Stronger Bell Inequalities for Werner states?

Cite this problem as Problem 19.


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


Werner states are  d\times d bipartite states of the form:


where P_{asym}, is the projector onto the anti-symmetric space in \mathbb{C}^d\otimes \mathbb{C}^d. In the qubit case (d=2), the definition reduces to

\rho_{AB}=p|\psi^{-}\rangle\langle \psi^{-}|+(1-p)\frac{\mathbb{I}}{4},

with |\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|0,1\rangle-|1,0\rangle).

The above state violates the CHSH inequality as long as p>\frac{1}{\sqrt{2}}. As noted in [1], when we restrict to Bell inequalities for dichotomic measurements, then the two-qubit Werner state violates a Bell inequality iff p>\frac{1}{K_G(3)}, where K_G(3) is Grothendieck’s constant of order 3 [2]. Unfortunately, the exact value of K_G(3) is unknown, and, at the time this problem was posed, the best lower bound on K_G(3) was K_G(3)\geq \sqrt{2}. This, once again, gives the critical visibility p>\frac{1}{\sqrt{2}}.

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.


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

I_{n,m}=\sum_{x=1}^{n}\sum_{y=1}^{m}\langle A_xB_y\rangle +\sum_{1\leq y\leq y'\leq m}\langle A_{yy'}B_{y}\rangle-\langle A_{yy'}B_{y'}\rangle+ \sum_{1\leq x\leq x'\leq n}\langle B_{xx'}A_{x}\rangle-\langle B_{xx'}A_{x'}\rangle.

Here A_x, A_{yy'} (B_y, B_{xx'}) denote Alice’s (Bob’s) dichotomic observables. On one hand, the maximum classical value of I_{n,n} can be shown equal to n^2. On the other hand, using see-saw optimization methods, it is possible to find numerical lower bounds for the maximum quantum value achievable for I_{n,n} in a singlet state. Doing this for n=100 and dividing the result by 100^2, we obtain the improved lower bound K_G(3)\geq 1.417241>\sqrt{2}. Note that, for n=100, the Bell inequality I_{n,n} has 465 settings.

Vértesi’s lower bound on K_G(3) was later improved, using different methods and inequalities, in [5], [6].


[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.