The Beta-Bound: Drift constraints for Gated Quantum Probabilities

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.

💡 Research Summary

The paper introduces a rigorous, interpretation‑neutral framework for analysing how conditional “projective gating” influences quantum probability assignments when the gate and the subsequent measurement do not commute. The central result is the β‑bound, an inequality that quantitatively limits the drift of probabilities caused by this non‑commutativity.

Consider a density operator ρ, a projector F (the gate), and an effect E (the read‑out). The probability that the system passes the gate is s = Tr(ρF). After successful gating the conditioned state is ρ_F = F ρ F / s, and the fully‑gated probability of E is p_ρ(E) = Tr(ρ_F E F)/s = Tr(ρ_F E). If the gate is applied only on one side of the effect, one obtains the left‑partial and right‑partial probabilities q_L(E) = Tr(ρ_F E)/s and q_R(E) = Tr(ρ E F)/s. The symmetric partial‑gating drift is defined as

Δp_F(E) = q_L(E) + q_R(E) − 2 p_ρ(E).

Theorem 4.1 (β‑bound) states that

|Δp_F(E)| ≤ 2 √