Maximally entangled states and the Bell inequality∗

Kar’s recent proof showing that a maximally entangled state of two spin-1/2 particles gives the largest violation of a Bell inequality is extended to N spin-1/2 particles (N ≥ 3). In particular, it is shown that all the states yielding a direct contradiction with the assumption of local realism do generally consist of a superposition of maximally entangled states. Recently, Kar (see Ref. [1], and references therein) has shown that a maximally entangled states of two spin-1/2 particles not only gives a maximum violation of the CHSH inequality [2] but also gives the largest violation attainable for any pairs of four spin observables, these pairs being noncommuting for both systems. To prove this, Kar made use of an elegant (and powerful) technique based on the determination of the eigenvectors and eigenvalues of the associated Bell operator [3]. In this Letter we would like to extend these results to the case in which N spin-1/2 particles (N ≥ 3) are considered. We will show that the most general N-particle state giving the largest violation of a Bell inequality does consist of a superposition of maximally entangled states. As expected, those states giving maximal departure from classical expectations correspond to the class of states introduced by Greenberger, Horne, and Zeilinger in proving Bell’s theorem without using inequalities [4, 5]. In order to look for a violation of local realism when dealing with N spin-1/2 particles it is necessary to consider correlation functions involving ∗An almost identical version of this paper was originally published in: J.L. Cereceda, Phys. Lett. A 212, 123 (1996).