Integrated Information in Relational Quantum Dynamics (RQD)

TL;DR

The paper defines a quantum integrated-information measure $\Phi(\rho)$ as the minimum quantum Jensen–Shannon distance to the nearest product state across all partitions, establishing that $\sqrt{D^{Q}_{JS}}$ is a metric and that $\Phi$ is monotone under CPTP maps. It proves that a unique bipartition suffices to achieve the minimum, yielding a unique closest product state $\sigma^*(\rho)$ and enabling a geometric projection viewpoint with a convex-set interpretation. This leads to a suite of results: a canonical entanglement witness $W_\rho=\sigma^*(\rho)-\rho$, an integration dendrogram that hierarchically decomposes correlations, and a Max-$\Phi$ observer principle identifying the optimal way to preserve holism under observation. The work also introduces a quantum Markov-blanket concept via the optimal partition boundary and discusses broader implications for IIT, decoherence, and quantum causality, while outlining computational and interpretational caveats and future directions.

Abstract

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.