Exact Schmidt Numbers and PPT Extremality of an Infinite Grid-State Family

Krebs and Gachechiladze [Phys. Rev. A 113, 062428 (2026)] introduced a family of positive-partial-transpose (PPT) generalized grid states in local dimension d = 2k − 1. They determined the Schmidt number for k ≤ 5 and proposed an all-order scaling pattern. For their particular explicitly specified family, we prove SN(R_k) = k = (d + 1)/2 for every k ≥ 2 using a universal signed-minor identity. We also prove that each R_k generates an extreme ray of the complex Hermitian PPT cone, and obtain exact ranks rank(R_k) = k² + k − 1 and rank(R_k^Γ) = k² + 2k − 3. Local filtering yields extremal trace-preserving PPT channels whose Choi matrices have Schmidt number k and whose partially transposed Choi matrices have Schmidt number two. Their threefold iterates are entanglement breaking by a theorem of Park (2026). The underlying states, their PPT decomposition, and other families with linear Schmidt-number growth were established in earlier work; the results here concern the specific 2026 grid-state family.