I believe the answer to this question is well-known; but, unfortunately, I don't know.
In quantum computing, we know that mixed states are represented by density matrices. And the trace norm of the difference of two density matrices characterizes the distinguishability of the two corresponding mixed states. Here, the definition of trace norm is the sum of all eigenvalues of the density matrix, with an extra multiplicative factor 1/2 (in accordance with statistical difference of two distributions). It is well-known that, when the the difference of two density matrices is one, then the corresponding two mixed states are totally distinguishable, while when the difference is zero, the two mixed states are totally indistinguishable.
My question is, does the trace norm of the difference of two density matrices being one imply these two density matrices can be simultaneously diagonalizable? If this is the case, then taking the optimal measurement to distinguish these two mixed states will behave like to distinguish two distributions over the same domain with disjoint support.