Pawel Horodecki

Quantum communication complexity advantage implies violation of a Bell inequality

Significance For many communication complexity problems the quantum strategies, distinguished by using Bell nonlocal correlations, provide exponential advantage over the best possible classical strategies. Conversely, for any Bell nonlocal correlations there exists a communication complexity problem that is solved more efficiently using the former. Despite many efforts, there were only two problems for which one could certify that any strategy that outperforms the classical one must harbor Bell nonlocal correlations. We prove that any large advantage over the best known classical strategy makes use of Bell nonlocal correlations. Thus, we provide the missing link to the fundamental equivalence between Bell nonlocality and quantum advantage. We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.