Storage and retrieval of von Neumann measurements

This work examines the problem of learning an unknown von Neumann measurement dimension $d$ from a finite number copies. To obtain faithful approximation given measurement, we are allowed to use it $N$ times. Our main goal is estimate asymptotic behavior maximum value average fidelity function ${F}_{d}$ for general $N\ensuremath{\rightarrow}1$ scheme. We show that ${F}_{d}=1\ensuremath{-}\mathrm{\ensuremath{\Theta}}(\frac{1}{{N}^{2}})$ arbitrary but fixed $d$. In addition that, compared various schemes $d=2$. observed scheme based on deterministic port-based teleportation asymptotically optimal performs poorly low $N$. particular, discovered parallel scheme, which despite its lack optimality, provides high values and uses only two-qubit entangled memory states.