# Integrated Information in Relational Quantum Dynamics (RQD)
## 4 Optimal Factorizations and Convex Geometry of \(\Phi\)
One can also argue from the negative-definite metric viewpoint that the distance-squared function \(\sigma\mapsto\delta^{2}(\rho,\sigma)\) is strictly convex on a geodesically convex domain of product states. Thus, the minimizer on each partition is unique. If there were multiple bipartitions achieving the same minimum value, one of their corresponding product states would still be closer in \(\delta\)-distance than any other state, so we may define \(\sigma^{*}(\rho)\) to be one of them. In generic cases, the optimal partition \((A^{*}|B^{*})\) is unique as well (ties are non-generic and can be broken arbitrarily). We emphasize the important consequence: the closest product-state approximation to \(\rho\) is unique. We denote this distinguished state by \(\sigma^{*}(\rho)=\rho_{A^{*}}\otimes\rho_{B^{*}}\), where \((A^{*}|B^{*})\) is an optimal bipartition. We may call \((A^{*}|B^{*})\) the minimum information partition (MIP) of \(\rho\), borrowing terminology from III. There is no degeneracy in identifying the \"best split\" of the system--a fact that contrasts with earlier approaches where one often had to choose among candidate partitions or deal with ties. Uniqueness comes from the convexity of the divergence and can be seen as a benefit of using \(D^{Q}_{\mathrm{JS}}\) as opposed to mutual information, for example, which might not identify a unique optimal split in some cases. With \(\sigma^{*}(\rho)\) in hand, we can reinterpret the integrated information as a **distance** from \(\rho\) to the set of disjoint-state products. In fact, Theorem 3 implies:  
\[\Phi(\rho)\ =\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\left\|\,\sigma^{*}(\rho) \right)\ =\ \delta^{2}\!\left(\rho,\,\sigma^{*}(\rho)\right)\,, \tag{12}\]  
since \(\delta(\rho,\sigma)=\sqrt{D^{Q}_{\mathrm{JS}}(\rho\|\sigma)}\). In words, \(\Phi(\rho)\) is the squared distance (in the \(\delta\) metric) from \(\rho\) to the **closed, convex set**  
\[\mathcal{P}_{2}=\{\rho_{A}\otimes\rho_{B}:A|B\;\text{a bipartition}\}\]  
of all bipartite product states. We can therefore leverage geometric intuition: computing \(\Phi(\rho)\) is performing a metric projection of the point \(\rho\) onto the set \(\mathcal{P}_{2}\). By standard results in convex geometry and metric spaces, this immediately yields a raft of powerful corollaries:  
* **Convexity of \(\Phi\). ** As an infimum (indeed minimum) of convex functions \(\sigma\mapsto D^{Q}_{\mathrm{JS}}(\rho\|\sigma)\), the distance-to-set function \[\rho\mapsto\inf_{\sigma\in\mathcal{P}_{2}}D^{Q}_{\mathrm{JS}}(\rho\|\sigma)\] (13) is convex on state space. Equivalently, for any two states \(\rho_{1},\rho_{2}\) and any \(0\leq t\leq 1\), \[\Phi\!\left(t\,\rho_{1}+(1-t)\,\rho_{2}\right)\ \leq\ t\,\Phi(\rho_{1})+(1-t)\,\Phi(\rho_{2})\,.\] (14) This **convexity** means that mixing states cannot increase the integrated information beyond the mixture of their individual \(\Phi\) values. In practical terms, a noisy or probabilistic mixture of two configurations will tend to have less (or equal) holistic structure than a pure configuration, which aligns with intuition. * **Lipschitz Continuity (Robustness). ** The function \(\Phi(\rho)\) is 1-Lipschitz continuous with respect to the metric \(\delta\). That is, \[|\Phi(\rho)-\Phi(\rho^{\prime})|\ \leq\ \delta(\rho,\rho^{\prime})\] (15) for all states \(\rho,\rho^{\prime}\). Small changes or errors in the state can only cause small (at most proportional) changes in the integrated information. This follows from a general fact: in any metric space, the distance from a point to a convex set is a 1-Lipschitz function (intuitively, if \(\rho\) and \(\rho^{\prime}\) are close, their nearest projections on a convex set cannot be very different in distance).