Towards a complete classification of holographic entropy inequalities

TL;DR

This work provides a deterministic, graph-theoretic framework to classify all holographic entropy inequalities (HEIs) by linking HEIs to contraction maps and partial cubes. By recasting the problem on hypercubes and employing a three-step algorithm (Partition generator, Graph contraction, Partial cube identifier), the authors show how to generate contraction maps and read off corresponding HEIs, with a formal claim of completeness within this contraction-map approach. A concrete example derives the monogamy of mutual information (MMI) from a star-graph contraction, illustrating the method’s power to generate known and potentially new HEIs. The paper also discusses implications for quantum entropy inequalities, computational complexity, and possible future directions to improve optimization, classify image-graph families, and connect these results to bulk geometry and holographic cones.

Abstract

We propose a deterministic method to find all holographic entropy inequalities that have corresponding contraction maps and argue the completeness of our method. We use a triality between holographic entropy inequalities, contraction maps and partial cubes. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the argued completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method. We also demonstrate interesting by-products, most notably, a procedure to generate candidate quantum entropy inequalities.

Figures (5)

• Figure 1: The diagram of the maps in the proof of proposition \ref{['pro:equivalence-bit-graph']}. The isometries $\iota_{M,N}$ and $\tilde{\iota}_{M,N}$ map between bitstrings $\{0,1\}^{M,N}$ and the set of vertices $V_{M,N}$ of hypercubes $H_{M,N}$. $f$ is a contraction map. $\phi^V$ is a vertex map from $H_M$ to $H_N$.
• Figure 2: Reading off a contraction map from the graph contraction mapping from $H_{M=3}$ to $H_{N=4}$. The blue dots are vertices. The black lines are the edges connecting the vertices. (a) Each vertex is labeled with $\{0,1\}^3$. The $\sigma$-th partition, for example, $\iota_M(V_{p_0^\sigma})=\{\iota_M(v_0)=000\}, \iota_M(V_{p_1^\sigma})=\{\iota_M(v_1)=001,\iota_M(v_2)=010,\iota_M(v_3)=100,\iota_M(v_7)=111\} ,\iota_M(V_{p_2^\sigma})=\{\iota_M(v_4)=011\},\iota_M(V_{p_3^\sigma})=\{\iota_M(v_5)=101\},\iota_M(V_{p_4^\sigma})=\{\iota_M(v_6)=110\}$ is chosen. The vertices labeled with the bitstrings in $\{001,010,100,111\}$ are enclosed by a rounded square. (b) After identifying the vertices based on the choice of the partition, there are three edges between every pair of the adjacent vertices $\chi_{p^\sigma_w}\in V_\sigma$ in the new graph $(V_\sigma, \mathbb{E}_\sigma)$. We obtain the graph $G_\sigma=(V_\sigma, E_\sigma)$ in (c) by removing two edges between every pair of the adjacent vertices. (c) The graph contraction with the choice of partition generates a star graph, a partial cube of isometric dimension $N=4$. Every vertex $\chi_{p^\sigma_w}\in V_\sigma$ gets labeled with a bitstring in a subset of $\{0,1\}^4$, e.g., $\{\iota_{N=4}(\chi_{p^\sigma_0}) = 0000,\iota_4(\chi_{p^\sigma_1}) =0001, \iota_4(\chi_{p^\sigma_2}) =0011,\iota_4(\chi_{p^\sigma_3}) =1001,\iota_4(\chi_{p^\sigma_4}) =0101\}$.
• Figure 3: The graphs associated with the first three members of cyclic inequalities.
• Figure 4: The star graphs corresponding to MMI (a) and SSA(b) with boundary conditions. One may start with (a), identify the vertices $A\leftrightarrow C$ and contract the edge with the central vertex of the star to get (b).
• Figure 5: The star graph constructed from $H_5$. All bitstrings with odd number of $1$s are identified with the center vertex. The corresponding contraction map is also labeled.

Theorems & Definitions (17)

• Definition 2.1: Hamming distance
• Theorem 2.1: 'Proof by contraction'
• Definition 2.2: A graph contraction map/weak graph homomorphism
• Definition 2.3: Partial cubeOvchinnikovWINKLER1984221
• Definition 2.4: Isometric dimension Ovchinnikov
• Theorem 2.2: 'Proof by graph contraction'
• Proposition 2.1: Equivalence between contraction maps and graph contraction maps
• proof
• Corollary 3.1
• Proposition 4.1: Lower bound on $N$