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

Armin Tavakoli1, Marek Żukowski2, and Časlav Brukner3,4

1Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland
2International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdansk, Poland
3Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
4Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

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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, [22]; 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

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[2] A. K. Pan, "Oblivious communication game, self-testing of projective and nonprojective measurements, and certification of randomness", Physical Review A 104 2, 022212 (2021).

[3] Marcin Wieśniak, "Symmetrized persistency of Bell correlations for Dicke states and GHZ-based mixtures", Scientific Reports 11 1, 14333 (2021).

[4] Zhih-Ahn Jia, Lu Wei, Yu-Chun Wu, and Guang-Can Guo, "Quantum Advantages of Communication Complexity from Bell Nonlocality", Entropy 23 6, 744 (2021).

[5] Joseph Ho, George Moreno, Samuraí Brito, Francesco Graffitti, Christopher L. Morrison, Ranieri Nery, Alexander Pickston, Massimiliano Proietti, Rafael Rabelo, Alessandro Fedrizzi, and Rafael Chaves, "Quantum communication complexity beyond Bell nonlocality", arXiv:2106.06552.

The above citations are from Crossref's cited-by service (last updated successfully 2021-09-23 15:00:10) and SAO/NASA ADS (last updated successfully 2021-09-23 15:00:12). The list may be incomplete as not all publishers provide suitable and complete citation data.