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 $M_{ijk}$?

  • $\begingroup$ 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. $\endgroup$
    – narip
    Commented Jan 12, 2022 at 3:10

1 Answer 1


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 $M_{213}$.

  • $\begingroup$ why $M_{j, ik}$ and not say $M_{i,jk}$? $\endgroup$
    – User101
    Commented Jan 12, 2022 at 11:04
  • $\begingroup$ I don't know. Write the authors, I'm sure they'll appreciate the interest. $\endgroup$ Commented Jan 12, 2022 at 16:00
  • $\begingroup$ @User101 If there are some replies, glad to know! :) $\endgroup$
    – narip
    Commented Jan 13, 2022 at 1:08
  • $\begingroup$ Actually I found this notation in a bunch of paper and wrote to all the authors. No replies so far! Will definitely share here if I get one. $\endgroup$
    – User101
    Commented Jan 13, 2022 at 8:28

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