The Beta-Bound: Drift constraints for Gated Quantum Probabilities
Jonathon Sendall
TL;DR
The paper addresses how gating and measurement noncommutativity distorts quantum probabilities by introducing the $\beta$-bound, which precisely bounds the drift $|\Delta p_F(E)|$ with $s=\mathrm{Tr}(\rho F)$ and $\varepsilon=\|[F,E]\|$ via $|\Delta p_F(E)| \le 2\sqrt{\frac{1-s}{s}}\,\varepsilon$. It integrates a coherence witness $W(\rho,F)=\|F\rho(I-F)\|_1$ and a record fidelity gap $\Delta_{\mathcal{T}}(\rho_F,R)$ to diagnose cross-boundary coherence and symmetry-sensitive readouts, respectively. The framework is shown to be operational and falsifiable through three experimental vignettes (Hong–Ou–Mandel, atomic energy-dephasing, and decoherence-induced classicality) and extended to general readouts via a tight bound $|\Delta_F(R)| \le 2\sqrt{\frac{1-s}{s}} \|[F,R]\|$. By connecting to twirl maps and decoherence theory, the authors place gating-induced drift within existing structural perspectives while providing concrete, testable predictions. Overall, the work offers a compact, quantitative package—beta-bound, coherence witness, and record fidelity gap—that any interpretation of quantum mechanics must accommodate, along with a practical template for experimental validation.
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 $β$-bound, an inequality that controls how much probability assignments can drift when gating and measurement fail to commute. For a density operator $ρ$, projector $F$, and effect $E$, with gate-passage probability $s = {\rm Tr}(ρF)$ and commutator norm $\varepsilon = \|[F, E]\|$, the symmetric partial-gating drift satisfies $|Δp_F(E)| \leq 2 \sqrt{(1 - s)/s} \cdot \varepsilon$. The constant 2 is sharp. We introduce two diagnostic quantities: the coherence witness $W(ρ, F) = \|F ρ(I - F)\|_1$, measuring cross-boundary coherence, and the record fidelity gap $Δ_T(ρ_F, R)$, 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.