Universal Cloning Is Optimal Exactly for Designs

Werner's universal cloner turns T copies of an arbitrary pure state in C^d into T+k approximate copies with optimal global fidelity D_T/D_{T+k}, where D_m = dim Sym^m(C^d). When the input is drawn from a restricted ensemble, a better cloner may exist. We show that, for optimal average global fidelity, it exists unless the ensemble is a design of order T+k: for every probability measure on pure states and all T, k ≥ 1, the optimal value equals D_T/D_{T+k} if and only if the measure is a complex projective (T+k)-design. Sufficiency follows from known arguments; the new content is necessity, which we prove from semidefinite-programming complementary slackness together with a rigidity lemma for operators acting on symmetric tensors. As corollaries, Werner's cloner is optimal for the uniform ensemble of n-qubit stabilizer states exactly when T+k ≤ 3 (for every n), for symmetric informationally complete ensembles exactly when T+k ≤ 2, and for complete sets of mutually unbiased bases exactly when T+k ≤ 3 in d = 2 and T+k ≤ 2 in d ≥ 3. We give exact rational certificates for the first strict gaps, and we contrast the result with structured state families, where matching moments is neither necessary nor sufficient for optimality. Files: the paper (PDF) and its LaTeX source; paper2_certificates.zip, the exact-arithmetic certificate scripts with their outputs, README and SHA-256 checksums (MIT licence); SHA256SUMS.txt lists the SHA-256 hashes of all files. Companion paper: doi:10.5281/zenodo.23428376.