LAUSR

We construct the convex set $$\mathcal{M}$$M of two-qutrit states, and the subset $$\mathcal{P}\subset \mathcal{M}$$P⊂M. We characterize the extremal points of rank one and rank two of $$\mathcal{M}$$M. We further show that the extremal points of $$\mathcal{P}$$P have rank not equaling to two, and at most six. We apply our results to a long-standing conjecture on the locally distinguishable subspace under local operations and classical communications. We prove that rank-one or rank-two matrices in $$\mathcal{M}$$M hold for the conjecture. We further simplify the conjecture, by showing that the conjecture holds if and only if it holds for states in $$\mathcal{P}$$P.