Causal Priority Theory part-4 Quantum Extension of ΔC via von Neumann Entropy — A BasisIndependent Formulation

Title: Causal Priority Theory part-4: Quantum Extension of \bm{\Delta C} via von Neumann Entropy - A Basis-Independent Formulation Description: [Overview] This paper is the fourth installment in the Causal Priority Theory (CPT) series. While previous papers (v1-v3) utilized Shannon information entropy to define causal change (\bm{\Delta C}), this work addresses the fundamental limitations of that approach—specifically basis-dependence and observer-dependence—by extending the theory into the quantum domain using von Neumann entropy. [Key Contributions of Part 4] 1. Identification of Shannon Entropy Limitations: Quantitatively demonstrates how Shannon-based \bm{\Delta C} fails to serve as a fundamental physical variable due to its dependence on the choice of basis and coarse-graining. 2. Quantum Redefinition of \bm{\Delta C}: Formally redefines causal update quantity (\bm{\Delta C}) using the von Neumann entropy \bm{S(\rho) = -\text{Tr}(\rho \log \rho)}. 3. Proof of Basis Independence: Provides a mathematical proof that the new definition is invariant under unitary transformations, ensuring that "causal change" is an objective property of the quantum state itself. 4. Resolution of the Coarse-Graining Problem: Establishes an operationally defined quantum framework that eliminates the ambiguity inherent in classical information-theoretic descriptions. 5. Consistency Check: Confirms that the results of Part 1 through Part 3 remain valid under this more rigorous quantum formulation. [Position in the Series] • Previous (Part 3): Examined the validity conditions of \bm{\Delta C} conservation in CPTP maps. • Current (Part 4): Completes the transition of the CPT framework from classical information theory to a robust quantum information-theoretic foundation. • Next (Part 5): Further elevates \bm{\Delta C} from an "entropy difference" to a "relative entropy (distance)," embedding causal directionality (the arrow of time) into the heart of the theory.