previous papers on quantum complexity [@Hei01; @Hei01b; @Hei04b] the analysis of reductions was somewhat cumbersome, since a certain discretization had to be applied in each particular case. Proposition \[pro:1\] simplifies the analysis and will be used for a number of reductions, in particular in sections 5 and 6.

Next we recall additivity properties of the quantum minimal error, see [@Hei01b], Corollary 2.

\[pro:3\] Let $p\in{{\mathbb N}}$ and let $S_l:F\to G$ $(l=1,\dots,p)$ be mappings such that $S(f)=\sum_{l=1}^p S_l(f)\quad(f\in F)$. Let $\nu_1,\dots,\nu_p\in {{\mathbb N}}$ be numbers satisfying $$\sum_{l=1}^p e^{-\nu_l/8}\le \frac{1}{4}.$$ Then for all $n_1,\dots,n_p\in{{\mathbb N}}_0$ $$e_{\sum_{l=1}^p \nu_l n_l}^{{\rm q }}(S,F)\le2\sum_{l=1}^p e_{n_l}^{{\rm q }}(S_l,F).$$

Given a subset $B\subseteq X$ of a normed space $X$, we denote by $\mathscr{C}(B)$ the set of all precompact subsets of $B$. A set $H\subset X^\#$ is called linearly independent over a non-empty set $B\subseteq X$, if the restrictions of elements of $H$ to ${\rm span}(B)$ form a linearly independent subset of $({\rm span}(B))^\#$.

Finally we state multiplicativity properties of the minimal quantum error.

\[pro:4\] Let $K={{\mathbb K}}$. Assume that $F$ is a subset of a normed space $Y$, that $\Lambda$ is a linearly independent subset of $Y^\#$ and $\sup_{f\in F}|f({{\lambda }})|<\infty$ for each ${{\lambda }}\in \Lambda$. Let $J:F\to 