Bell-Inequality Violation for Continuous, Non-Projective Measurements

TL;DR

This paper develops a theoretical framework to demonstrate Bell inequality violations using continuous, non-projective measurements in solid-state quantum systems, enabling nonlocality certification without traditional projective measurements.

Contribution

It introduces a method to extract Bell-CHSH inequality violations from continuous measurement data without assuming collapse models, applicable to systems with weak, non-demolition readouts.

Findings

Bell inequality violation can be detected from continuous measurement data.

The method agrees with traditional projective measurement tests in simulations.

Classical correlations do not violate Bell inequalities, confirming quantum nonlocality.

Abstract

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 from continuous time-series data. Here we develop a general theoretical framework showing that Bell-CHSH inequality violation can be extracted from continuous, non-projective measurements without assuming any specific collapse model or phase distribution. We show that sufficiently long continuous measurements of a single entangled pair sample its internal phase-probability structure, enabling effective dichotomic observables to be constructed through phase-sensitive projections and coarse-graining. The resulting Bell correlator is governed by two experimentally accessible resources: intrinsic single-qubit phase spread and nonlocal phase locking between qubits. We benchmark the resulting estimator against conventional projective-measurement CHSH tests implemented via quantum-circuit simulations using Qiskit, finding quantitative agreement in the Bell-violating regime without parameter fitting. Classical deterministic correlations cannot violate the CHSH bound, whereas quantum phase-locked systems recover the nonlinear angular dependence characteristic of entanglement. Our results provide a practical route to demonstrating Bell nonlocality in platforms where measurements are inherently continuous and weak.