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A diagonal matrix entry uses the same row and column index. The diagonal entries of a lower triangular matrix determine its determinant by their product.

A matrix entry whose row and column indices differ. These entries lie outside the diagonal matrix entries. A diagonal matrix has all its off-diagonal entries equal to zero. Cancellation of these entries when averaging density matrices does not require the individual preparations to have zero entries there.

Repeatedly applying the previous product rule givesFormally, the result holds for $p=1$; multiplication by $B$ preserves the zero off-diagonal matrix entries and multiplies each diagonal matrix entry by $B_{ii}$, proving the induction step.

The diagonal matrix entry is $D_{ii}={\sum }_{k=1}^N{B}_{ik}{C}_{ki}$. Only $k=i$ can contribute, so $D_{ii}=B_{ii}{C}_{ii}$. This also shows that the product is a diagonal matrix, with its diagonal obtained by entrywise multiplication of the two diagonals.