Joint measurement of incompatible observables and the bell inequalities

Willem M. de Muynck, Willy De Baere, Hans Martens

Michael M. Wolf, David Pérez-Garcı́a, Carlos García-Gutiérrez Fernández

It is well known that jointly measurable observables cannot lead to a violation of any Bell inequality--independent of the state and the measurements chosen at the other site. In this Letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the Clauser-Horne-Shimony-Holt inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case.

Joseph B. Altepeter, Evan R. Jeffrey, Paul G. Kwiat

Diederik Aerts, Massimiliano Sassoli de Bianchi

The view exists that Bell-tests would only be about local incompatibility of quantum observables and that quantum non-locality would be an unnecessary concept in physics. In this note, we emphasize that it is not incompatibility at the local level that is important for the violation of Bell-CHSH inequality, but incompatibility at the non-local level of the joint measurements. Hence, non-locality remains a necessary concept to properly interpret the outcomes of certain joint quantum measurements.

Shasanka Mohan Roy