Integrated Information in Relational Quantum Dynamics (RQD)

We introduce a quantum integrated-information measure Φ for multipartite states within the Relational Quantum Dynamics (RQD) framework. Φ(ρ) is defined as the minimum quantum Jensen–Shannon distance between an n-partite density operator ρ and any product state over a bipartition of its subsystems. We prove that its square root induces a genuine metric on state space and that Φ is monotonic under all completely positive trace-preserving maps. Restricting the search to bipartitions yields a unique optimal split and a unique closest product state. From this geometric picture we derive a canonical entanglement witness directly tied to Φ and construct an integration dendrogram that reveals the full hierarchical correlation structure of ρ. We further show that there always exists an “optimal observer”—a channel or basis—that preserves Φ better than any alternative. Finally, we propose a quantum Markov blanket theorem: the boundary of the optimal bipartition isolates subsystems most effectively. Our framework unites categorical enrichment, convex-geometric methods, and operational tools, forging a concrete bridge between integrated information theory and quantum information science.