smoothness and geodesic structure of semi-classical bulk geometries.
• Establish that any inequality failing the contraction map test is guaranteed to be violated by some holographic geometry, effectively ruling out the existence of "valid but unprovable" linear HEIs within the rational coefficient framework.

The authors note that their proof relies on the existence of a graph model corresponding to smooth bulk geometries (via RT surfaces) and is restricted to linear inequalities with rational coefficients. They do not rule out the possibility of non-linear inequalities or cases where the bulk dual is not semi-classical, but within the defined scope, the contraction map method is proven to be complete.