Measurement incompatibility and quantum advantage in communication

TL;DR

This paper demonstrates that measurement incompatibility is essential for quantum advantage in communication and introduces a class of tasks to witness incompatibility of quantum measurements, providing bounds and characterizations for various measurement sets.

Contribution

It establishes the equivalence of classical and quantum strategies restricted to compatible measurements, linking measurement incompatibility directly to quantum communication advantage, and introduces new communication tasks for witnessing incompatibility.

Findings

Measurement incompatibility is necessary for quantum advantage in communication.

A class of communication tasks can witness measurement incompatibility.

Identified all sets of three incompatible qubit measurements that can be witnessed.

Abstract

Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems assisted with shared randomness is the same as the set obtained from d-dimensional quantum strategies restricted to compatible measurements with shared randomness in any communication scenario. Thus, measurement incompatibility is necessary for quantum advantage in communication, and any quantum advantage (with or without shared randomness) in communication acts as a witness to the incompatibility of the measurements at the receiver's end in a semi-device-independent way. We introduce a class of communication tasks - a general version of random access codes - to witness incompatibility of an arbitrary number of quantum measurements with arbitrary outcomes acting on d-dimensional systems, and provide generic upper bounds on the success metric of these tasks for compatible measurements. We identify all sets of three incompatible rank-one projective qubit measurements that random access codes can witness. Finally, we present the generic relationship between different sets of probability distributions - classical, quantum with or without shared randomness, and quantum restricted to compatible measurements with or without shared randomness - produced in communication scenarios.