# Integrated Information in Relational Quantum Dynamics (RQD)
## 7 Toward a Quantum Markov Blanket
In other words, among all choices of \(Z\) of the same size, \(Z=X\) yields the smallest residual \(\Phi\) (or Jensen-Shannon correlation) between \(Y\) and \(Z\)'s complement when one conditions on \(Z\). Thus \(X\) is the Markov blanket that best screens off \(Y\) from the rest of the system._  
Proof (Sketch). : A full proof is technical and will be provided elsewhere, but the intuition is the following: For \(X\) being the optimal partition side, \(\Phi(\rho)=D_{\mathrm{JS}}^{Q}(\rho_{XY}\|\rho_{X}\otimes\rho_{Y})\) is minimal. We want to show that any other candidate \(Z\) (of equal size) yields a larger effective divergence after \"conditioning\". Using the properties of the Petz recovery channel, one can show a data-processing inequality in reverse: if \(X\) is the minimizing set, then for any \(Z\) of the same size,  
\[D_{\mathrm{JS}}^{Q}(\rho_{XY}\|\rho_{X}\otimes\rho_{Y})\leq D_{\mathrm{JS}}^ {Q}(\rho_{ZY}\|\rho_{Z}\otimes\tilde{\rho}_{Y|Z})\,. \tag{28}\]  
The right-hand side is essentially the \(\Phi\) of splitting the system into \(Z\) vs. the rest after optimally conditioning on \(Z\) (via Petz). This inequality relies on two facts: (i) monotonicity of \(D_{\mathrm{JS}}^{Q}\) under the specific CPTP map that projects onto \(Z\) and uses Petz recovery (ensuring no loss of relevant info for \(Z\)), and (ii) the definition of \(X\) as giving the smallest baseline \(D_{\mathrm{JS}}^{Q}(\rho_{XY}\|\rho_{X}\otimes\rho_{Y})\). With these, one concludes that the optimal \(Z\) is \(Z=X\). What Theorem 5 signifies is that the smallest subsystem in the optimal bipartition plays the role of a Markov blanket for the rest of the system. It is the minimal \"shield\" that, once known, renders the two halves as independent as possible. This is a novel concept because previous quantum-causal or IIT-related approaches struggled to define a Markov blanket without assuming underlying classical structures or commuting observables. Here, the notion falls out naturally from \(\Phi\)--the same quantity that measures holism also identifies the boundary that best isolates that holism. This quantum Markov blanket idea has several potential applications:  
* **Quantum Causal Discovery:** In analogy to classical causal discovery algorithms that search for Markov blankets of variables to infer graph structure, one could scan over each subsystem \(i\) (or group of interest) in a quantum network, compute its Markovblanket \(X_{i}\) via the \(\Phi\) minimization, and use those blankets to infer an underlying interaction structure or causal graph. Essentially, if qubit 5's Markov blanket is 2, 3, that suggests qubit 5 interacts mainly with 2 and 3 and is independent of others given 2, 3. Repeating for all yields a causal adjacency structure. * **Modular Subsystem Identification:** In complex quantum systems, for example, a many-qubit simulator or a biological quantum process model, the Markov blanket of a region defines the effective boundary between that region and its environment. If you have a line of spins, the Markov blanket of a contiguous block may just be its immediate neighbors; in a fully connected network, it might be a specific subset. Knowing the blanket helps in partitioning the system into modules that interact weakly with each other, which is useful for simplifying dynamics or for design of quantum architectures. * **Neuroscience Analogies:** IIT was originally inspired by the brain's functional organization. If one models neurons or brain regions as quantum subsystems (a speculative but intriguing idea) [23], \(\Phi\) could identify which set of neurons constitutes a functional cluster (high internal integration) and what the Markov blanket is (the interface neurons that connect that cluster to the rest of the brain). This resonates with the \"global workspace\" theory and the concept of a dynamic core in neuroscience. Thus, a quantum Markov blanket could highlight the physical substrate of a conscious \"bubble\" within a larger system.