The Beta-Bound: Drift constraints for Gated Quantum Probabilities

Abstract

Quantum mechanics provides extraordinarily accurate probabilistic predictions, yet the framework remains silent on what distinguishes quantum systems from definite measurement outcomes. This paper develops a measurement-theoretic framework for projective gating. The central object is the $\beta$-bound, an inequality that controls how much probability assignments can drift when gating and measurement fail to commute. For a density operator $\rho$, projector $F$, and effect $E$, with gate-passage probability $s = {\rm Tr}(\rho F)$ and commutator norm $\varepsilon = \|[F, E]\|$, the symmetric partial-gating drift satisfies $|\Delta p_F(E)| \leq 2 \sqrt{(1 - s)/s} \cdot \varepsilon$. The constant 2 is sharp. We introduce two diagnostic quantities: the coherence witness $W(\rho, F) = \|F \rho (I - F)\|_1$, measuring cross-boundary coherence, and the record fidelity gap $\Delta_T(\rho_F, R)$, measuring expectation-value change under symmetrisation. Three experimental vignettes demonstrate falsifiability: Hong--Ou--Mandel interferometry, atomic energy-basis dephasing, and decoherence-induced classicality. The framework is operational and interpretation-neutral, compatible with Everettian, Bohmian, QBist, and collapse approaches. It provides quantitative structure that any interpretation must accommodate, along with a template for experimental tests.