# Do non-necessarily symmetric, minimal IC-POVM exist in all dimensions?

In dimension $$d$$, a POVM $$\left\{M_i\right\}_i$$ with $$d^2$$ outcomes consisting of linearly independent rank-1 elements is a minimal, informationally complete POVM. If we add the condition that the scalar products must satisfy: $$\mathrm{Tr}\left[M_iM_j\right]=\frac{1+d\delta_{i,j}}{d+1}$$ Then the POVM is called a SIC-POVM.

It is unknown whether SIC-POVM exist for all values of $$d$$. However, if we remove this symmetry assumption, do we know whether such "MIC-POVMs" exist in every dimension? What are examples of such POVMs?

I would assume they do, since they are the only extremal POVM with $$d^2$$ non-trivial elements, but I fail to see a general example that would work for any $$d$$.

Yes. Take $$d^2$$ random rank-1 projectors $$P_i$$ and assume that they sum to $$A\ge 0$$:
$$\sum_{i=1}^{d^2} P_i = A.$$
Then operators $$\{A^{-1/2}P_iA^{-1/2}\}$$ form a minimal IC-POVM, since with high probability $$\{P_i\}$$ are linearly independent in the space of $$d \times d$$ matrices.
One could also take $$d+1$$ random orthonormal bases. They form an IC-POVM, not minimal, however. But this could be easier to implement in practice.