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

Question

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 $d$.

Answer 1

Yes, the thesis Informationally Complete Measurements and Optimal Representations of Quantum Theory. John B. DeBrota has examples of constructions for all dimensions.

Answer 2

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.