Sidon Sets, Additive Energy, and the Optimal Cloning of Finite Phase-State Codebooks

Phase states |ψ_x⟩ = d^(−1/2) Σ_j ω^(x k_j) |j⟩, indexed by x ∈ Z_q and a key set K = {k_1,…,k_d} ⊂ Z_q, are the state family behind quantum fingerprinting and quantum hashing. We ask how well an adversary holding T copies of |ψ_x⟩ can produce T+k copies, and when a finite codebook is exactly as hard to clone as the continuous family of all phase states (the torus), whose optimal cloning fidelity was recently determined. We show that the answer is governed by additive combinatorics of K. (i) The optimal 1→2 global cloning fidelity equals (2d−1)/d² if and only if K is a Sidon set. (ii) A B_{T+k} key set attains the torus optimum exactly in the cases we prove: analytically for d = 2 (all T) and for T = 1 (all d, k), and for T = 2, 3 (all d, k) by a proof whose finitely many computational steps are verified by standalone exact and interval-arithmetic checkers; we conjecture it for all T. (iii) The optimal fidelity of fusing |ψ_x⟩⊗|ψ_y⟩ into |ψ_{x+y}⟩ equals E_q(K)/d³, where E_q(K) is the additive energy. Small Fourier bias, the standard design criterion for quantum hashing, does not control cloning: numerically, for d ≥ 3 the Sidon sets we tested that are not B_3 already fail at T+k = 3, and arithmetic progressions are worst in most cases tested. For finite orbits of non-abelian groups we prove a window theorem giving a dual-feasibility criterion, an obstruction, and a representation-theoretic selection rule for when matching the (T+k)-th moment suffices, and we give certified counterexamples for SU(3) and SU(2) showing that moment matching alone does not. Files: the paper (PDF) and its LaTeX source; ca_appendix.zip, the computer-assisted certificates with standalone exact/interval/ball-arithmetic checkers; coverage_check.py, the region-coverage audit; qh-lab-v1.0.0.zip, an independent re-implementation with 46 tests (MIT licence). SHA256SUMS.txt lists the SHA-256 hashes of all files.