A new proof confirms the conjecture of Yu, Lai and Oh on a rounded family of strengthened quantum Hamming bounds, which interpolates between Hamming-type bounds and the Singleton bound. The result holds for arbitrary exact quantum subspaces of dimension K > 1, for every integer local dimension p ≥ 2, every integer t ≥ 1, both distances 2t + 1 and 2t + 2, and every interpolation parameter e ∈ {0, ..., t}.

Yu, Lai and Oh had previously established 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. The conjecture remained open for the remaining combinations of parameters, and its resolution closes a gap in the theory of upper bounds for quantum error-correcting codes.

The proof rests on a more general inequality: for p ≥ 3, it is shown that K f(0) ≤ pⁿ 𝔼 f(X), where X follows a binomial distribution Bin(n, 1 − p⁻²), for every real polynomial f of degree at most D, with 2 ≤ D < d, nonnegative on {0, ..., n} and positive at zero. Here n and d are the code length and distance. This polynomial formulation encompasses the family of bounds as particular cases.

The advance has direct implications for the construction of quantum codes: Hamming and Singleton bounds set the theoretical maximum of logical qubits a code can store given its length and distance, and knowing their exact form allows a rigorous assessment of how close known codes come to being optimal. The interpolating family also offers a continuum of restrictions that may guide the search for new exact quantum codes.

It remains to be seen whether similar techniques can extend the result to non-integer local dimensions or to non-exact codes, and whether the underlying polynomial inequality leads to still tighter bounds for quantum code parameters yet to be determined.