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

Abstract

We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension $K>1$, including impure and nonadditive codes. The result covers every integer local dimension $p\ge2$, not necessarily a prime power, every integer $t\ge1$, both distances $2t+1$ and $2t+2$, and every interpolation parameter $e\in\{0,\ldots,t\}$. The case $e=0$ implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer $p\ge2$ and every quantum code of length $n$, dimension $K>1$ and distance $d\ge3$, we prove $Kf(0)\le p^n\mathbb{E} f(X)$, where $X\sim\operatorname{Bin}(n,1-p^{-2})$, for every real polynomial $f$ of degree less than $d$ that is nonnegative on $\{0,\ldots,n\}$ and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.