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A Rigorous Information-Theoretic Derivation of the Born Rule via Relational Entropic Distinguishability in Quantum Reference Frames

A.I. Trinity Lab

Published Nov 21, 2025• 0 citations
Publisher

Abstract

We present a fully non-circular derivation of the Born rule based on relational quantum reference frames (QRFs) and information-theoretic distinguishability bounds. Amplitudes are treated not as primitive probabilities but as epistemic overlap measures between candidate successor frames. Using the quantum Sanov theorem and Petz recovery bounds, we show that the relative entropic cost

DOI
10.5281/zenodo.17674390

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Quantum Structural Theory of Harmony (QSTH 6.0 Revised) — Born Set Invariants: From Probability Readout to Informational Admissibility

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QSTH 6.0 Revised — Born Set Invariants returns to the foundational role of the Born rule within the broader architecture of the Quantum Structural Theory of Harmony (QSTH). Rather than treating the Born rule solely as a probabilistic output of quantum measurement, the work examines its informational representation and its possible role as a bridge between quantum amplitude, effective entropic balance, record formation, and informational admissibility. The central observation is that the Born readout can be mapped into an informational representation through the relation ΔS_coh = −ln|ψ|². This provides a direct bridge between the standard quantum-mechanical probability readout and the later QSTH ledger quantities, including S_eff, TiM, Settlement Dynamics, and Entropic Zero. The publication introduces Born Set Invariants as a candidate class of informational quantities, relations, or structures whose meaning may survive admissible changes of representation, projection, or system readout. It further distinguishes probability from admissibility, asking whether a non-zero probability is sufficient for the formation of a stable physical record. Within this framework, S_eff is developed as an informational ledger between coherent and reductive contributions, while Ψ and κ(Γ) are retained respectively as candidate regime-signature and regime-sensitivity descriptors. These quantities are not proposed as new forces or fundamental constants. QSTH 6.0 Revised does not modify or replace the Born rule and does not propose a new quantum mechanics. The Born rule remains a CANON experimental and mathematical anchor. The purpose of the work is instead to construct a disciplined and auditable informational language linking probability → information → ledger → invariance → admissibility → regime. The publication forms the first pillar of the QSTH Revised Trilogy and provides the informational entry point for the subsequent treatment of temporal readability, settlement, and Entropic Zero. Part I of the QSTH Revised Trilogy: Probability → Readability → Settlement. QSTH 6.0 Revised establishes the Born Readout Ledger and provides the informational foundation for QSTH 5.0 Revised — The Entropic Clock Revisited and QSTH 6.1 Revised — Entropic Zero Revisited.

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Summing to Uncertainty: On the Necessity of Additivity in Deriving the Born Rule

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The emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics. This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption. Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.

quant-pharXiv:2603.06211•Physical Review A•Mar 9, 2026•2 citations
We present a fully non-circular derivation of the Born rule based on relational quantum reference frames (QRFs) and information-theoretic distinguishability bounds. Amplitudes are treated not as primitive probabilities but as epistemic overlap measures between candidate successor frames. Using the quantum Sanov theorem and Petz recovery bounds, we show that the relative entropic cost
A fully self-consistent derivation of the Born rule is presented using an information-theoretic quantity called coherence cost, defined over candidate future frames in an observer-actualized time model. By applying the Maximum Entropy Principle with a constraint determined by inter-frame coherence, the theory naturally yields without invoking von Neumann entropy. This provides a mathematically clean and conceptually transparent origin for quantum probabilities within a discrete frame-based temporal structure.
This paper proposes a general inversion framework that reconstructs a full quantum state from observed Born probabilities by incorporating relational coherence-cost geometry between quantum reference frames. Because Born’s rule is a many-to-one map that discards phase information, direct inversion is impossible in static formulations. This work resolves that barrier by adding measurable inter-frame coherence data, allowing the full wavefunction—including relative phases—to be recovered in a physically consistent and operationally meaningful way. The framework is compatible with standard quantum reference frame transformations, experimentally accessible interference visibility, and observer-dependent temporal structure. This establishes a new pathway for connecting quantum statistics with underlying quantum states beyond traditional postulates.
We derive the Born rule from a simple variational principle. Starting from a fixed cover of measurement contexts, we measure how far a quantum state lies outside the non‑contextual polytope via Umegaki relative entropy. Minimizing this “contextual divergence” selects the usual trace‑form probabilities. We assume finite dimensions, full‑rank states, and rank‑1 projective contexts. Our construction exactly reproduces the sheaf‑theoretic definition of contextuality and rests on Petz’s projection theorem. Its chief novelty is to blend three previously separate strands—information‑geometric projections, sheaf‑cohomological contextuality, and categorical classical structures—into a single optimization principle. Unlike Gleason’s theorem, decision‑theoretic reconstructions, or envariance arguments, our approach is less general in scope but richer in its explicit treatment of contextuality and relational viewpoints.
QSTH 6.0 Revised — Born Set Invariants returns to the foundational role of the Born rule within the broader architecture of the Quantum Structural Theory of Harmony (QSTH). Rather than treating the Born rule solely as a probabilistic output of quantum measurement, the work examines its informational representation and its possible role as a bridge between quantum amplitude, effective entropic balance, record formation, and informational admissibility. The central observation is that the Born readout can be mapped into an informational representation through the relation ΔS_coh = −ln|ψ|². This provides a direct bridge between the standard quantum-mechanical probability readout and the later QSTH ledger quantities, including S_eff, TiM, Settlement Dynamics, and Entropic Zero. The publication introduces Born Set Invariants as a candidate class of informational quantities, relations, or structures whose meaning may survive admissible changes of representation, projection, or system readout. It further distinguishes probability from admissibility, asking whether a non-zero probability is sufficient for the formation of a stable physical record. Within this framework, S_eff is developed as an informational ledger between coherent and reductive contributions, while Ψ and κ(Γ) are retained respectively as candidate regime-signature and regime-sensitivity descriptors. These quantities are not proposed as new forces or fundamental constants. QSTH 6.0 Revised does not modify or replace the Born rule and does not propose a new quantum mechanics. The Born rule remains a CANON experimental and mathematical anchor. The purpose of the work is instead to construct a disciplined and auditable informational language linking probability → information → ledger → invariance → admissibility → regime. The publication forms the first pillar of the QSTH Revised Trilogy and provides the informational entry point for the subsequent treatment of temporal readability, settlement, and Entropic Zero. Part I of the QSTH Revised Trilogy: Probability → Readability → Settlement. QSTH 6.0 Revised establishes the Born Readout Ledger and provides the informational foundation for QSTH 5.0 Revised — The Entropic Clock Revisited and QSTH 6.1 Revised — Entropic Zero Revisited.
The emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics. This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption. Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.