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 $C$.