The Quantum Hamming Bound in Arbitrary Local Dimension

Abstract

The quantum Hamming bound is the finite-length sphere-packing count for exact quantum error correction: the code dimension times the number of correctable local error patterns must fit inside the ambient Hilbert space. For nondegenerate codes this follows from disjoint error spheres. Degeneracy is the only obstruction, because distinct physical errors can coincide on the code subspace and turn sphere packing into an overcount. The central finite-length question has been whether this overcount can ever invalidate the Hamming inequality. Earlier linear-programming, asymptotic, and structural results left a pointwise finite-length problem for arbitrary exact subspace codes. Writing $Q=q^2$ for the Hamming-scheme alphabet, the nonbinary range begins at $Q=9$; here we prove the bound for every $q\ge3$, while the binary endpoint is governed by a distinct $Q=4$ charging geometry. For every nontrivial exact subspace code in this range, any possible violation reduces to a two-center Hamming-ball intersection inequality normalized by the Lloyd response. For $q\ge4$, the Lloyd-square linear program has a uniform half-gap after reduction to alphabet $Q\ge16$ and critical length $n=4t+1$. Qutrits form the boundary: the half-gap disappears, but the bridge is handled by a quadratic-filtered Lloyd square, an exact coefficient-certificate reduction, and a Stein-tangent positivity argument. Thus degeneracy may merge error sectors, but not enough to beat the Hamming count. This proves the nonbinary part of the finite-length quantum Hamming bound; together with the independent binary endpoint theorem, it gives the result in arbitrary local dimension.