# Integrated Information in Relational Quantum Dynamics (RQD)
## 3 Monotonicity and Metric Properties of \(D_{\text{JS}}^{Q}\)  
We first establish two fundamental properties of the quantum JSD that underpin the integrated information measure: a data-processing inequality and a metric structure. These ensure that \(\Phi\) behaves sensibly under observers and has a well-defined geometry. **Theorem 1** (**Data-Processing Monotonicity**). : _For any two states \(\rho,\sigma\) and any CPTP map (quantum channel) \(\mathcal{E}\), the quantum Jensen-Shannon divergence cannot increase under processing:_  
\[D_{\text{JS}}^{Q}(\mathcal{E}(\rho)\,\|\,\mathcal{E}(\sigma))\ \leq\ D_{\text{JS}}^{Q}(\rho\|\sigma)\,. \tag{4}\]  
_In particular, if an \"observer\" interacts with or measures the system (modeled by \(\mathcal{E}\)), the integrated information of the post-interaction state cannot exceed that of the pre-interaction state \(\Phi(\mathcal{E}(\rho))\leq\Phi(\rho)\)._  
Proof (**Sketch**). : This is the quantum analog of the classical data-processing inequality, and it holds because \(D_{\text{JS}}^{Q}\) belongs to the class of contractive divergences. Indeed, the JSD can be expressed in terms of the quantum relative entropy \(D(\rho\|\sigma)=\text{Tr}[\rho(\log\rho-\log\sigma)]\) as  
\[D_{\text{JS}}^{Q}(\rho\|\sigma)=\underset{\pi}{\text{min}}\Big{(}\tfrac{1}{2} \,D(\rho\|\pi)\ +\ \tfrac{1}{2}\,D(\sigma\|\pi)\Big{)} \tag{5}\]  
(the minimal relative entropy to an intermediary state)--an expression which inherits the monotonicity of \(D\) under CPTP maps [14,15]. Alternatively, one can invoke the joint convexity of \(D_{\text{JS}}^{Q}\) and the complete positivity of \(\mathcal{E}\) to show  
\[\mathcal{E}(\frac{\rho+\sigma}{2})=\frac{\mathcal{E}(\rho)+\mathcal{E}(\sigma)} {2} \tag{6}\]  
and apply monotonicity of von Neumann entropy under partial trace. The inequality \(\Phi(\mathcal{E}(\rho))\leq\Phi(\rho)\) then follows immediately: for any partition that attains (or approximates) the minimum for \(\rho\), applying \(\mathcal{E}\) to both \(\rho\) and each product component cannot increase the divergence, so the minimal divergence for \(\mathcal{E}(\rho)\) is bounded by that of \(\rho\). This theorem formalizes the intuitive idea that observation cannot create integration. Any act of coarse-graining, measurement, or decoherence will tend to lose correlations or entanglement, never to introduce new irreducible correlations that were not already present. In category-theoretic terms, one can frame this as an enriched functor property: each physical process or observation defines a mapping between state spaces that does not expand distances. More precisely, we can view each system's state space as a metric space \((\mathcal{S},\delta)\) with \(\delta(\rho,\sigma)=\sqrt{D_{\text{JS}}^{Q}(\rho\|\sigma)}\) (as introduced below). Then any CPTP map \(F:\mathcal{S}_{1}\rightarrow\mathcal{S}_{2}\) is a \([0,\infty]\)-enriched functor between these metric spaces, meaning \(\delta_{2}(F(\rho),F(\sigma))\leq\delta_{1}(\rho,\sigma)\) for all states (Lawvere metric space enrichment). This functorial perspective identifies an \"observer\" or dynamics with a structure-preserving map in an information metric space. By Theorem 1, all such observers are non-expansive in the \(\delta\) metric, and hence cannot increase \(\Phi\). This categorical formulation helps unify the notion of observers across classical and quantum domains, but for the remainder of this paper, we will generally work in standard information-theoretic terms. Practical Implications of Monotonicity: Theorem 1 implies a powerful experimental guarantee. In most tomographic protocols, one acquires raw measurement frequenciesand then applies a maximum-likelihood (ML) estimator to project onto the physical state space. That ML step is itself a CPTP map, so by monotonicity, the post-processed \(\Phi\) can only be smaller than the true \(\Phi\) of the prepared state. In other words, any non-zero \(\Phi\) we report is a conservative lower bound on the system's intrinsic holism.