# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix C Proof of Proposition 1 (Canonical Optimal Witness)  
### Preliminaries  
**Quantum Jensen-Shannon Divergence (QJSD).** For two density operators (quantum states) \(\rho\) and \(\sigma\) on the same Hilbert space, the QJSD \(D_{JS}^{Q}(\rho\|\sigma)\) is defined as the quantum analog of the classical Jensen-Shannon divergence. One convenient expression is given by the von Neumann entropy \(S(\cdot)\) (with log defined base 2 for convenience):  
\[D_{JS}^{Q}(\rho\|\sigma)\ =\ g\Big{(}\frac{\rho+\sigma}{2}\Big{)}\ -\ \frac{1}{2}\,S(\rho)\ -\ \frac{1}{2}\,S(\sigma)\,. \tag{11}\]  
This divergence is non-negative and symmetric (\(D_{JS}^{Q}(\rho\|\sigma)=D_{JS}^{Q}(\sigma\|\rho)\)), and it is bounded as \(0\leq D_{JS}^{Q}(\rho\|\sigma)\leq\ln 2\). In particular, \(D_{JS}^{Q}(\rho\|\sigma)=0\) if and only if \(\rho=\sigma\). We will not need properties beyond those of a well-behaved distance measure, but note that \(\sqrt{D_{IS}^{Q}}\) defines a true metric on state space.  
**Entropy identities.** We will use the identity for the derivative of entropy: for a density operator \(X\),  
\[\frac{\partial}{\partial X}S(X)\ =\ -\ \log X\ -\ I\,,\]  
where \(I\) is the identity on \(X\)'s Hilbert space. Equivalently, for any small variation \(\delta X\), the first-order change in entropy is \(\delta S(X)=-\text{Tr}[(\log X+I),\delta X]\). We will also use the additivity of entropy on product states: if \(X_{A}\) and \(X_{B}\) are states on subsystems \(A\) and \(B\), then \(S(X_{A}\otimes X_{B})=S(X_{A})+S(X_{B})\).  
**Hilbert-Schmidt inner product.** For operators \(A,B\) on the same space, denote \(\langle A,B\rangle_{HS}:=\text{Tr}(A^{\dagger}B)\). In particular, since \(\rho\) and \(\sigma\) are Hermitian, \(\langle\rho,\sigma\rangle_{HS}=\text{Tr}(\rho,\sigma)\). We will frequently work with gradients of scalar functions with respect to this inner product; note that if \(F(X)\) is a differentiable function on states, the condition \(\
abla_{X}F=0\) (the zero operator) is equivalent to \(\text{Tr}[(\
abla_{X}F),\delta X]=0\) for all variations \(\delta X\) in the domain.  
**Product-state manifold.** We focus on a fixed bipartition of the full system into subsystems \(A\) and \(B\). Let \(\mathcal{P}*A|B\) denote the set of all bipartite product states on this cut. An element \(\pi\in\mathcal{P}*A|B\) can be written as \(\pi=\pi_{A}\otimes\pi_{B}\), where \(\pi_{A}\) and \(\pi_{B}\) are density operators on \(A\) and \(B\), respectively. Importantly, \(\mathcal{P}*A|B\) is _not_ a convex set (convex combinations of product states yield separable states with classical correlations in general), but it is a smooth manifold (essentially the Cartesian product of the manifold of density matrices on \(A\) with that on \(B\), constrained by normalization on each). We will be minimizing \(D*JS^{Q}(\rho\|\sigma)\) over \(\sigma\in\mathcal{P}_{A|B}\), where \(\rho\) is a fixed state outside this set (an entangled or otherwise correlated state across \(A|B\)). By definition, the _integrated information_\(\Phi(\rho)\) is this minimum divergence:  
\[\Phi(\rho)\ =\ \min_{\sigma\in\mathcal{P}_{A|B}}D_{JS}^{Q}(\rho\|\sigma)\,.\]  
Because \(D_{JS}^{Q}\) is non-negative and vanishes exactly on factorable states, \(\Phi(\rho)\) quantifies how far \(\rho\) is from being a product state on the \(A|B\) split. A state \(\rho\) is integrable (completely uncorrelated across \(A|B\)) if and only if \(\Phi(\rho)=0\); otherwise, \(\Phi(\rho)>0\) measures the irreducible entanglement or correlation across the bipartition.  
Proposition 1 states that there is a unique product state \(\sigma^{*}=\rho_{A}^{*}\otimes\rho_{B}^{*}\) attaining this minimum, and furthermore that the Hermitian operator \(W_{\rho}:=\sigma^{*}-\rho\) serves as an optimal entanglement witness for the \(A|B\) cut, satisfying \(\text{Tr}[W_{\rho},\pi]\geq 0\) for all product states \(\pi\) on \(A|B\) while \(\text{Tr}[W_{\rho},\rho]<0\) exactly by an amount \(-\Phi(\rho)\). In the following sections, we provide a rigorous proof of these claims.