The strengthened quantum Hamming bound of Yu, Lai, and Oh

Abstract

We prove a conjecture of Yu, Lai, and Oh: their rounded family of strengthened quantum Hamming bounds, which interpolates between Hamming-type bounds and the Singleton bound, holds for arbitrary exact quantum subspaces of dimension $K>1$. The result covers every integer local dimension $p\ge2$, every integer $t\ge1$, both distances $2t+1$ and $2t+2$, and every interpolation parameter $e\in\{0,\ldots,t\}$. Yu, Lai, and Oh proved these bounds for pure codes and for general codes of distance three or four; the case $e=0$ contains the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. More generally, for $p\ge3$, we prove the inequality $Kf(0)\le p^n\mathbb{E}f(X)$, where $X\sim\operatorname{Bin}(n,1-p^{-2})$, for every real polynomial $f$ of degree at most $D$, with $2\le D<d$, that is nonnegative on $\{0,\ldots,n\}$ and positive at zero; here $n$ and $d$ are the code length and distance.