Junhyeong An

Understanding the relationship between operational entanglement criteria and the dimensional structure of entanglement remains a central problem in quantum information theory. In this work, we analyze the geometry of states satisfying the reduction criterion in bipartite systems. We prove that, for two-qutrit states, the union of the two reduction-positive cones is contained in the cone of states with Schmidt number at most two. Our proof relies on an explicit dual-cone computation together with structural results on positive maps, similar to the theorem of Yang–Leung–Tang stating that every 2-positive map on the set of all 3×3 complex matrices is decomposable. Our work clarifies the geometric role of the reduction criterion in detecting high-dimensional entanglement.