# Integrated Information in Relational Quantum Dynamics (RQD)
## 1 Introduction  
Modern developments in quantum physics and theories of consciousness increasingly suggest that _relations_ and _information_, rather than isolated material objects, are fundamental to reality. Recent proposals cast consciousness itself as a state of matter, suggesting a natural home for IIT in quantum-mechanical systems [1]. In particular, integrated information theory (IIT) posits that the level of intrinsic awareness or consciousness of a physical system corresponds to the quantity of integrated information (denoted \(\Phi\)) present in its state [2, 3]. IIT argues that a system whose information is integrated, i.e., not decomposable into independent parts, has an irreducible subjective existence for itself [4]. This aligns with interpretations of quantum mechanics emphasizing that quantum states are relationalrather than absolute [5]. [PERSON]'s famous dictum _\"every it derives its significance from bits\"_ encapsulates the view that what fundamentally exists are acts of observation or informational relations, not static isolated entities [6]. Relational Quantum Mechanics (RQM) [5;7] formalizes this idea, proposing that the state of a quantum system is nothing more than the information one physical system has about another. Embracing this relational stance, we further posit that the \"being\" of a quantum system is constituted by the network of quantum information it shares with the rest of the world. Under this hypothesis, consistent with the idealistic interpretation of IIT (IIT 4.0's \"idealistic ontology\" [8]), any system with non-zero integrated information (\(\Phi>0\)) possesses intrinsic existence (a rudimentary point of view \"for itself\"), whereas \(\Phi=0\) indicates complete reducibility (no intrinsic unity beyond its parts). Building on RQM, Relational Quantum Dynamics (RQD) elevates the primacy of informational relations by construing quantum states as inherently contextual and observer-relative, thereby embedding the IIT notion of integrated information within the formal structure of quantum theory. Under RQD, each system's \"being\" arises from a network of dynamical information flows--quantified by \(\Phi\)--connecting its constituent subsystems, and classical spacetime and observations emerge from patterns of these relational correlations. By forgoing absolute state assignments in favor of context-dependent relational states, RQD unifies quantum measurement, spacetime emergence, and observerhood in a single coherent ontology [9]. To explore these ideas rigorously, we develop a formal measure \(\Phi(\rho)\) of integrated information for an \(n\)-partite quantum system in state \(\rho\). We require a quantitative gauge of how holistically correlated or irreducible the state \(\rho\) is. In classical IIT, measures based on Kullback-Leibler divergence or mutual information have been proposed to quantify the loss of information upon splitting a system [10;11]. Here, a natural choice for the quantum case is the quantum Jensen-Shannon divergence (\(D_{\mathrm{JS}}^{\mathrm{Q}}\)), a symmetrized and bounded measure of distance between quantum states [12;13]. Intuitively, \(D_{\mathrm{JS}}^{\mathrm{Q}}\) will serve as an \"integration distance\"; it vanishes if \(\rho\) can be exactly factorized into independent local states (no integration) and grows as \(\rho\) becomes more entangled or correlated across subsystems (more integration). Specifically, we define \(\Phi(\rho)\) as the minimum \(D_{\mathrm{JS}}^{\mathrm{Q}}\) between \(\rho\) and any product state obtained by partitioning the system. Let \(P_{1},\ldots,P_{k}\) be a partition of the \(n\) subsystems into \(k\) disjoint groups (blocks), and define the corresponding product state \(\rho_{P_{1}}\otimes\cdots\otimes\rho_{P_{k}}\) as the tensor product of the reduced density operators on each block.