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

TL;DR

This paper investigates how indefinite causal structures can be used to learn and approximate unknown von Neumann measurements, demonstrating improved fidelity over traditional quantum networks for qubits with multiple copies.

Contribution

It introduces a formalism using process matrices to store and reproduce von Neumann measurements, providing a method to optimize measurement approximation fidelity.

Findings

Maximum average fidelity approaches 1 as 1 - 1/N^2

Indefinite causal structures outperform quantum networks for qubits with N ≥ 3

Provides SDP program for fidelity computation

Abstract

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 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 $F_d(N) = 1 - \Theta \left( \frac{1}{N^2}\right)$ for arbitrary but fixed dimension $d$. Furthermore, we present the SDP 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 $N \ge 3$.