Thank you for your response H.H. Rugh! However, I am forbidden to use any orthogonalization method such as Gram-Schmidt, because here the vectors are physically "Quantum States" which cannot be disturbed and turned into Orthogonal vectors. Perhaps, I found one solution to the problem by proposing a set of POVM here, but the upper bound on the probability of distinctions isn't proved. For these vectors, only a POVM can be applied which corresponds to a measurement device for a quantum state (these 4 vectors in the Hilbert Space). So, still a general solution to my problem is unknown.

Yes, they take that form with different eigenvectors but some degenerate eigenvalues $\lambda_j$'s for each $\rho_i$ eigen expansion. But the 4 matrices I have, do not have all orthogonal subspaces (there are intersecting subspaces), hence I need an optimization to operate over them to achieve distinction with maximum probability. How to construct the $\phi_i$ that you initially pointed out?

This is a theorem I found which says something similar, but how do I construct the $\mathcal{P}_{\mathcal{S_i}}$ for each $\rho_i$?

Let $\{\rho_i,1\leq i\leq m\}$ be a set of linearly independent density operators $\rho_i$ with prior probabilities $p_i$. Then the optimal measurement is a von Neumann measurement with measurement operators $\{\Pi_i=\mathcal{P}_{\mathcal{S_i}},1\leq i\leq m\}$ where $\mathcal{P}_{\mathcal{S_i}}$ is an orthogonal projection onto an $r_i$-dimensional subspace $\mathcal{S_i}$ of $\mathcal{H}$ (Hilbert space of $\rho_i$'s) with $r_i= rank(\rho_i)$ and $\mathcal{P}_{\mathcal{S_i}}\mathcal{P}_{\mathcal{S_j}}=\delta^i_j\mathcal{P}_{\mathcal{S_i}}$ (orthonormality of operators).

Yes, mathematically that will work, but how do I guarantee the maximization the probability of distinction through that?

By distinguishing I mean applying the operators to these 4 vectors gives me one of them uniquely and makes the other vanish or at least gives a linear combination of the 4 with very high probability of one and vanishingly of the other ones (this is optimization distinguishability) when they are not orthogonal.