Higher quantum bound for the Vértesi-Bene-Bell inequality and the role of positive operator-valued measures regarding its threshold detection efficiency

Abstract

Recently, V\'ertesi and Bene [Phys. Rev. A 82, 062115 (2010)] derived a two-qubit Bell inequality ${I}_{\mathrm{CH}3}$, which they show to be maximally violated only when more general positive operator-valued measures (POVMs) are used instead of the usual von Neumann measurements. Here we consider a general parametrization for the three-element POVM involved in the Bell test and obtain a higher quantum bound for the ${I}_{\mathrm{CH}3}$ inequality. With a higher quantum bound for ${I}_{\mathrm{CH}3}$, we investigate whether there is an experimental setup that can be used for observing that POVMs give higher violations in Bell tests based on this inequality. We analyze the maximum errors supported by the inequality to identify a source of entangled photons that can be used for the test. Then we study whether POVMs are also relevant in the more realistic case where partially entangled states are used in the experiment. Finally, we investigate the required efficiencies of the ${I}_{\mathrm{CH}3}$ inequality, and the type of measurements involved, for closing the detection loophole. We obtain the result that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal required detection efficiency (in the case of two qubits) of the Clauser-Horne-Bell inequality.