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 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 (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 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 C: Formally redefines causal update quantity (C) using the von Neumann entropy S () = -Tr (). 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 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 C from an "entropy difference" to a "relative entropy (distance), " embedding causal directionality (the arrow of time) into the heart of the theory.