# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix C Proof of Proposition 1 (Canonical Optimal Witness)
### Experimental Recipe for \(W_{\rho}\)  
The results above not only prove Proposition 1 rigorously, but also suggest a practical procedure for _constructing and testing the witness \(W_{\rho}\) in an experiment_. Here we outline a self-contained protocol:  
1. **Compute the closest product state \(\sigma^{*}\). ** Given the state \(\rho\) (obtained, say, via quantum state tomography on subsystems \(A\) and \(B\)), one needs to find \(\sigma^{*}=\rho_{A}\otimes\rho_{B}\) that minimizes \(D^{\mathrm{Q}}_{JS}(\rho||\sigma)\). In practice, since we have shown the minimizer must use the true marginals of \(\rho\), this amounts to setting \(\rho_{A}^{*}=\mathrm{Tr}_{B}[\rho]\) and \(\rho_{B}^{*}=\mathrm{Tr}_{A}[\rho]\). (If \(\rho_{A}\) and \(\rho_{B}\) are not known a priori, they can be computed from the tomographic data for \(\rho\).) One may also verify that this choice indeed yields the smallest divergence by evaluating \(D^{\mathrm{Q}}_{JS}(\rho||\rho_{A}\otimes\rho_{B})\) and comparing with nearby alternatives. In more complex scenarios, for example, many-partite systems where \(\Phi\) involves a search over many splits, numerical convex optimization routines can be used to project \(\rho\) onto the product-state manifold. 2. **Construct the witness operator \(W_{\rho}\). ** Once \(\sigma^{*}=\rho_{A}\otimes\rho_{B}\) is determined, form the operator \(W_{\rho}=\sigma^{*}-\rho\). Because both \(\sigma^{*}\) and \(\rho\) are density operators (positive semidefinite with unit trace), \(W_{\rho}\) is Hermitian and has zero trace. Diagonalize \(W_{\rho}\) if needed to inspect its spectrum: it will have both positive and negative eigenvalues, with the negative part corresponding to the subspace of \(\rho\)'s entanglement. For experimental implementation, it is often convenient to decompose \(W_{\rho}\) into a sum of locally measurable observables. Noting that \(\sigma^{*}=\rho_{A}\otimes\rho_{B}\) and assuming \(\rho\) is known (from tomography), one can expand each of \(\rho_{A},\rho_{B}\), and \(\rho\) in a product basis (for instance, Pauli operators on qubit systems). Then \(W_{\rho}\) is expressed as a linear combination of tensor-product operators on \(AB\). The number of terms in this decomposition is manageable--it equals the number of terms needed to describe \(\rho\) and \(\sigma^{*}\) separately, so essentially no overhead beyond measuring \(\rho\) itself. In summary, prepare an apparatus to measure the observable \(W_{\rho}\) on the two subsystems jointly. 3. **Witness test and verification. ** Perform measurements of \(W_{\rho}\) on many copies of the state \(\rho\) to estimate the expectation value \(\langle W_{\rho}\rangle_{\rho}=\mathrm{Tr}[W_{\rho},\rho]\). According to our theory, this should come out negative, and in fact equal to \(-\Phi(\rho)\). A negative mean value certifies the presence of entanglement (integrated information) across the partition. For a quantitative test, one can independently compute \(\Phi(\rho)\) (either from the divergence formula or by diagonalizing \(W_{\rho}\) and summing the negative part of its spectrum) and check that \(-\langle W_{\rho}\rangle_{\rho}\) matches \(\Phi(\rho)\). Additionally, to confirm that \(W_{\rho}\) is behaving as a valid witness, one should also measure \(\langle W_{\rho}\rangle_{\pi}\) on various product-state inputs \(\pi=\pi_{A}\otimes\pi_{B}\). In an experiment, these could be product states prepared by isolating \(A\) and \(B\) (with no entanglement). Our derivations guarantee \(\mathrm{Tr}[W_{\rho},\pi]\geq 0\) for all such states; observing this (within error bars) validates that the construction of \(W_{\rho}\) was successful. In practice, it suffices to test \(W_{\rho}\) on a tomographically complete set of product states or on the extremal product states (e.g., pure product states aligning with \(W_{\rho}\)'s extremal eigenvectors) to build confidence that no false negatives occur.