# Integrated Information in Relational Quantum Dynamics (RQD)
## 4 Optimal Factorizations and Convex Geometry of \(\Phi\)  
We now turn to analyzing the minimization in  
\[\Phi(\rho)=\min_{P_{i}}D^{Q}_{\mathrm{JS}}(\rho\|\bigotimes_{i}\rho_{P_{i}}). \tag{9}\]  
Two fundamental questions arise: (a) Which partition achieves the minimum? (b) Is the minimum achieved by a unique state? We will see that thanks to the metric structure and convexity properties of \(D^{Q}_{\mathrm{JS}}\), the answer is remarkably neat: it is always a bipartition (a split into \(k=2\) blocks) that attains the minimum, and moreover the closest product state is unique. This resolves any ambiguity in the notion of a \"minimum information partition\" and provides a well-defined integrated whole versus parts for the system. **Theorem 3** (Bipartition Sufficiency and Existence of Optimal Product State). : _For any state \(\rho\) on \(n\) subsystems, the minimum in Definition 1 is attained on some bipartition \((A|B)\) of the system. In other words,_  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]  
_where the minimum is over all splits of \(1,\ldots,n\) into two disjoint groups \(A\) and \(B\). Furthermore, there exists at least one product state \(\rho_{A}\otimes\rho_{B}\) that achieves this minimum, and for each minimizing bipartition, the optimal product state on that cut is unique. In particular, there is a unique closest product state \(\sigma^{*}(\rho)=\rho_{A^{*}}\otimes\rho_{B^{*}}\), where \((A^{*}|B^{*})\) is the optimal partition._  
Proof (Sketch). : The fact that one can restrict to \(k=2\) blocks without loss of generality follows from a simple inequality: for any partition with \(k>2\) blocks, one can show that merging any two blocks cannot increase the divergence. Intuitively, splitting into more than two parts introduces additional independent components, which can only make it harder for a single product state to approximate \(\rho\). More formally, consider three disjoint subsets \(X,Y,Z\) of subsystems; one can show  
\[D^{Q}_{\mathrm{JS}}\!\!\left(\rho\middle\|\rho_{X}\otimes\rho_{Y}\otimes\rho _{Z}\right)\geq D^{Q}_{\mathrm{JS}}\!\!\left(\rho\middle\|\rho_{XY}\otimes \rho_{Z}\right), \tag{11}\]  
because \(\rho_{XY}\otimes\rho_{Z}\) (where \(\rho_{XY}=\mathrm{Tr}_{Z}\rho\)) allows correlations between \(X\) and \(Y\) that the fully factorized version \(\rho_{X}\otimes\rho_{Y}\otimes\rho_{Z}\) forbids, and thus it is closer to \(\rho\) (lower divergence). Iterating this argument, any partition with \(k>2\) can be coarse-grained into a bipartition that yields an equal or smaller \(D^{Q}_{\mathrm{JS}}\). Hence the minimum occurs at some \(A|B\). Next, because for a fixed bipartition \((A|B)\) the set of product states \(\rho_{A}\otimes\rho_{B}\) is a compact convex subset of state space and \(D^{Q}_{\mathrm{JS}}\!\!\left(\rho\middle\|\sigma\right)\) is a continuous function of \(\sigma\), the infimum over \(\sigma=\rho_{A}\otimes\rho_{B}\) is actually a minimum (attained at some \(\sigma\)). This uses the compactness of the set of marginal states \(\rho_{A}\) and \(\rho_{B}\); any minimizing sequence has a convergent subsequence in the product space by compactness, and by continuity, the limit is a minimizer. Finally, for a given bipartition, the uniqueness of the minimizing \(\rho_{A}\otimes\rho_{B}\) follows from the strict convexity of \(D^{Q}_{\mathrm{JS}}\) in its second argument. Quantum JSD is jointly convex [13] and in particular, if two distinct product states \(\sigma_{1}=\rho_{A}^{1}\otimes\rho_{B}^{1}\) and \(\sigma_{2}=\rho_{A}^{2}\otimes\rho_{B}^{2}\) both yield the same divergence  
\[D^{Q}_{\mathrm{JS}}\!\!\left(\rho\middle\|\sigma_{1}\right)=D^{Q}_{\mathrm{JS }}\!\!\left(\rho\middle\|\sigma_{2}\right)=m,\]  
then any mixture \(\tilde{\sigma}=t,\sigma_{1}+(1-t),\sigma_{2}\) (which is still a valid separable state on \(AB\)) would produce \(D^{Q}_{\mathrm{JS}}\!\!\left(\rho\middle\|\tilde{\sigma}\right)<m\) for \(0<t<1\), contradicting the minimality.