Integral zeros of Krawtchouk polynomials and quantum entanglements

arXiv:2611.08518v1 Announce Type: new Abstract: In Heo and Kiem (2019), the authors raised a problem relating integral zeros of binary Krawtchouk polynomials, parametrized by quadruples $(m,n,k,\ell)$ with $m\le n$ and $k\le \ell$, to the existence of pairs of subspaces satisfying the range criterion in quantum information theory, and settled the symmetric case $k=\ell$. In this paper, we solve the problem for the following three asymmetric families: $(m,n,k,\ell)=(3,t^2-1,t(t-1)/2,t(t+1)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3t^2+2t-1,(3t^2-t)/2,(3t^2+5t+2)/2)$, $t\ge2$; $(m,n,k,\ell)=(4,3s^2+4s,(3s^2+s)/2,(3s^2+7s+4)/2)$, $s\ge1$. The remaining cases are sporadic zeros, with only finitely many for each $m\ge5$.