# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix C Proof of Proposition 1 (Canonical Optimal Witness)
### Dual-Cone Separation and Supporting Hyperplane  
Having characterized \(\sigma^{*}=\rho_{A}\otimes\rho_{B}\) as the unique closest product state to \(\rho\), we now adopt a convex geometric perspective. Although the set \(\mathcal{P}*A|B\) of product states is not convex, consider its convex hull--the set of all states that are mixtures of product states across \(A|B\). This convex hull is exactly the set of separable states (unentangled states) on the bipartition. Denote by \(\mathcal{S}*A|B\) the closed convex set of all separable density matrices on \(AB\) (with respect to the \(A|B\) split). By assumption, \(\rho\) lies outside \(\mathcal{S}*A|B\) (since \(\Phi(\rho)>0\)), while \(\sigma^{*}\) lies on the boundary of \(\mathcal{S}*A|B\). Indeed, \(\sigma^{*}\) is a pure product state, an extreme point of the convex set. In this scenario, the supporting hyperplane theorem guarantees that there exists a hyperplane that cleanly separates the point \(\rho\) from the convex set \(\mathcal{S}*A|B\), touching \(\mathcal{S}*A|B\) at the boundary point \(\sigma^{*}\). The normal (perpendicular direction) of this hyperplane in operator space is unique up to scaling, and we will see that it corresponds to the canonical witness \(W_{\rho}\). To formalize this, define the dual cone of \(\mathcal{S}_{A|B}\) (based at \(\sigma^{*}\)) as  
\[\mathcal{W}\ :=\ \left\{W=W^{\dagger}:\ \mathrm{Tr}[W\,\pi]\ \geq\ 0\ \ \mathrm{for\ all}\ \pi\in\mathcal{S}_{A|B}\right\}.\]  
Elements of \(\mathcal{W}\) are precisely the Hermitian operators that have a non-negative expectation value on every separable state (hence on every product state in particular). In quantum information, any non-zero \(W\in\mathcal{W}\) with \(\mathrm{Tr}[W,\rho]<0\) is an entanglement witness for the entangled state \(\rho\). The set \(\mathcal{W}\) is a convex cone (closed under positive linear combinations). By construction, any supporting hyperplane separating \(\rho\) from \(\mathcal{S}*A|B\) is characterized by some \(W\in\mathcal{W}\): namely the hyperplane \(X:\mathrm{Tr}[WX]=c\) for an appropriate constant \(c\). Since \(\sigma^{*}\) lies on the boundary, we can choose the constant as \(c=\mathrm{Tr}[W,\sigma^{*}]\). Then the hyperplane \(H:=X:\mathrm{Tr}[WX]=\mathrm{Tr}[W,\sigma^{*}]\) supports \(\mathcal{S}*A|B\) at \(\sigma^{*}\) (meaning \(H\) contains \(\sigma^{*}\) and \(\mathrm{Tr}[W,\pi]\geq\mathrm{Tr}[W,\sigma^{*}]\) for all \(\pi\in\mathcal{S}_{A|B}\)), and \(\rho\) lies strictly on the other side: \(\mathrm{Tr}[W,\rho]<\mathrm{Tr}[W,\sigma^{*}]\). In fact, by shifting \(W\) by a scalar multiple of the identity, we can usually arrange \(\mathrm{Tr}[W,\sigma^{*}]=0\) for convenience, in which case \(\mathrm{Tr}[W,\pi]\geq 0\) for all separable \(\pi\) and \(\mathrm{Tr}[W,\rho]<0\). This \(W\) is then an entanglement witness detecting \(\rho\). Geometrically, there is a unique supporting hyperplane to \(\mathcal{S}*A|B\) that is _orthogonal_ to the vector \(\rho-\sigma^{*}\) (the line segment joining \(\rho\) to \(\mathcal{S}*A|B\)). The normal vector to this hyperplane can be taken as  
\[W_{\rho}\ :=\ \sigma^{*}\ -\ \rho\,.\]  
(This choice indeed yields \(\mathrm{Tr}[W_{\rho},\sigma^{*}]=\mathrm{Tr}[\sigma^{*}\sigma^{*}]-\mathrm{ Tr}[\rho,\sigma^{*}]\). We will shortly see that \(\mathrm{Tr}[W_{\rho},\sigma^{*}]\) equals zero under a suitable normalization, confirming \(W_{\rho}\) as the correctly calibrated witness). By construction, \(W_{\rho}\) is Hermitian and lies in the dual cone \(\mathcal{W}\). In fact, \(W_{\rho}\) is the unique (up to scaling) dual-cone element defining the supporting hyperplane at \(\sigma^{*}\). We may call \(W_{\rho}\) the canonical witness direction for the \(A|B\) entanglement in \(\rho\). Furthermore, by treating the QJSD as a Bregman divergence of the von Neumann entropy, one obtains the identity  
\[\mathrm{Tr}\big{[}(\sigma^{*}-\rho)\,\rho\big{]}\ =\ -\Phi(\rho)\,,\]  
in complete analogy with [PERSON] and [PERSON]'s information-geometric formulation of Bregman divergences [27].