# Integrated Information in Relational Quantum Dynamics (RQD)
## 3 Monotonicity and Metric Properties of \(D_{\text{JS}}^{Q}\)
Combined with the 1-Lipschitz bound from Section 4, which shows that, for trace-norm errors \(\leq\varepsilon\), the resulting uncertainty in \(\Phi\) is at most \(\Delta\Phi\leq\varepsilon\), we see that standard tomography fidelities (\(\sim\)2%) translate into \(\Delta\Phi\lesssim 0.02\) bits. Thus, even imperfect state reconstruction cannot spuriously inflate \(\Phi\), making it a robust witness of genuine multipartite integration. \(\Phi\) as a Resource Monotone: Beyond its operational significance, monotonicity under all CPTP maps elevates \(\Phi\) to the status of a resource-theoretic monotone. In entanglement theory, one demands that valid entanglement measures never increase under the free operations of LOCC; analogously, here, any allowed quantum channel can only degrade holism. This endows \(\Phi\) with the same foundational role for \"quantum holism\" that entanglement measures play for nonlocality:  
\[\Phi\big{(}\mathcal{E}(\rho)\big{)}\ \leq\ \Phi(\rho), \tag{7}\]  
for every CPTP map \(\mathcal{E}\). Consequently, \(\Phi\) quantifies a resource that cannot be generated by noise or local processing, and it fits naturally into the burgeoning framework of quantum resource theories. Next, we establish that the quantum JSD endows state space with a true metric, not just a divergence. It is known that the classical Jensen-Shannon divergence yields a legitimate distance metric via its square root. The quantum version inherits a similar property:  
**Theorem 2** (Metric Structure of Quantum JSD). : _Define \(\delta(\rho,\sigma):=\sqrt{D^{Q}_{\mathrm{JS}}(\rho\|\sigma)}\) for density operators \(\rho,\sigma\). Then \(\delta(\rho,\sigma)\) is a metric on the space of quantum states. In other words, it satisfies positivity, symmetry, and the triangle inequality. In particular, \(\delta(\rho,\sigma)=0\) if and only if \(\rho=\sigma\), and for any three states \(\rho,\sigma,\tau\) we have_  
\[\delta(\rho,\tau)\leq\delta(\rho,\sigma)+\delta(\sigma,\tau). \tag{8}\]  
Proof (Sketch). : The non-negativity and symmetry of \(\delta\) are immediate from the properties of \(D^{Q}_{\mathrm{JS}}\). The non-degeneracy (\(\delta(\rho,\sigma)=0\iff\rho=\sigma\)) holds because \(D^{Q}_{\mathrm{JS}}(\rho\|\sigma)=0\) iff \(\rho=\sigma\). The crux is the triangle inequality. We employ a known characterization: a divergence \(D(\cdot\|\cdot)\) is of the negative type (or negative definite) if and only if its square root defines a metric [16]. Recent works have shown that the (classical) Jensen-Shannon divergence is of a negative type, allowing the construction of an isometric embedding of probability distributions into a real Hilbert space [17]. The quantum Jensen-Shannon divergence shares this property [18]. Specifically, one can show \(D^{Q}_{\mathrm{JS}}\) is negative-definite on quantum states, for example, by expressing it as an \(L^{2}\) distance in an appropriate purified representation or by verifying the inequality \(\sum_{ij}a_{i}a_{j}\cdot D^{Q}_{\mathrm{JS}}(\rho_{i}\|\rho_{j})\leq 0\) for any choices of states \(\rho_{i}\) and real coefficients \(a_{i}\) summing to 0 (a hallmark of negative-type functions). Given negative definiteness, [PERSON]'s theorem [16] guarantees that \(\delta(\rho,\sigma)=\sqrt{D^{Q}_{\mathrm{JS}}(\rho\|\sigma)}\) satisfies the triangle inequality. Hence, \(\delta\) is a metric on state space. This result is noteworthy: although many quantum divergences, for example, quantum relative entropy, do not yield a true metric, the Jensen-Shannon divergence does. Thus, the set of density matrices can be treated as a metric space with distance \(\delta\). Geometrically, \((\mathcal{S},\delta)\) is not a Euclidean space but can be embedded isometrically into a (potentially infinite-dimensional) inner-product space. Intuitively, one can think of each quantum state as a point in some high-dimensional \"feature space\" such that \(\delta(\rho,\sigma)\) is the Euclidean distance between those points. This will allow us to leverage convex geometry within the space of quantum states.