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arXiv:1007.2578 · Physical Review A

Two-qubit Bell inequality for which positive operator-valued measurements are relevant

Tamás Vértesi, Erika Bene

Published Dec 27, 2010• 52 citations
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Abstract

A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed, rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin [in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony, edited by W. C. Myrvold and J. Christian (Springer, The Netherlands, 2009), pp. 125--138].

DOI
10.1103/physreva.82.062115

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A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed, rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin [in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony, edited by W. C. Myrvold and J. Christian (Springer, The Netherlands, 2009), pp. 125--138].
Recently, V\'ertesi and Bene [Phys. Rev. A 82, 062115 (2010)] derived a two-qubit Bell inequality ICH3{I}_{\mathrm{CH}3}ICH3​, 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 ICH3{I}_{\mathrm{CH}3}ICH3​ inequality. With a higher quantum bound for ICH3{I}_{\mathrm{CH}3}ICH3​, 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 ICH3{I}_{\mathrm{CH}3}ICH3​ 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.
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We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) MA{\mathcal{M}}_{A}MA​ with the following property. Considering a bipartite Bell test where Alice uses MA{\mathcal{M}}_{A}MA​, then for any possible shared entangled state $\ensuremath{\rho}$ and any set of (possibly infinitely many) POVMs NB{\mathcal{N}}_{B}NB​ performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.
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