If I have the density matrix $\rho$ for a bipartite system (each system is two-dimensional), what set of observables should I choose so that I obtain the maximum possible violation of the Bell-CHSH inequality?
Say $\rho$ can be written as: $$\rho = \frac{1}{4} \sum_{n,m = 0}^{3} M_{nm} (\sigma_n \otimes \sigma_m)$$
Where $\sigma_i$ belongs to the vector $\vec \sigma = \{1_2, \sigma_1, \sigma_s, \sigma_3 \}$, i.e. the $2 \times 2$ identity matrix followed by the Pauli matrices. Then, according to section IV.B of this review by Horodecki et al., the maximum possible violation is $2 \sqrt{\lambda_1^2 + \lambda_2^2}$, where $\lambda_1, \lambda_2$ are the two largest singular values of the matrix $M_{ij}$, where ${i, j = 1, 2, 3}$ (i.e the lower right $3 \times 3$ block). But I have not been able to find an algorithm to write down what operators will actually achieve this maximum violation.
How do I find these operators? Or is there any resource which explains the procedure to find these optimal observables?