Karol Bartkiewicz, Bohdan Horst, Karel Lemr, Adam Miranowicz
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here, we describe extremal states that have the maximal entanglement measured by the concurrence, negativity and relative entropy of entanglement for a given degree of the CHSH violation. We give an explicit expression for these states which happen to be the same for these three entanglement measures. For finding the extremal states we use a generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found states together with the states providing the lower bound on these entanglement measures for a given CHSH violation define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. Furthermore, we describe an efficient experimental method to determine the degree of the CHSH violation for arbitrary two single-photon polarization qubits using six discrete measurement settings instead of nine settings required for obtaining a complete correlation matrix.
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http://arxiv.org/abs/1306.6504
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