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

TL;DR

This paper derives a higher quantum bound for the Vértesi-Bene Bell inequality using POVMs, demonstrating that they enable lower detection efficiency thresholds for loophole-free Bell tests with entangled photons.

Contribution

It provides a new quantum bound for the $I_{CH3}$ inequality considering POVMs and analyzes their impact on detection efficiency thresholds in Bell experiments.

Findings

POVMs yield higher violations of the $I_{CH3}$ inequality.

Lower detection efficiency thresholds are achievable with POVMs.

The threshold efficiency is comparable to the minimal required for the Clauser-Horne inequality.

Abstract

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 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 $I_{CH3}$-inequality. With a higher quantum bound for $I_{CH3}$, we investigate if 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 if POVMs are also relevant in the more realistic case that partially entangled states are used in the experiment. Finally, we investigate which are the required efficiencies of the $I_{CH3}$-inequality, and the type of measurements involved, for closing the detection loophole. We obtain that POVMs allow for the lowest threshold detection efficiency, and that it is comparable to the minimal (in the case of two-qubits) required detection efficiency of the Clauser-Horne-Bell-inequality.