On the optimal certification of von Neumann measurements

TL;DR

This paper investigates the optimal certification of von Neumann measurements in quantum systems, establishing conditions for their distinguishability and certification with statistical significance, and exploring connections to $q$-numerical range.

Contribution

It introduces the first comprehensive framework for certifying von Neumann measurements, extending quantum hypothesis testing to measurement procedures and establishing conditions for perfect distinguishability.

Findings

Derived conditions for perfect distinguishability of quantum states and measurements.

Established the connection between measurement certification and $q$-numerical range.

Provided single-shot and parallel certification protocols for von Neumann measurements.

Abstract

In this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of $q$-numerical range.