CHSH inequality test via disturbance-free measurement

We propose disturbance-free measurement using a "weak-value" scheme, in which a weakly measured quantum system is post-selected (to the initial state) to confirm that there is no disturbance. The probability of obtaining the non-disturbed state is asymptotically close to unity. We theoretically show that outcomes of the disturbance-free measurement for a two-qubit state satisfy the Clauser--Horne--Shimony--Holt inequality. We experimentally demonstrate the test for a typical (maximally entangled) two-qubit state based on a linear optical system. In experiments, polarization-entangled photon-pairs generated by the spontaneous parametric down-conversion process are measured by instruments such as strength-variable polarization-measurement apparatuses and a fiber-based Bell state analyzer.