Cite this problem as **Problem 19**.

**Problem**

Find Bell Inequalities which are stronger than the CHSH inequalities in the sense that they are violated by a wider range of Werner states.

**Background**

Werner states are bipartite states of the form:

,

where , is the projector onto the anti-symmetric space in . In the qubit case (), the definition reduces to

,

with .

The above state violates the CHSH inequality as long as . As noted in [1], when we restrict to Bell inequalities for dichotomic measurements, then the two-qubit Werner state violates a Bell inequality iff , where is Grothendieck’s constant of order [2]. Unfortunately, the exact value of is unknown, and, at the time this problem was posed, the best lower bound on was . This, once again, gives the critical visibility .

Attempts at using other facets for the local polytope with a higher number of settings failed to improve this value. In 2003, Daniel Collins and Nicolas Gisin [3] found a Bell Inequality and states which violate the new but not the CHSH inequalities. Alas, the range of Werner states violating the new inequality is smaller than that for the CHSH setting.

**Solution**

The problem was solved by Tamás Vértesi in [4], where he presents a family of Bell inequalities of the form:

.

Here () denote Alice’s (Bob’s) dichotomic observables. On one hand, the maximum classical value of can be shown equal to . On the other hand, using see-saw optimization methods, it is possible to find numerical lower bounds for the maximum quantum value achievable for in a singlet state. Doing this for and dividing the result by , we obtain the improved lower bound . Note that, for , the Bell inequality has 465 settings.

Vértesi’s lower bound on was later improved, using different methods and inequalities, in [5], [6].

**References**

[1] A. Acín, N. Gisin and B. Toner, Phys. Rev. A 73, 062105 (2006).

[2] J. L. Krivine, Adv. Math. 31, 16 (1979).

[3] D. Collins, N.Gisin, quant-ph/0306129 (2003).

[4] T. Vértesi, Phys. Rev. A 78, 032112 (2008).

[5] B. Hua, M. Li, T. Zhang, C. Zhou, X. Li-Jost, S.-M. Fei, Journal of Physics A, 48 (6), p. 065302 (2015).

[6] S. Brierley, M. Navascués and T. Vértesi, arXiv:1609.05011.