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PreprintCC BY 4.0arXiv

Local geometry for Schmidt number witnesses

  • Young-Hoon Kiem
  • Seung-Hyeok Kye
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Abstract

Suppose that FE is the face of the convex set of all m⊗ n bi-partite states which consists of states with ranges contained in a subspace E. By the result in [Phys. Rev. A {\bf 112} (2025), 032426], there exist Schmidt number ℓ witnesses outside of the face FE⊥ if and only if ℓ≤μ, where μ is the minimum of Schmidt ranks of nonzero vectors in E. In this paper, we show that every neighborhood of the projection state located at the center of FE⊥ contains Schmidt number ℓ witnesses if and only if ℓ≥μ. In order to understand the local shape of the convex sets of k-blockpositive matrices around this projection state, we introduce principal supporting hyperplanes, and show that they correspond to vectors in E with Schmidt rank ≤ k. Using techniques from algebraic geometry, we show that the inclusions between these convex sets are detected by principal supporting hyperplanes through the projection state for generic subspaces E, while for non-generic subspaces they may not be detected by any supporting hyperplane.

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Cite

BibTeXRIS
BibTeX
@misc{kiem2026local,
  title = {{Local geometry for Schmidt number witnesses}},
  author = {Young-Hoon Kiem and Seung-Hyeok Kye},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.14199},
  eprint = {2608.14199},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14199},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Local geometry for Schmidt number witnesses
AU  - Young-Hoon Kiem
AU  - Seung-Hyeok Kye
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.14199
UR  - https://arxiv.org/abs/2608.14199
AB  - Suppose that F_E is the face of the convex set of all m⊗ n bi-partite states which consists of states with ranges contained in a subspace E. By the result in [Phys. Rev. A {\bf 112} (2025), 032426], there exist Schmidt number ℓ witnesses outside of the face F_E^⊥ if and only if ℓ≤μ, where μ is the minimum of Schmidt ranks of nonzero vectors in E. In this paper, we show that every neighborhood of the projection state located at the center of F_E^⊥ contains Schmidt number ℓ witnesses if and only if ℓ≥μ. In order to understand the local shape of the convex sets of k-blockpositive matrices around this projection state, we introduce principal supporting hyperplanes, and show that they correspond to vectors in E with Schmidt rank ≤ k. Using techniques from algebraic geometry, we show that the inclusions between these convex sets are detected by principal supporting hyperplanes through the projection state for generic subspaces E, while for non-generic subspaces they may not be detected by any supporting hyperplane.
ER  -