Exact Optimal Cloning of Stabilizer States

We determine the optimal average global fidelity of T→T+k approximate cloning for the uniform ensemble of n-qubit stabilizer states, for every number of qubits n, every k ≥ 1 and every 1 ≤ T ≤ 3. With D = 2^n, the optimum is C(D+T−1, T)·FT+k−κ(k), where F_s is the stabilizer frame potential and κ(k) is the largest dimension of a doubly-even binary code of length k. The result rests on an exact formula for the largest eigenvalue of the mixed stabilizer moment operators: for all a+b ≥ 1 and all n it equals Fa+b−w, where w = min(a,b) + κ(|a−b|) is the Witt index of an associated Z4-valued quadratic form. The proof works in the algebra of stochastic Lagrangian relations of Gross, Nezami and Walter and is analytic. Consequences: the universal cloner is optimal exactly when T+k ≤ 3; requesting one more copy (k → k+1) leaves the optimum unchanged exactly when κ increases, i.e. when k+1 ≡ 0, 4, 6 or 7 (mod 8); and as k → ∞ the optimum converges to the optimal probability of identifying the state from T copies. Files: the paper (PDF) and its LaTeX source (main3.tex, refs3.bib, main3.bbl); paper3_code.zip, certificate scripts and outputs plus an independent implementation of the argument's key steps; SHA256SUMS.txt.