Abstract
We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.
Highlights
Bell enthusiastically supported the proposal of Clauser, Horne, Shimony, and Holt, which is based on a new scheme and the CHSH inequality [4]
We rigorously prove that under Alice and Bob spin measurements, the least upper bound 32 on the violation of the original Bell inequality holds for all two-qubit and all two-qutrit states exhibiting perfect correlations/anticorrelations
In Secton 6, we summarize the main results and stress that description of general density operators ensuring perfect correlations or anti-correlations for spin or polarization observables may simplify performance of a hypothetical experiment on violation of the original Bell inequality
Summary
The recent loophole free experiments [1,2,3] demonstrated violations of classical bounds for the wide class of the Bell-type inequalities which derivations are not based on perfect (anti-) correlations, for example, the Clauser–Horne–Shimony–Holt (CHSH) inequality [4] and its further various generalizations [5,6,7,8,9,10,11,12,13,14]. For the violation FOBd of the original Bell inequality by a two-qudit state ρd exhibiting perfect correlations/anticorrelations, the CHSH inequality implies for all d ≥ 2 the upper bound (2 2 − 1) (see in Section 3) and the latter upper bound is more than the least upper bound 32 proved [26,32] for the two-qubit singlet. We rigorously prove that under Alice and Bob spin measurements, the least upper bound 32 on the violation of the original Bell inequality holds for all two-qubit and all two-qutrit states exhibiting perfect correlations/anticorrelations. Experimenters need not prepare an ensemble of systems in the singlet state since, by Proposition 2 and Theorem 1, for such experiments, a variety of two-qubit states, pure and mixed, can be used and it might be easier to prepare some of such states
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