  We prove that for a programmable measurement device that approximates every
POVM with an error $\le \delta$, the dimension of the program space has to grow
at least polynomially with $\frac{1}{\delta}$. In the case of qubits we can
improve the general result by showing a linear growth. This proves the
optimality of the programmable measurement devices recently designed in [G. M.
D'Ariano and P. Perinotti, Phys. Rev. Lett. \textbf{94}, 090401 (2005)].
