# Integrated Information in Relational Quantum Dynamics (RQD)
## 1 Introduction
Then we define  
\[\Phi(\rho)\ =\ \min_{\mathrm{partitions}\ \{P_{i}\}}\ D_{\mathrm{JS}}^{ \mathrm{Q}}\!\!\left(\rho\ \middle\|\ \bigotimes_{i}\rho_{P_{i}}\right). \tag{1}\]  
In other words, \(\Phi(\rho)\) quantifies the closest that \(\rho\) can be (in terms of quantum Jensen-Shannon divergence) to an uncorrelated state. If \(\rho\) is itself a product state under some partition, then \(\Phi(\rho)=0\) (no integrated information). If \(\rho\) is highly entangled or correlated across all subsystems, \(\Phi(\rho)\) will be large, indicating strong irreducible correlations. This construction extends to the quantum domain that the integrated information measures use in classical IIT [10;11], with \(D_{\mathrm{JS}}^{\mathrm{Q}}\) replacing classical divergence or mutual information measures. Importantly, as we will show, \(D_{\mathrm{JS}}^{\mathrm{Q}}\) enjoys several properties (data-processing inequalities, convexity, a true metric structure) that make \(\Phi(\rho)\) analytically tractable and well-behaved even for high-dimensional or rank-deficient states. In developing our framework, we prove four main results about \(\Phi\) and \(D_{\mathrm{JS}}^{\mathrm{Q}}\) as outlined above: (1) \(\Phi\) is monotonic under processing (no observer can increase it), (2) the square-root of \(D_{\mathrm{JS}}^{\mathrm{Q}}\) is a metric on state space, (3) it suffices to consider bipartitions to attain the minimum in \(\Phi(\rho)\), and (4) the observer can be formalized as a metric-space functor that is non-expansive. These results lay the groundwork for analyzing integrated information in quantum systems. After presenting these, we introduce a set of novel convex-geometric and operational insights that arise from viewing \(\Phi(\rho)\) as a distance to the convex set of product states. This geometric perspective reveals that every state has a unique nearest product state (defining a canonical minimum information partition for \(\rho\)), and it unlocks an array of corollaries including convexity and robustness of \(\Phi\), a natural gradient-flow that \"dis-integrates\" a state, and even a direct construction of an entanglement witness from \(\Phi\). Furthermore, we leverage the unique optimal partition to build a hierarchical decomposition of any multipartite state into successively independent components, an integration dendrogram that maps out the structure of correlations in the state. Finally, we explore two broader implications: an observer selection principle stating that one measurement basis or channel maximally preserves a system's integrated information (suggesting a first-principles derivation of preferred pointer bases in decoherence theory), and a prospective quantum Markov blanket theorem, which identifies, for any subsystem, the minimal \"boundary\" that most effectively screens it off from the rest of the system. To our knowledge, this is the first fully metric-geometric construction of a quantum-IIT measure that (i) is provably a Lawvere metric enrichment of CPM, (ii) collapses to bipartitions by convexity, (iii) yields a canonical entanglement witness, and (iv) admits a functorial observer-selection principle.