EPR-Bell Tests with Unsharp Observables and Relativistic Quantum Measurement

In this contribution I will review the analysis of the Einstein-Podolsky-Rosen argument [19], Bell’s inequalities [5] and of associated experiments for spins in terms of positive operator valued measures (in short: povms). Specifically, I will explore the relation between the Clauser-Horne-Shimony-Holt (CHSH) [16] inequality and a fundamental classicality property of observables – their coexistence; this leads to the question whether for macroscopic systems a relatively ‘small’ amount of unsharpness may suffice to ensure (and explain) the practical impossibility of exhibiting nonclassical features such as those represented by Bell-type inequalities. I will present a derivation of Bell’s inequalities for unsharp spins which follows a reconstruction by Mittelstaedt and Stachow [32] of the original EPR argument. In this treatment, the Bell inequalities follow from a conjunction of two assumptions, (unsharp) reality and locality, applied in the context of the quantum mechanics of an entangled pair of spins. Since the reality assumption can be consistently incorporated into the quantum formalism, it is locality that is incompatible with the latter. However, a contradiction only arises when the degree of unsharpness of the spins is not too high; otherwise the nonlocality of quantum mechanics cannot be detected with such observables. The contradiction can be resolved if the locality assumption is weakened so as to allow for a benign form of nonlocality: one has to accept that (unsharp) objectification can occur over spacelike distances or between dynamically separated parts of a system. Note that this argument is not about the supplementation of quantum mechanics with hidden variables but exhibits only the inevitability and nature of quantum nonlocality. But it does raise the question of a consistent description of the process of measurement for extended, entangled systems and for localised measurements in spacelike separated regions of spacetime. A note on terminology may be in place. The term ‘Bell inequality’ refers, strictly speaking, to the inequality originally exhibited by Bell for pair probabilities associated with a triple of spin observables. Bell’s argument made explicit use of the strict correlation between certain pairs of observables in the spin singlet state and thereby ignored the unavoidable experimental imprecisions. In order