Sparse anisotropic positive maps for qutrit entanglement: exact indecomposability and PPT geometry

Abstract

Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters $w$ and $z$. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square $0\le w,z\le2/3$, complete positivity on the smaller square $0\le w,z\le1/3$, and decomposability is lost precisely outside a quarter circle in the corner $w,z\ge1/3$. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint $W_*=W(2/3,2/3)$ we construct a four-parameter family of PPT edge states of rank type $(5,5)$, derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although $W_*$ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.