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Integral zeros of Krawtchouk polynomials and quantum entanglements

  • Jaeseong Heo
  • Young-Hoon Kiem
  • Chul-hee Lee
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Abstract

In Heo and Kiem (2019), the authors raised a problem relating integral zeros of binary Krawtchouk polynomials, parametrized by quadruples (m,n,k,ℓ) with m≤ n and k≤ ℓ, to the existence of pairs of subspaces satisfying the range criterion in quantum information theory, and settled the symmetric case k=ℓ. In this paper, we solve the problem for the following three asymmetric families: (m,n,k,ℓ)=(3,t2−1,t(t−1)/2,t(t+1)/2), t≥2; (m,n,k,ℓ)=(4,3t2+2t−1,(3t2−t)/2,(3t2+5t+2)/2), t≥2; (m,n,k,ℓ)=(4,3s2+4s,(3s2+s)/2,(3s2+7s+4)/2), s≥1. The remaining cases are sporadic zeros, with only finitely many for each m≥5.

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Cite

BibTeXRIS
BibTeX
@misc{heo2026integral,
  title = {{Integral zeros of Krawtchouk polynomials and quantum entanglements}},
  author = {Jaeseong Heo and Young-Hoon Kiem and Chul-hee Lee},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08518},
  eprint = {2611.08518},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08518},
}
RIS
TY  - UNPB
TI  - Integral zeros of Krawtchouk polynomials and quantum entanglements
AU  - Jaeseong Heo
AU  - Young-Hoon Kiem
AU  - Chul-hee Lee
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.08518
UR  - https://arxiv.org/abs/2611.08518
AB  - In Heo and Kiem (2019), the authors raised a problem relating integral zeros of binary Krawtchouk polynomials, parametrized by quadruples (m,n,k,ℓ) with m≤ n and k≤ ℓ, to the existence of pairs of subspaces satisfying the range criterion in quantum information theory, and settled the symmetric case k=ℓ. In this paper, we solve the problem for the following three asymmetric families: (m,n,k,ℓ)=(3,t^2−1,t(t−1)/2,t(t+1)/2), t≥2; (m,n,k,ℓ)=(4,3t^2+2t−1,(3t^2−t)/2,(3t^2+5t+2)/2), t≥2; (m,n,k,ℓ)=(4,3s^2+4s,(3s^2+s)/2,(3s^2+7s+4)/2), s≥1. The remaining cases are sporadic zeros, with only finitely many for each m≥5.
ER  -