# Integrated Information in Relational Quantum Dynamics (RQD)
## 4 Optimal Factorizations and Convex Geometry of \(\Phi\)
** From Lemma 1 and  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]2  
we get in \(\mathcal{H}\)  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]3Pulling this back via the Frechet derivative  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]4  
the stationarity condition \(\
abla_{\phi(\sigma^{*})}\delta^{2}(\rho,\sigma^{*})=0\) yields  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]5  
so that (up to a positive scalar) \(W_{\rho}=\sigma^{*}-\rho\) is precisely the normal operator defining the separating hyperplane in the original operator space. **Proposition 1** (Canonical Entanglement Witness). : _Let \(\sigma^{*}(\rho)=\rho_{A^{*}}\otimes\rho_{B^{*}}\) be the closest product state to \(\rho\), achieved on the optimal bipartition \((A^{*}|B^{*})\). Define an observable (Hermitian operator) \(W_{\rho}:=\sigma^{*}(\rho)-\rho\,.\) Then \(W_{\rho}\) is an **entanglement witness** for the bipartition \(A^{*}|B^{*}\). In particular, \(W_{\rho}\) has a non-negative expectation value on all product states factorized across \(A^{*}\) and \(B^{*}\), yet has a strictly negative expectation on \(\rho\) itself. Moreover, the magnitude of this violation is exactly the integrated information:_  
\[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]6  
_In fact, \(W_{\rho}\) is the optimal (most \"detecting\") witness for the entanglement between \(A^{*}\) and \(B^{*}\)._  
Proof. : By Lemma 1 and the metric-gradient calculation, the minimizing condition \(\
abla_{\phi(\sigma^{*})}\delta^{2}(\rho,\sigma^{*})=0\) pulls back to \(\log\tfrac{\rho+\sigma^{*}}{2}-\log\rho=0\), and hence \(W_{\rho}=\sigma^{*}-\rho\) is the geodesic-normal. Positivity on any product state \(\pi=\pi_{A}\otimes\pi_{B}\) follows because \(\sigma^{*}\) is the closest product state. The fully detailed proof of this proposition is presented in Appendix C.  
In summary, \(W_{\rho}=\sigma^{*}(\rho)-\rho\) is a constructive, canonical witness to the entanglement or correlation that makes \(\rho\) irreducible across its optimal split. It detects exactly the entanglement corresponding to \(\Phi(\rho)\) and no more. In particular, \(\operatorname{Tr}[W_{\rho},\rho]=-\Phi(\rho)\) quantifies the \"violation\" of \(\rho\) being separable: the more integrated \(\rho\) is, the more \(W_{\rho}\) yields a negative expectation on it, with the gap equal to \(\Phi(\rho)\). One could say this provides an operational meaning to \(\Phi\): it is the magnitude by which \(\rho\) fails the best possible separability test. This is a novel connection between integrated information and entanglement theory -- it bridges a geometrical measure of holistic correlation (\(\Phi\)) with the traditional notion of an entanglement witness from quantum information. In practice, once one computes \(\sigma^{*}(\rho)\) (by any method), one immediately obtains \(W_{\rho}\), which is an observable whose expectation value on \(\rho\) is \(-\Phi(\rho)\) and on any fully factorized state (on that cut) is non-negative. Experimentally, measuring the set of local observables that constitute \(W_{\rho}\), which will generally be a difference of two reduced density operators, could verify the presence of entanglement and even quantify it in terms of integrated information.