The uncertainty principle does not entirely determine the non-locality of quantum theory

Ravishankar Ramanathan, Dardo Goyeneche, Piotr Mironowicz, Paweł Horodecki

Published 2015-06-16Version 1

One of the most intriguing discoveries regarding quantum non-local correlations in recent years was the establishment of a direct correspondence between the quantum value of non-local games and the strength of the fine-grained uncertainty relations in \textit{Science, vol. 330, no. 6007, 1072 (2010)}. It was shown that while the degree of non-locality in any theory is generally determined by a combination of two factors - the strength of the uncertainty principle and the degree of steering allowed in the theory, the most paradigmatic games in quantum theory have degree of non-locality purely determined by the uncertainty principle alone. In this context, the fundamental question arises: is this a universal property of optimal quantum strategies for all non-local games? Indeed, the above mentioned feature occurs in surprising situations, even when the optimal strategy for the game involves non-maximally entangled states. However, here we definitively prove that the answer to the question is negative, by presenting explicit counter-examples of non-local games and fully analytical optimal quantum strategies for these, where a definite trade-off between steering and uncertainty is absolutely necessary. We provide an intuitive explanation in terms of the Hughston-Jozsa-Wootters theorem for when the relationship between the uncertainty principle and the quantum game value breaks down.