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.
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.
Abstract: In Causal Priority Theory (CPT), the ΔC conservation law has previously been treated as a fundamental axiom. This paper (v3) provides a rigorous examination of this law, explicitly identifying the classes of Completely Positive Trace-Preserving (CPTP) maps for which this conservation law holds and those for which it does not. The paper presents: 1. Counterexamples: Four specific cases demonstrating that ΔC conservation does not hold for general CPTP maps. 2. Necessary Conditions: Derivation of four essential conditions required for the conservation law to be valid. 3. Main Theorem: A formal proof that in a closed quantum system on a finite-dimensional Hilbert space, the total ΔC is conserved under unitary evolution. 4. Application to Black Holes: A structural establishment of ΔC_env = 0 using Hilbert space decomposition, upgrading the axiomatic assumptions of the previous papers (v1 and v2) to conditional theorems. This work serves as a critical bridge between the heuristic descriptions of CPT and its formal quantum mechanical foundation, setting the stage for the quantum extension via von Neumann entropy in v4 and relativ
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.
This paper clarifies when time comparisons are physically meaningful in Causal Priority Theory (CPT), introducing a structured notion of comparability. In CPT, time is defined as the accumulation of a causal update quantity along a history, and is therefore inherently dependent on event partitioning. As a result, time is not globally unique, raising the question of when different histories can be meaningfully compared. To address this, the paper introduces three key structures: coarse-graining classes (describing how distinguishability is reduced), stable order (identifying when temporal ordering is preserved), and admissible partition classes (defining domains of robust comparison). These lead to a central conclusion: comparability is not absolute but physically constrained by information loss and redistribution. The monotonicity of the update quantity under CPTP maps induces a contraction structure that organizes derived time into distinct comparability regimes. CPT thus replaces the assumption of globally comparable time with a notion of layered comparability, consisting of history-dependent time, coarse-graining-dependent comparison, and robust comparison within admissible classes. This framework establishes the physical scope of time comparison and sets the stage for Part-9, where the necessity of quantum relative entropy is justified. --- Update (v2): Added description for clarity and discoverability.
This paper presents the "Causal Priority Theory," a new unified framework that redefines causal change \bm{\Delta C} as the sole fundamental variable of the universe. It addresses the logical incompatibility between relativity theory and quantum mechanics regarding the definition of time. By deriving spacetime geometry, the Lorentz factor, and the uncertainty principle from the primacy of causal updates, this theory provides a consistent solution to the black hole information paradox and the violation of Bell's inequality. It also introduces a "Will Operator" (\bm{W}) to describe the observer's role in quantum collapse, offering falsifiable predictions for future experiments.
This paper investigates the relation between continuous state descriptions and derived time within the framework of Causal Priority Theory (CPT). In CPT, time is treated as a history-dependent quantity constructed from causally ordered quantum state updates quantified by quantum relative entropy. Key contributions: • Structural non-automaticity: The paper shows that the existence of a continuous state description does not by itself guarantee a canonically assigned continuous derived time. • Two levels of non-automaticity: • Existence level: Continuity does not ensure finite or convergent partition-sum derived time. • Asymptotic-assignment level: The leading asymptotic time scale is not determined by the state image alone in general, as shown by an explicit same-image counterexample. • Conditional existence: The paper formulates conditions under which continuous derived-time candidates exist, while showing that such candidates remain description-dependent. This part (Part-11) provides the continuous-description layer of the broader CPT program and prepares the transition to the subsequent structural analysis of breakdown regimes and non-comparability.
This paper formulates "Event Partition Dependence" in Causal Priority Theory (CPT)—the principle that the flow of derived time depends on how an observer partitions events. • Partition Dependence: Formalizes time as a quantity dependent on the resolution of observation. • Stability of Order: Identifies the physical conditions under which the time order remains stable despite information loss (coarse-graining). • Conclusion: Proves that the "arrow of time" is robustly preserved under specific uniformity conditions, regardless of the partition. This work extends the unified framework (Episodes 0–6) to address the hierarchical nature of physical observation.