Bound on Bell inequalities by fraction of determinism and reverse triangle inequality

Abstract

It is an established fact that entanglement is a resource. Sharing an entangled state leads to nonlocal correlations and to violations of Bell inequalities. Such nonlocal correlations illustrate the advantage of quantum resources over classical resources. In this paper, we quantitatively study Bell inequalities with $2\ifmmode\times\else\texttimes\fi{}n$ inputs. As found in Gisin et al. [Int. J. Quantum. Inform. 05, 525 (2007)], quantum mechanical correlations cannot reach the algebraic bound for such inequalities. Here we uncover the heart of this effect, which we call the fraction of determinism. We show that any quantum statistics with two parties and $2\ifmmode\times\else\texttimes\fi{}n$ inputs exhibit a nonzero fraction of determinism, and we supply a quantitative bound for it. We then apply it to provide an explicit universal upper bound for Bell inequalities with $2\ifmmode\times\else\texttimes\fi{}n$ inputs. As our main mathematical tool, we introduce and prove a reverse triangle inequality, stating in a quantitative way that if some states are far away from a given state, then their mixture is also. The inequality is crucial in deriving the lower bound for the fraction of determinism, but is also of interest on its own.