Certifying measurement incompatibility in prepare-and-measure and Bell scenarios

TL;DR

This paper investigates the certification of measurement incompatibility in quantum prepare-and-measure scenarios, establishing a link with Bell scenarios and introducing a hierarchy of incompatibility notions with concrete examples.

Contribution

It provides a general theorem connecting measurement incompatibility certification in PM and Bell scenarios, and introduces a hierarchy of incompatibility notions with optimal examples for qubits.

Findings

Incompatibility can be certified in PM scenarios iff in Bell scenarios with maximally entangled states.

A hierarchy of incompatibility notions is proposed, increasing classical simulation power.

An example shows qubit incompatibility can be certified against trit simulations, the strongest for qubits.

Abstract

We consider the problem of certifying measurement incompatibility in a prepare-and-measure (PM) scenario. We present different families of sets of qubit measurements which are incompatible, but cannot lead to any quantum over classical advantage in PM scenarios. Our examples are obtained via a general theorem which proves a set of qubit dichotomic measurements can have their incompatibility certified in a PM scenario if and only if their incompatibility can be certified in a bipartite Bell scenario where the parties share a maximally entangled state. Our framework naturally suggests a hierarchy of increasingly stronger notions of incompatibility, in which more power is given to the classical simulation by increasing its dimensionality. For qubits, we give an example of measurements whose incompatibility can be certified against trit simulations, which we show is the strongest possible notion for qubits in this framework.