Continuation of the question here, what is going to happen if we change the some of the conditions. I write it as a quote from here and change the appropriate places which are underlined:
I need to construct $\underline{k\leq n^2}$ positive semi-definite matrices, say $\{P_i\}_{i=1}^k$. Entries of these matrices are complex numbers.
1> Each matrices are of dimension $n^2\times n^2$.
2> I only know the $n\times n$ block at top left corner of each matrix.
3> $P_i$ is a projection operator. For time being, let us assume that they are of rank $\underline{r_i\leq n}$ for each $i$. Moreover they are mutually orthonormal.
4> $\underline{\sum_{i=1}^{k}P_i=I_{n^2}}$ is the identity operator.
One possible simpler case is when $r_i=n$ for all $i$. I am having doubt that the answer is not going to extend for the above cases.
Take the spectral decomposition of the matrix and use the method stated in the answer of the above question.