to be $J$ and ${\varphi }$, respectively, considered as mappings into $E$. Define $A_E= ((A_l)_{l=0}^{k-1},(b_l)_{l=0}^{k-1},{\varphi }_E)$. Then $A_E$ is a quantum algorithm from $F$ to $E$ with $n_q(A_E)\le n_1$. By (\[QT5\]) and (\[KC4\]), $$e_{n_1}^{{\rm q }}(J_E,F)\le e(J_E,A_E,F)\le {\sigma }+\delta=e_{n_1}^{{\rm q }}(J,F)+\delta.$$ Note that, since $F\subseteq E$, $\Lambda$ is linearly independent over $E$. Furthermore, since $B_E=\mathcal{E}$ is bounded in $Y$, the restriction of $T$ to $E$ is a bounded linear operator from $E$ to $G$. Applying now the first part of Proposition \[pro:4\], we get $$e^{{\rm q }}_{\nu_1 n_1+2\nu_2 n_2}(S,F)\le 
4 e^{{\rm q }}_{n_1}(J_E,F)\,e^{{\rm q }}_{n_2}(T,B_E)\le 4 (e^{{\rm q }}_{n_1}(J,F)+\delta)\,e^{{\rm q }}_{n_2}(T,\mathcal{E}),$$ which gives the desired result, since $\delta>0$ was arbitrary.

Let $e_n^{{\rm det}}(S,F)$ denote the (classical) $n$-th minimal deterministic error, that is, the minimal error among all deterministic, adaptive algorithms using at most $n$ informations (see, e.g., [@Hei05a], section 4). We want to apply relations (\[AJ4\]) and (\[QT3\]) with $e^{{\rm q }}_{n_1}(J,F)$ replaced by $e^{{\rm det}}_{n_1}(J,F)$. An estimate of $e_n^{{\rm q }}(S,F)$ by $e_n^{{\rm det}}(S,F)$ is not obvious, since classical deterministic algorithms can use information with values in ${{\mathbb K}}$ directly, while quantum algorithms can use them only through a finite encoding. We therefore supply the following

\[lem:3\] Let $K={{\mathbb 