Theorem: Tolerance schedule

Fix a packet with finite ${\nu }_h$ and a requested mean tolerance $\epsilon >0$. Any admissible model parameters satisfying

give

In the family of idealized models allowing arbitrary finite windows, capture transmissivity approaching one, arbitrarily accurate independent preparation, no fixed readout or control error floor, and the stated selective controls for every finite $K$, such parameters exist.

Proof. The first assertion is (eq:modelbounds).

For existence in the stated model family, choose $N$ by geometric decay; if ${\nu }_h=0$, the ideal bias already vanishes. Then choose finite $K\ge N$ with ${q_h}^{K+1}$ sufficiently small. Choose $L$ large enough that ${\mu }_L$ is sufficiently small, using Lemmas lem:window–lem:modeerror. For this finite $L$, $E_f<\infty$. Choose ${\tau }_{\ast }<1$ sufficiently close to one that $\ell (E_f,{\tau }_{\ast })$ is small. The corresponding capture coupling is smooth with finite peak for each chosen parameter.

Choose $m_0,p_Q$ and the bounded pulse deviations sufficiently small; exact pulses are included in the family. Choose the readiness, pointer-preparation, damping, basis, cross-talk, pointer-control, and assignment parameters so that (eq:readbudget) meets the final allowance. There are finitely many error terms, so these choices meet the two required budgets.