Monogamy of Mutual Information in Graph States

TL;DR

The paper tackles the problem of when the monogamy of mutual information (MMI) fails in graph and stabilizer states by translating MMI into algebraic constraints on graph adjacency and tableau subspaces. It develops a framework that connects MMI violations to structural graph motifs, proving a forbidden-subgraph conjecture for a concrete family of star-like graphs and extending the analysis to generalized star topologies through column-space/distributivity criteria. An explicit, exhaustive study up to $n=8$ qubits shows star-graph structures capture many, but not all, MMI-violating instances, suggesting a broader landscape of violations beyond the initial scope. The work deepens our understanding of entanglement distribution in stabilizer and graph states and has potential implications for quantum networks and holographic entropy inequalities, while outlining clear directions for extending these insights to higher qubit counts and more general settings.

Abstract

The monogamy of mutual information (MMI) is a quantum entropy inequality that enforces the non-positivity of tripartite information. We investigate the failure of MMI in graph states as a forbidden-subgraph phenomenon, conjecturing that every MMI-violating graph state is local-Clifford equivalent to one whose graph contains a four-star subgraph. We construct a family of star-like graphs whose states fail a specific class of MMI instances, and extend this analysis to general star topologies. Deriving adjacency matrix constraints that fix the MMI evaluation for these instances and interpreting them physically, we prove the forbidden-subgraph conjecture for this family of graphs. Finally, through an exhaustive search over graph representatives for all $8$-qubit stabilizer entropy vectors, we establish that MMI failure is not reducible to the cases within our scope.

Figures (13)

• Figure 1: Left: star graph on five vertices $K_5$. Right: the result of locally complementing $K_5$ on vertex $1$, a complete graph on five vertices. The neighborhood of vertex $1$, $N(1)$, is shaded in both cases. Notice that locally complementing vertex $1$ of the complete graph recovers $K_5$.
• Figure 2: Graph Representation for ${|{K_4}\rangle}$, a state $LC$ equivalent to ${|{GHZ}\rangle}_4$, which violates all $3$-party instances of MMI.
• Figure 3: Graph Representation for ${|{K^{1,1,2}_{4}}\rangle} = CZ_{4,5}\left( {|{K_4}\rangle}\otimes{|{+}\rangle}\right)$, with a four-star $K_4$ induced by vertices $\{1,2,3,4\}$. While ${|{K^{1,1,2}_{4}}\rangle}$ is not a $GHZ$ state, it nevertheless fails $MMI_{\{ 1,3,45 \}}$ due to its star topology.
• Figure 4: Local information submatrix $\Gamma^{\mathcal{L}}$, for $\mathcal{L}=\{ 1,3,4,5\}$ in ${|{K^{1,1,2}_{4}}\rangle}$, given in block matrix form. The state ${|{K^{1,1,2}_{4}}\rangle}$ is partitioned such that $C\sqcup I\sqcup J\sqcup K = \{ 1,2,3,4,5\}$, and shared information elements are colored. Notice that $S_A$ for $A\subseteq \mathcal{L}$ is independent of the self adjacency blocks $CC$, $II$, and $JJ$, which are shaded gray.
• Figure 5: Generalized star graph $G$ which satisfies the conditions of subgraph disjointness, graph partitioning, subgraph anchoring, and inter-subgraph disconnectedness. The subsystems $\{c\},\ I,\ $$J$, and $K$ give a valid partition that defines $MMI_{cIJ}$, as in Eq. \ref{['eq:MMI-Fail-G_m']}, where any path between the subsystems $I$, $J$ and $K$ must include the central vertex $c$. Any graph state represented by a graph of type $G$ will have an entropy vector that violates $MMI_{cIJ}$.

Theorems & Definitions (7)

• Proposition 1
• proof
• Conjecture 1: Forbidden-Subgraph
• Proposition 2
• proof
• Proposition 3
• proof