# A notational confusion in a Bell like inequality

In the tripartite Bell type inequality know as Svetlichny inequality, given in this (freely available) article. The quantity $$M_{ijk} = Tr [\rho(\sigma_i \otimes \sigma_j \otimes \sigma_k)]$$, $$i,j,k\in \{1,2,3 \}$$, where $$\sigma_i$$ represents Pauli matrix, is defined below equation 6.

My question: After this quantity, they define the matrix $$M_{j, ik}$$, but it is not clear to me how this matrix is related to $$M_{ijk}$$. In other words, how do I obtain $$M_{j,ik}$$ from $$M_{ijk}$$?

• I guess they are nearly the same, $M_{ijk}$ is the element of the matrix $M$ by some manipulation of the order of the index. Commented Jan 12, 2022 at 3:10

The standard meaning of this notation is that you're using $$n$$-ary encoding of the indices. In this case $$n=3$$, so \begin{align}11 \sim 1\\ 12 \sim 2 \\ 13 \sim 3 \\ 21 \sim 4 \\ 22 \sim 5 \\ 23 \sim 6 \\ 31 \sim 7 \\ 32 \sim 8 \\ 33 \sim 9 \end{align} The matrix element $$M_{1,6}$$ is then represented as $$M_{1,23}$$ which is given by $$M_{213}$$.
• why $M_{j, ik}$ and not say $M_{i,jk}$? Commented Jan 12, 2022 at 11:04