# Distributed sampling, quantum communication witnesses, and measurement incompatibility

## TL;DR

This paper investigates how quantum communication can be certified in prepare-and-measure experiments with untrusted devices, revealing that measurement incompatibility is essential for quantum advantage in distributed sampling tasks.

## Contribution

It establishes a link between quantum communication certification and measurement incompatibility, providing an operational interpretation and a generalized notion of incompatibility.

## Key findings

- Quantum communication can be certified via measurement-device-independent witnesses.
- Quantum advantage in distributed sampling requires incompatible measurements.
- The work offers fundamental insights into measurement incompatibility and quantum communication.

## Abstract

We study prepare-and-measure experiments where the sender (Alice) receives trusted quantum inputs but has an untrusted state-preparation device and the receiver (Bob) has a fully-untrusted measurement device. A distributed-sampling task naturally arises in such scenario, whose goal is for Alice and Bob to reproduce the statistics of his measurements on her quantum inputs using a fixed communication channel. Their performance at such task can certify quantum communication (QC), and this is formalised by measurement-device-independent QC witnesses. Furthermore, we prove that QC can provide an advantage (over classical communication) for distributed sampling if and only if Bob's measurements are incompatible. This gives an operational interpretation to the fundamental notion of measurement incompatibility, and motivates a generalised notion of it. Our findings have both fundamental and applied implications.