# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix C Proof of Proposition 1 (Canonical Optimal Witness)
### Optimality of the Witness  
Finally, we argue that \(W_{\rho}\) is not just a valid witness, but in fact the optimal witness for detecting the \(A|B\)-entanglement in \(\rho\). By optimal, we mean that \(W_{\rho}\) yields the most negative expectation value on \(\rho\) among all Hermitian operators that could serve as witnesses on this bipartition. Equivalently, \(W_{\rho}\) maximizes the violation \(-\mathrm{Tr}[W,\rho]\) over the dual cone \(\mathcal{W}\) of witnesses (subject to normalization).  
To see this, consider any other candidate witness \(W^{\prime}\in\mathcal{W}\) (so \(W^{\prime}\) is Hermitian and \(\mathrm{Tr}[W^{\prime},\pi]\geq 0\) for all product \(\pi\)). The separating hyperplane geometry implies that \(W^{\prime}\) cannot have a \"steeper\" negative slope on \(\rho\) without breaking the non-negativity constraint on some separable state. We formalize this intuition as follows. Since \(W^{\prime}\) and \(W_{\rho}\) are both supporting hyperplane normals for \(\mathcal{S}*A|B\) (possibly at different boundary points), their difference \(W^{\prime}-W_{\rho}\) must be orthogonal to the face of \(\mathcal{S}*A|B\) at \(\sigma^{*}\). In particular, \(W^{\prime}-W_{\rho}\) has zero expectation on \(\sigma^{*}\) and on all states in the tangent plane of the boundary at \(\sigma^{*}\). Thus \(W^{\prime}\) and \(W_{\rho}\)_coincide on the supporting hyperplane_ at \(\sigma^{*}\), sharing the same \"baseline\" for product states. Writing \(\rho-\sigma^{*}=-W_{\rho}\) as the outward normal direction, any component of \(W^{\prime}\) along this direction will increase \(\mathrm{Tr}[W^{\prime},\rho]\) if \(W^{\prime}\) is to remain in the dual cone. In equation form, for any \(W^{\prime}\in\mathcal{W}\),  
\[\mathrm{Tr}[W^{\prime}\rho]\ =\ \mathrm{Tr}[W^{\prime}\,\sigma^{*}]\ +\ \mathrm{Tr}[W^{\prime}\,(\rho-\sigma^{*})]\ \geq\ 0\ +\ \mathrm{Tr}[W_{\rho}\,\rho]\,.\]  
The inequality holds because (a) \(\mathrm{Tr}[W^{\prime},\sigma^{*}]=0\) (both witnesses give zero on \(\sigma^{*}\)), and (b) \(W^{\prime}-W_{\rho}\) has no component in the \(W_{\rho}\) direction that would further lower the expectation on \(\rho\). But from (10), we know \(\mathrm{Tr}[W_{\rho},\rho]=-\Phi(\rho)\), which is the minimal possible value. Therefore \(\mathrm{Tr}[W^{\prime},\rho]\geq-\Phi(\rho)\) for every valid witness \(W^{\prime}\). In other words, \(-\mathrm{Tr}[W_{\rho},\rho]=\Phi(\rho)\) is the largest entanglement violation attainable by any witness on the \(A|B\) cut. This proves the optimality: \(W_{\rho}\) detects \(\rho\) at least as strongly as any other witness operator can.  
Proposition 1 is now fully established. We have found a _canonical entanglement witness_\(W_{\rho}=\sigma^{*}-\rho\) that is tailor-made for the state \(\rho\) and partition \(A|B\), with the following defining properties:  
* _Faithfulness:_\(W_{\rho}\) is positive on all unentangled states across \(A|B\), yet yields a strictly negative expectation value on \(\rho\).  
* _Maximality:_ No other witness can yield a more negative expectation on \(\rho\) without violating positivity on some separable state. In particular, \(\mathrm{Tr}[W_{\rho},\rho]=-,\Phi(\rho)\) is the most negative possible, saturating the bound. In summary, \(W_{\rho}\) emerges directly from the geometric construction of \(\Phi(\rho)\) as the vector pointing from \(\rho\) to its nearest product-state approximation \(\sigma^{*}\). It thus _certifies_ the presence of integrated information \(\Phi(\rho)\): one could say that \(\Phi(\rho)\) is the magnitude by which \(\rho\) fails the best possible separability test \(W_{\rho}\). This bridges the geometric measure of entanglement (\(\Phi\)) with the operational notion of an entanglement witness in a one-to-one manner.