# Integrated Information in Relational Quantum Dynamics (RQD)
## 5 Hierarchical Decomposition: The Integration Dendrogram  
Thus far, we have focused on identifying a single \"critical cut\" \((A^{*}|B^{*})\) that yields the minimal integrated information. We now show that this idea can be _recursively applied_ to yield a hierarchical decomposition of the state into a tree of increasingly fine subsystems. The result is a binary tree (dendrogram) that represents the multi-scale structure of correlations in \(\rho\). This construction parallels hierarchical clustering in classical data analysis but here it is based on quantum information relationships intrinsic to \(\rho\) itself. The procedure is as follows:1. **Level 0 (Root):** Start with the full system as one set \(S=1,2,\ldots,n\). Compute \(\Phi(\rho)\) and find the unique optimal bipartition \((A^{*}|B^{*})\) of \(S\) that achieves it. This is the root split of the dendrogram, and the value \(\Phi(\rho)\) will label the root node as a measure of how hard it is to tear the entire system into two parts. 2. **Level 1:** Now take the two blocks \(A^{*}\) and \(B^{*}\) separately. For each block (subsystem group) considered its own subsystem, compute its integrated information \(\Phi(\rho_{A^{*}})\) within that block, i.e., allow partitions internal to \(A^{*}\). Find the optimal bipartition of \(A^{*}\) that achieves \(\Phi(\rho_{A^{*}})\), and similarly partition \(B^{*}\) optimally. This yields splits \(A^{*}\to(A_{1}|A_{2})\) and \(B^{*}\to(B_{1}|B_{2})\) at the next level. Attach these as children of the respective nodes in the tree, and label the nodes \(A^{*}\) and \(B^{*}\) with \(\Phi(\rho_{A^{*}})\) and \(\Phi(\rho_{B^{*}})\). 3. **Continue recursively:** At each subsequent level, for every current leaf node of the tree (which corresponds to some subset of qubits or subsystems), if that subset contains more than one elementary subsystem, compute its optimal bipartition and \(\Phi\)-value, and split it. Continue until every leaf is an individual elementary subsystem (which cannot be split further). The recursion will terminate after at most \(n-1\) levels (when all subsystems are singletons). The outcome of this algorithm is a full binary tree whose leaves are the individual subsystems \(1,\ldots,n\), and whose internal nodes represent larger groupings that were optimally split. We call this tree the integration dendrogram of \(\rho\) (see Appendix B). Each internal node is labeled by the \(\Phi\)-value of that grouping, i.e., how much integrated information had to be \"broken\" to split it into two parts. To illustrate the usefulness of this dendrogram, consider its properties:
* **Uniqueness and Stability:** Because each bipartition at each step is unique (Theorem 3) and changes continuously with \(\rho\) (Lipschitz continuity), the entire dendrogram is uniquely determined by \(\rho\) and is robust to small perturbations. Small changes in \(\rho\) will only gradually change the \(\Phi\) values and possibly slightly adjust the splits, but the overall hierarchical order (which splits occur at which scale) will not wildly reshuffle. This is in stark contrast to some heuristic clustering methods that can have unstable hierarchies. Here the hierarchy is rooted in strict convex optimal cuts at each step, making it canonical for each state. * **Multi-Scale Summary of Correlations:** The dendrogram provides a **multi-scale map** of the entanglement/correlation structure in \(\rho\). The top of the tree tells you the largest-scale division (where the weakest global link in the system lies). Further down, you see progressively smaller modules and sub-modules, down to individual units. Each node's height (the \(\Phi\) value for that subset) quantifies how strongly that subset resists factorization. A high \(\Phi\) at a certain node means that the subset is very integrated and only separable with a large loss, whereas a low \(\Phi\) node indicates a relatively weakly bound cluster that could almost be split without much information loss.