Storage and retrieval of von Neumann measurements via indefinite causal order structures

This work presents the problem of learning an unknown von Neumann measurement of dimension $d$ using indefinite causal structures. In the considered scenario, we have access to $N$ copies of the measurement. We use the formalism of process matrices to store information about the given measurement that later will be used to reproduce its best possible approximation. Our goal is to compute the maximum value of the average fidelity function ${F}_{d}(N)$ of our procedure. We prove that ${F}_{d}(N)=1\ensuremath{-}\mathrm{\ensuremath{\Theta}}(\frac{1}{{N}^{2}})$ for arbitrary but fixed dimension $d$. Furthermore, we present the SDP program (semi-definite program) for computing ${F}_{d}(N)$. Basing on the numerical investigation, we show that for the qubit von Neumann measurements using indefinite causal learning structures provide better approximation than quantum networks, starting from $N\ensuremath{\ge}3$.