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Recently, V\'{e}rtesi and Bene [Phys. Rev. A. {\bf 82}, 062115 (2010)]derived a two-qubit Bell inequality, $I_{CH3}$, which they show to be maximallyviolated only when more general positive operator valued measures (POVMs) areused instead of the usual von Neumann measurements. Here we consider a generalparametrization for the three-element-POVM involved in the Bell test and obtaina higher quantum bound for the $I_{CH3}$-inequality. With a higher quantumbound for $I_{CH3}$, we investigate if there is an experimental setup that canbe used for observing that POVMs give higher violations in Bell tests based onthis inequality. We analyze the maximum errors supported by the inequality toidentify a source of entangled photons that can be used for the test. Then, westudy if POVMs are also relevant in the more realistic case that partiallyentangled states are used in the experiment. Finally, we investigate which arethe required efficiencies of the $I_{CH3}$-inequality, and the type ofmeasurements involved, for closing the detection loophole. We obtain that POVMsallow for the lowest threshold detection efficiency, and that it is comparableto the minimal (in the case of two-qubits) required detection efficiency of theClauser-Horne-Bell-inequality.

A higher quantum bound for the V\'ertesi-Bene-Bell-inequality and the role of POVMs regarding its threshold detection efficiency