Title: Geometry of Schmidt number witnesses for bi-partite quantum states
Abstract: The Schmidt number of a bipartite quantum state is an important tool for measuring the degree of entanglement. To determine it, we rely on Schmidt number k witnesses, which constitute the set difference between the Choi matrices of (k-1)-positive maps and k-positive maps. In this talk, we will specify the locations of Schmidt number k witnesses on the affine space of all Hermitian matrices of trace one by utilizing the correspondence between faces and subspaces.
For a given face, we may assign a number k with the following property: there exist Schmidt number i-witnesses outside of the face if and only if i ≤ k. The corresponding face of a subspace E has this property if and only if the minimum of the Schmidt ranks of vectors in the orthogonal complement of E is precisely k. For generic subspaces E of a specific dimension, this number k depends only on the dimension of the subspace. Furthermore, we show that there exist Schmidt number i-witnesses for i > k in any neighborhood of the projection state located at the center of the face.
This talk is based on two papers:
- Global locations of Schmidt number witnesses, Phys. Rev. A 112 (2025), 032426 (with Kyung Hoon Han)
- Local geometry for Schmidt number witnesses, arXiv 2608.14199 (with Young-Hoon Kiem)