Exact Robustness Frontiers and Fidelity Limits for Qubit Memories

We determine exact resource limits for a qubit memory with retained classical records. For every qubit noise channel A, the largest output generalized entanglement-breaking robustness under an input budget R_G ≤ r is a semidefinite program in two 4 × 4 positive matrices, attained with at most five classical outcomes. For amplitude damping we solve the full budget curve, including two mixed-state regimes, and attain it with one classical bit. Its contraction coefficient is one, approached with vanishing input resource, although every fixed positive resource budget contracts strictly. A different argument controls memories prepared from a perfect qubit: for prescribed independent noise channels A_1, …, A_n, arbitrary adaptive instruments and finite classical records obey R_G(Γ_n) ≤ R(Γ_n) ≤ ∏_j ν(A_j), where ν(A) = max{0, 2λ_max J_A − 1} and J_A is normalized. The proof propagates a positive 2 × 2 operator over the branches. For amplitude-damping survivals with product b, both robustness optima equal b; the optimal average and worst-case pure-state fidelities are (2+b)/3, the optimal Bell fidelity is (1+b)/2, and the minimum half-diamond error is (1−b)/2. Finite protocols attain all four operational optima simultaneously. Thus classical control can improve fidelity while leaving optimal robustness unchanged. The model permits one persistent qubit and excludes quantum bypass, environment access, and postselection. A visual guide records the reasoning routes, their joint premises, and the distinction between prior tools and the results developed here.