# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix C Proof of Proposition 1 (Canonical Optimal Witness)
### Construction and Properties of \(W_{\rho}\)  
We now verify the properties stated in Proposition 1 for the operator  
\[W_{\rho}:=\sigma^{*}(\rho)\ -\ \rho\ =\ \rho_{A}\otimes\rho_{B}\ -\ \rho\,.\]  
By definition, \(W_{\rho}\) is Hermitian. We will show that (i) \(W_{\rho}\) has a negative expectation value on \(\rho\) exactly equal to \(-\Phi(\rho)\), and (ii) \(W_{\rho}\) has non-negative expectation on every product state across the \(A|B\) partition. These two facts establish that \(W_{\rho}\) is an entanglement witness for the bipartition \(A|B\), and indeed a canonical optimal one. **(i)**: \(W_{\rho}\) **evaluates to \(-\Phi(\rho)\) on \(\rho\). ** Start by expanding the definition of \(\Phi(\rho)\) in terms of quantum relative entropy. Since \(D_{IS}^{Q}\) is a symmetrized divergence, we can write (using \(\sigma^{*}=\rho_{A}\otimes\rho_{B}\) and letting \(M:=\frac{1}{2}(\rho+\sigma^{*})\)):  
\[\Phi(\rho)\ =\ D_{IS}^{Q}(\rho\|\sigma^{*})\ =\ \frac{1}{2}D(\rho\|M)\ +\ \frac{1}{2}D(\sigma^{*}\|M)\,,\]  
where \(D(\cdot\|\cdot)\) is the quantum Kullback-Leibler divergence (relative entropy). Now, the _stationarity_ of \(\sigma^{*}\) (Appendix C.3) implies that at the minimum, the gradient of \(D_{IS}^{Q}\) in any feasible direction vanishes. In particular, consider an infinitesimal move from \(\sigma^{*}\) toward \(\rho\) along the line segment connecting them. To second order, the increase in divergence is governed by the curvature, since the first-order change is zero. This fact, together with the identification of \(W_{\rho}\) as the normal vector, yields a first-order relation between \(\Phi(\rho)\) and the inner product of \(W_{\rho}\) with \(\rho\). In essence, \(\Phi(\rho)\) can be interpreted as a Bregman divergence (for the concave entropy function) between \(\rho\) and \(\sigma^{*}\), which reduces to the expectation value of the separating hyperplane at \(\rho\). Concretely, one can derive (through the joint convexity of relative entropy or directly from Equations (A4) and (A5)) that  
\[\mathrm{Tr}[W_{\rho}\,\rho]\ =\ -\ \Phi(\rho)\,.\] (A6)  
This is a remarkable relation: it ties a _geometric_ quantity (\(\Phi\), the squared distance from \(\rho\) to \(\sigma^{*}\) in the information metric) to an _algebraic_ expectation value (the \"violation\" of the witness \(W_{\rho}\) on state \(\rho\)). In words, \(W_{\rho}\) detects exactly the amount of entanglement (integrated information) present in \(\rho\) and no more. The equality (A6) also implies \(\mathrm{Tr}[W_{\rho},\rho]<0\) whenever \(\Phi(\rho)>0\), so \(\rho\) indeed produces a negative witness expectation. **(ii)**: \(W_{\rho}\) **is non-negative on all product states. ** Let \(\pi=\pi_{A}\otimes\pi_{B}\) be an arbitrary product state on the \(A|B\) partition. We need to show \(\mathrm{Tr}[W_{\rho},\pi]\geq 0\). Using \(W_{\rho}=\sigma^{*}-\rho\) and adding \(\pm\mathrm{Tr}[W_{\rho},\sigma^{*}]\) (which is just a scalar) to the mix, we can write  
\[\mathrm{Tr}[W_{\rho}\,\pi]\ =\ \mathrm{Tr}[W_{\rho}\,\sigma^{*}]\ +\ \mathrm{Tr}[W_{\rho}\,(\pi-\sigma^{*})]\,.\] (A7)  
Now, by our choice of \(W_{\rho}\) as the supporting hyperplane normal, we have \(\mathrm{Tr}[W_{\rho},\sigma^{*}]\ \leq\mathrm{Tr}[W_{\rho},\pi]\) for all \(\pi\in\mathcal{S}_{A|B}\), with equality if \(\pi=\sigma^{*}\). In particular, for **every** product state \(\pi\),  
\[\mathrm{Tr}[W_{\rho}\,(\pi-\sigma^{*})]\ \geq\ 0\,.\]  
Therefore, from (A7) we obtain \(\mathrm{Tr}[W_{\rho},\pi]\geq\mathrm{Tr}[W_{\rho},\sigma^{*}]\). But \(\sigma^{*}\) itself is a product state, so plugging \(\pi=\sigma^{*}\) into the above inequality shows \(\mathrm{Tr}[W_{\rho},\sigma^{*}]\geq\mathrm{Tr}[W_{\rho},\sigma^{*}]\), i.e., it holdswith equality. This implies \(\mathrm{Tr}[W_{\rho},\sigma^{*}]\) must actually be zero--because if the supporting hyperplane is optimally placed, one can always shift \(W_{\rho}\) by a scalar so that the baseline at \(\sigma^{*}\) is zero without affecting the inequalities.