# Does violation of a Bell inequality always imply quantum advantage in a communication complexity problem?

## TL;DR

This paper investigates whether violating Bell inequalities always leads to quantum advantages in communication complexity problems, finding that certain quantum correlations do not imply such advantages under specific conditions.

## Contribution

It demonstrates that Bell inequality violations do not necessarily guarantee quantum advantages in CCPs when classical models are not restricted by no-signaling constraints.

## Key findings

- Violations of correlation-type Bell inequalities do not always imply CCP advantages.
- Quantum correlations from small Bell violations may not provide any CCP advantage.
- Existence of quantum correlations violating $I_{3322}$ without enabling CCP advantages.

## Abstract

Quantum correlations which violate a Bell inequality are presumed to power better-than-classical protocols for solving communication complexity problems (CCPs). How general is this statement? We show that violations of correlation-type Bell inequalities allow advantages in CCPs, when communication protocols are tailored to emulate the Bell no-signaling constraint (by not communicating measurement settings). Abandonment of this restriction on classical models allows us to disprove the main result of, inter alia, [Brukner et. al., Phys Rev. Lett. 89, 197901 (2002)]; we show that quantum correlations obtained from these communication strategies assisted by a small quantum violation of the CGLMP Bell inequalities do not imply advantages in any CCP in the input/output scenario considered in the reference. More generally, we show that there exists quantum correlations, with nontrivial local marginal probabilities, which violate the $I_{3322}$ Bell inequality, but do not enable a quantum advantange in any CCP, regardless of the communication strategy employed in the quantum protocol, for a scenario with a fixed number of inputs and outputs