Integrated Information in Relational Quantum Dynamics (RQD)

Abstract: We introduce a quantum integrated-information measure $\Phi$ for multipartite states within the Relational Quantum Dynamics (RQD) framework. $\Phi(\rho)$ is defined as the minimum quantum Jensen-Shannon distance between an n-partite density operator $\rho$ 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 $\Phi$ 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 $\Phi$ and construct an integration dendrogram that reveals the full hierarchical correlation structure of $\rho$. We further show that there always exists an "optimal observer"-a channel or basis-that preserves $\Phi$ 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.