Optimality of programmable quantum measurements

We prove that for a programmable measurement device that approximates every POVM with an error $⩽\delta$, 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. 94, 090401 (2005).