Causal Priority Theory Part-9: On the Necessity and Minimality of Quantum Relative Entropy as the Causal Update Quantity

This paper establishes the necessity and minimality of quantum relative entropy as the causal update quantity in Causal Priority Theory (CPT). In CPT, time is defined as the accumulation of an information-theoretic update along causal histories. However, the choice of this update quantity has not previously been given a systematic justification. This paper addresses that gap. A set of structural requirements is introduced, including non-negativity, identity vanishing, direction sensitivity, compatibility with product composition, monotonicity under CPTP maps, basis independence, consistency with the classical Kullback–Leibler limit, and compatibility with history accumulation and correlation redistribution. Representative alternative candidates—such as entropy difference, trace distance, fidelity, and Bures distance—are shown to fail to satisfy these requirements or to support the central structural roles required in CPT. Quantum relative entropy (Umegaki form) is then identified as the canonical minimal choice: not uniquely required, but the standard candidate that satisfies the full requirement set with the fewest additional assumptions while maintaining coherence with the existing CPT framework. The result does not provide a new numerical prediction but instead imposes a theoretical constraint on the class of admissible causal update quantities in CPT. It strengthens the foundations of derived time, comparability, and coarse-graining stability developed in earlier parts of the series. Questions concerning the behavior of derived time under refinement limits and its connection to continuous dynamical descriptions are not treated here, and are instead addressed in subsequent parts of this series.