Exact Finite-Time Limits of Controlled Qubit Memories

We determine the exact performance of a qubit memory subjected to a prescribed finite sequence of independent unital noise channels. The controller may use adaptive instruments, fresh ancillas, unlimited classical records, and conditional recovery, but only one qubit persists between noise steps. If s(j,1) ≥ s(j,2) ≥ s(j,3) are the Bloch singular values of step j and P = Σ(i=1 to 3) Π(j=1 to m) s(j,i), both the optimal average and worst-case pure-state fidelity equal max{2/3, 1/2 + P/6}. We also obtain the exact entanglement fidelity, diamond-distance error, and a concurrence ceiling max{0, (P − 1)/2} valid even for selective branches. Every optimum is attained by a finite protocol. The proof bounds the singular values of each noise derivative in the Klein hyperbolic metric, composes the bounds through arbitrary filters, and combines the resulting normal-form inequality with established two-qubit filtering results. It follows that selective adaptive control cannot extend the entanglement lifetime beyond unitary axis alignment. Two separate consequences concern heralded control of positivity-improving nonunital noise and qualitative entanglement-source placement. The deterministic storage theorem does not cover general nonunital noise, additional persistent qubits, or correlated noise. A visual guide records the reasoning routes, their joint proof premises, and the distinction between inherited tools and the present results.