Storage and retrieval of von Neumann measurements
Paulina Lewandowska, Ryszard Kukulski, Łukasz Pawela, Zbigniew Puchała · Physical Review A · 2022
This work examines the problem of learning an unknown von Neumann measurement of dimension $d$ from a finite number of copies. To obtain a faithful approximation of the given measurement, we are allowed to use it $N$ times. Our main goal is to estimate the asymptotic behavior of the maximum value of the average fidelity function ${F}_{d}$ for a general $N\ensuremath{\rightarrow}1$ learning scheme. We show that ${F}_{d}=1\ensuremath{-}\mathrm{\ensuremath{\Theta}}(\frac{1}{{N}^{2}})$ for arbitrary but fixed dimension $d$. In addition to that, we compared various learning schemes for $d=2$. We observed that the learning scheme based on deterministic port-based teleportation is asymptotically optimal but performs poorly for low $N$. In particular, we discovered a parallel learning scheme, which despite its lack of asymptotic optimality, provides a high value of the fidelity for low values of $N$ and uses only two-qubit entangled memory states.