Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi conjectured that it is the limit of the sequence

$$a_n = \max_{c_i\in[0,1]} \lambda(M_{n})$$ where $\lambda(\cdot)$ is the maximal eigenvalue, and $M_{n}$ is the $(n+1)\times (n+1)$ tridiagonal matrix

$$\begin{pmatrix} c_0-c_0^2 & \frac1{\sqrt2}\sqrt{1-c_0^2} & & \\ \frac1{\sqrt2}\sqrt{1-c_0^2} & c_1c_0+\frac{c_1-c_0}{2} & \frac12\sqrt{1-c_1^2}\\ & \frac12\sqrt{1-c_1^2}& \ddots & \ddots\\ & & \ddots & \ddots & \frac12\sqrt{1-c_{n-1}^2}\\ & & & \frac12\sqrt{1-c_{n-1}^2} & c_nc_{n-1}+\frac{c_n-c_{n-1}}{2} \end{pmatrix}$$

Can one get an analytic expression for it?

This expression is nice for calculating $a_n$ numerically, but solving it exactly is a nightmare. I managed to do it for $a_1$, Mathematica did it for $a_2$, but after that there are only numerics. The first few values are

  • $a_1 = \frac{1}{16} \left(5 + 5 \sqrt{5}+\sqrt{50 \sqrt{5}-106}\right) \approx 1.161835$
  • $a_2 \approx 1.224739 $
  • $a_3 \approx 1.238024 $
  • $a_{100} \approx 1.250875$

Update: The asymptotic behaviour of the optimal solutions seems to be rather simple. Ignoring boundary effects, numerical evidence suggests that the $c_i$ converge quickly to a limiting value $C \approx 0.878273$, and that the coefficients of the optimal eigenstate decay exponentially with $i$. Assuming that both these behaviours do happen, elementary arguments show that $$\lim_{n\to\infty} a_n = \frac{4C^4-C^2+1}{4C^2-1}$$ so the problem reduces to calculating $C$.

  • 1
    $\begingroup$ What are diagonals? Unclear for me. $\endgroup$ – Fedor Petrov Jul 4 '16 at 14:19
  • $\begingroup$ Sorry, the matrix with properly written diagonals was too big. I rewrote the question to make it more clear. $\endgroup$ – Mateus Araújo Jul 4 '16 at 17:32

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.