Storage and retrieval of von Neumann measurements

TL;DR

This paper investigates the asymptotic behavior of learning unknown von Neumann measurements using finite copies, establishing the fidelity's decay rate and comparing different learning schemes for qubits.

Contribution

It derives the asymptotic fidelity scaling for measurement learning and compares the performance of various schemes, including port-based teleportation and a new parallel approach.

Findings

Fidelity approaches 1 at a rate of 1/N^2 for fixed dimension d

Port-based teleportation scheme is asymptotically optimal but slow for small N

A new parallel scheme achieves high fidelity with minimal entanglement for low N

Abstract

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 $N \rightarrow 1$ learning scheme. We show that $F_d = 1 - \Theta\left(\frac{1}{N^2}\right)$ 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.