# Integrated Information in Relational Quantum Dynamics (RQD)
## Appendix B
### Comparison with Standard Entanglement Measures  
To place our integrated-information measure \(\Phi\) in context, we compared it against two well-known multipartite entanglement quantifiers over an ensemble of random pure states on \(n=6\) qubits: 1. Global entanglement entropy and 2. Generalized multipartite concurrence.  
We generated 500 Haar-random 6-qubit pure states \(|\psi_{j}\rangle_{j=1}^{500}\), computed \(\Phi(\rho_{j})\) for each (with a full bipartition search), and evaluated for each state:  
**Global entanglement entropy:**:  
\[E_{\text{global}}(\psi_{j})\ =\ \frac{1}{6}\sum_{k=1}^{6}S\big{(} \operatorname{Tr}_{\{1,\ldots,6\}\setminus k}\rho_{j}\big{)},\quad\rho_{j}=| \psi_{j}\rangle\langle\psi_{j}|,\ S(\sigma)=-\operatorname{Tr}(\sigma\log_{2} \sigma)\,.\]  
This is the average one-qubit entropy, a standard measure of how each qubit is entangled with the rest.  
**Multipartite concurrence**: [25]  
We used the generalized \(n\)-qubit concurrence  
\[C_{\text{multi}}(\psi_{j})=2^{1-\frac{n}{2}}\sqrt{\big{(}2^{n}-2 \big{)}-\sum_{A\subset\{1,\ldots,n\},A\
eq\varnothing,\{1,\ldots,n\}} \operatorname{Tr}(\rho_{j;A}^{2})}\,\]where \(\rho_{j,A}=\operatorname{Tr}_{A}\rho_{j}\). This quantity vanishes exactly on fully separable pure states and grows with genuine multipartite entanglement.  
Figure 11 displays two scatter plots (500 points each): (a) \(\Phi\) vs. \(E_{\text{global}}\) and (b) \(\Phi\) vs. \(C_{\text{multi}}\).  
**Correlation Metrics:** To quantify the relationship, we compute the Spearman rank-correlation coefficient \(\rho_{s}\) between \(\Phi\) and each measure:  
* \(\rho_{s}(\Phi,;E_{\text{global}})\approx 0.81\).
* \(\rho_{s}(\Phi,;C_{\text{multi}})\approx 0.79\).  
Both correlations are highly significant (\(p<10^{-8}\)), indicating that states with higher average one-qubit entropy or higher multipartite concurrence also tend to have larger integrated information. However, the scatter also reveals that \(\Phi\) captures aspects of the correlation structure not fully reflected by these measures: some states with moderate concurrence can have relatively low \(\Phi\), and vice versa.  
**Discussion:**  
* **Complementary insights.** Global entanglement entropy quantifies how mixed individual qubits are but is insensitive to the pattern of correlations (e.g., it cannot distinguish between a GHZ-like global superposition and two independent Bell pairs if both yield similar one-qubit entropies). Multipartite concurrence detects genuinely \(n\)-party entanglement, but its dependence on all reduced-purity terms can obscure hierarchical structures. Integrated information \(\Phi\), by contrast, explicitly seeks the most separable bipartition, highlighting structural \"weak links\" in the correlation network.
* **Practical takeaways.** A high \(\Phi\) almost always implies large \(E_{\text{global}}\) and \(C_{\text{multi}}\), but the converse is not guaranteed: one must inspect \(\Phi\) to find the optimal split that reveals hidden modularity. In algorithm design, one could use \(E_{\text{global}}\) or \(C_{\text{multi}}\) as _pre-filters_to identify highly entangled states, then compute \(\Phi\) only on these to pinpoint their internal structure.
* **Applications.** For state classification tasks (e.g., distinguishing cluster from GHZ-family states), combining \(\Phi\) with conventional measures improves accuracy: \(\Phi\) flags the natural cut, while \(E_{\text{global}}\) and \(C_{\text{multi}}\) rank overall entanglement strength. In variational ansatze for quantum many-body simulations, \(\Phi\) can guide the selection of tensor network bonds: one allocates larger bond dimensions across high-\(\Phi\) cuts, using global entropy or concurrence to identify candidate states and then refining with \(\Phi\).