Bell-inequality violation for continuous, non-projective measurements

Many solid-state quantum platforms do not permit sharp, projective measurements but instead yield continuous voltage or field traces under weak, non-demolition readout. In such systems, standard Bell tests based on dichotomic projective measurements are not directly applicable, raising the question of how quantum nonlocality can be certified directly from continuous time-series data. Here we develop a general theoretical framework showing that Bell-CHSH inequality violations can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that continuous measurement records statistically probe the internal phase structure of an entangled pair, enabling effective dichotomic observables to be constructed through phase-sensitive coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between subsystems. We benchmark the framework against conventional projective-measurement CHSH tests, finding quantitative agreement in the Bell-violating regime without parameter fitting. We formally derive the local realistic bound appropriate to this continuous setting and demonstrate that phase-locked quantum systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical and principled route to certifying Bell nonlocality in quantum platforms where measurement is inherently continuous and weak.