# Integrated Information in Relational Quantum Dynamics (RQD)
## 4 Optimal Factorizations and Convex Geometry of \(\Phi\)
More concretely, one can use the triangle inequality \[\Phi(\rho)=\delta(\rho,\sigma^{*}(\rho))\leq\delta(\rho,\rho^{\prime})+\delta (\rho^{\prime},\sigma^{*}(\rho)).\] (16)But \(\delta(\rho^{\prime},\sigma^{*}(\rho))\geq\Phi(\rho^{\prime})\) because \(\sigma^{*}(\rho)\) might not be the optimal product for \(\rho^{\prime}\), so \[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]0 (17) Thus, \(\Phi(\rho)-\Phi(\rho^{\prime})\leq\delta(\rho,\rho^{\prime})\). Swapping \(\rho\leftrightarrow\rho^{\prime}\) gives the two-sided Lipschitz bound. This robustness is crucial for empirical or experimental scenarios: it means \(\Phi\) will not fluctuate wildly due to small perturbations or noise in the state, making it a reliable quantity to estimate. * **Gradient Flow towards Dis-integration. ** By viewing \(\Phi(\rho)=\delta^{2}(\rho,\mathcal{P}*2)\) as a squared-distance function in a (formal) Riemannian space, one can define a gradient descent dynamical system that flows \(\rho\) toward its nearest product state. Concretely, we can write a continuous time equation \[\Phi(\rho)\ =\ \min_{A|B}\ D^{Q}_{\mathrm{JS}}\!\!\left(\rho\,\middle\|\,\rho_{A} \otimes\rho_{B}\right), \tag{10}\]1 (18) which generates a trajectory \(\rho(t)\) that decreases \(\Phi\) monotonically and converges to the projection \(\sigma^{*}(\rho)\) as \(t\to\infty\). While here \(\
abla*\rho\) denotes a gradient with respect to the information geometry induced by \(D^{Q}_{\mathrm{JS}}\), one can intuitively think of this as the state \"pulling itself apart\" into independent pieces. In practice, this gradient-flow could be implemented by a family of CPTP maps that progressively erode the holistic correlations. The advantage of framing it as a gradient flow is that it provides a principled algorithm for finding \(\sigma^{*}(\rho)\) (the best factorized approximation) by following the steepest descent of integrated information, rather than using ad hoc separability criteria. Analyzing the specific form of \(\
abla_{\rho}\Phi\) is beyond our scope here, but it may be related to applying gentle local decoherence to remove entanglement at the fastest rate. All these properties flow naturally from the convex geometric view of \(\Phi\) as a projection distance. This perspective has not, to our knowledge, been applied in previous integrated information literature, and it unlocks powerful tools from convex optimization and metric geometry to study and compute \(\Phi\). We stress that \(\sigma^{*}(\rho)\) and \((A^{*}|B^{*})\) provide more information than just the scalar \(\Phi(\rho)\): they tell us exactly how the state can be optimally split and how far it is from such a split. In the next result, we show that \(\sigma^{*}(\rho)\) can be used to construct a special entanglement witness that \"certifies\" the integrated information. Lemma 1 (RKHS embedding of QJSD): _Define \(k(\rho,\sigma)\ =\ \ln 2\ -\ D^{Q}_{\mathrm{JS}}(\rho\|\sigma)\). Then \(k\) is a positive-definite kernel on the space of density operators. Hence by [PERSON]'s theorem [16; 18], there exists a real reproducing-kernel Hilbert space \((\mathcal{H},\langle\cdot,\cdot\rangle_{\mathcal{H}})\) and a feature map \(\phi:\rho\mapsto k(\,\cdot\,,\rho)\in\mathcal{H},\quad\text{with}\quad\langle \phi(\rho),\phi(\sigma)\rangle_{\mathcal{H}}=k(\rho,\sigma)\)._  
Proof (Sketch): Since \(D^{Q}_{\mathrm{JS}}\) is bounded (\(0\leq D^{Q}_{\mathrm{JS}}<\ln 2\)) and symmetric, the shifted kernel \(k=\ln 2-D^{Q}_{\mathrm{JS}}\) satisfies [PERSON]'s criterion for negative-type metrics, which guarantees a Hilbert-space embedding. **Metric-gradient and operator pullback.