I'm reading Nielsen and Chuang and I read quantum tomography process given by N&C (box 8.5), which provides an algorithm for determining $\chi$ in terms of block matrices and density matrices. And I'm pretty confused about the last two equation enter image description here

because the matrix $\beta $ seems not be necessary to calculate $\chi$ in this case just use the matrix $\Lambda$ and I don't know why we can express $\chi$ in this form?

The entire process to get $\beta$ matrix is detailed in this paper



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