bibkey: fuentes2025mmigraphstates
authors: Jesus Fuentes, Cynthia Keeler, William Munizzi, Jason Pollack
year: 2025
title: "Monogamy of Mutual Information in Graph States"
doi: 10.48550/arXiv.2511.19585
url: https://arxiv.org/abs/2511.19585v1
claim: "The paper reads MMI violation in graph states as a forbidden-subgraph phenomenon and conjectures (Conjecture 1, Forbidden-Subgraph) that any graph state that violates MMI has a graph representation that is LC-equivalent to a graph H containing a K_4 subgraph, the four-vertex star, with H a generalized star graph with respect to some partition; it verifies the conjecture exhaustively up to 8 qubits."
strata_touched:
  - D5/S3/Quantum/Entanglement/GraphStateMonogamyForbiddenSubgraph
license: citation-only
triage: anchor

Monogamy of Mutual Information in Graph States

The paper computes the entanglement entropy of a graph state from the adjacency matrix of its graph (eq. EEadj):

For a graph $G$ with vertex set $V=\{n\}$ and adjacency matrix ${\Gamma }_G$, let $\{A,\overline{A}\}$ be any bipartition of $V$. The entanglement entropy of $A$, in $|G⟩$, is calculated from ${\Gamma }_G$ as $S_A(|G⟩)={\text{rank}}_{{\mathbb{Z}}_2}({\Gamma }_G^{A\overline{A}})$.

and states MMI (eq. MMI) as

For an $n$-party system with pairwise-disjoint subsystems $I,J,$ and $K$, MMI requires $S_{IJ}+S_{IK}+S_{JK}\ge S_I+S_J+S_K+S_{IJK}$.

Local Clifford operations act on graph states as local complementations: “we use $LC$ to refer interchangeably to local Clifford equivalence and local complementation equivalence in graph states”. The symbol $K_4$ denotes the four-vertex star: “the graph state $|K_4⟩$, representable by the $4$-vertex star graph $K_4$”.

A generalized star graph $\mathcal{G}$ has a central subgraph $C$ connected to $k$ non-empty subgraphs ${\mathcal{G}}^1,\dots ,{\mathcal{G}}^k$ that are vertex-disjoint and mutually disconnected, with $\{C,V^1,\dots ,V^k\}$ a partition of $V$. After eq. K4Inclusion the paper notes: “the existence of a $K_4$ subgraph guarantees that there exists a partition under which the graph is (nontrivially) of type $\mathcal{G}$, since we can place the central vertex of $K_4$ in $C$, place each leaf of $K_4$ in $I,J,$ and $K$, respectively, and place all remaining vertices of the graph in $C$.”

The conjecture reads

Conjecture 1 (Forbidden-Subgraph). Any graph state that violates MMI has a graph representation that is $LC$-equivalent to a graph $H$ containing a $K_4$ subgraph. $H$ is $\mathcal{G}$ with respect to some partition.

and the discussion describes it as positing “the necessity of an induced four-star in the LC orbit of any graph whose associated graph state violates MMI”. The paper reports “We exhaustively verify this conjecture up through $n=8$ qubits”.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2511.19585
• URL: https://arxiv.org/abs/2511.19585v1
• Version and location: arXiv:2511.19585v1 (2025-11-24), the only version listed by the arXiv API on 2026-09-27; source file paper.tex, eq. MMI (line 405), the LC paragraph (line 546), eq. EEadj (lines 590–600), the generalized star definition (lines 1452–1466), eq. K4Inclusion and the partition remark (lines 1974–1980), Conjecture 1 (lines 2400–2403) and the discussion (line 2435).