Bound on Bell Inequalities by Fraction of Determinism and Reverse Triangle Inequality

TL;DR

This paper introduces a new mathematical bound based on the fraction of determinism and a reverse triangle inequality to quantify limitations on Bell inequality violations with 2 by n inputs.

Contribution

It establishes a universal upper bound on Bell inequalities for 2 by n inputs using the concept of fraction of determinism and a novel reverse triangle inequality.

Findings

Quantum correlations cannot reach algebraic bounds for 2 by n Bell inequalities.

Any quantum statistics with 2 parties and 2 by n inputs has a nonzero fraction of determinism.

Derived a universal upper bound for Bell inequality violations in this setting.

Abstract

It is an established fact that entanglement is a resource. Sharing an entangled state leads to non-local correlations and to violations of Bell inequalities. Such non-local correlations illustrate the advantage of quantum resources over classical resources. Here, we study quantitatively Bell inequalities with $2\times n$ inputs. As found in [N. Gisin et al., Int. J. Q. Inf. 5, 525 (2007)] quantum mechanical correlations cannot reach the algebraic bound for such inequalities. In this paper, we uncover the heart of this effect which we call the {\it fraction of determinism}. We show that any quantum statistics with two parties and $2 \times n$ inputs exhibits nonzero fraction of determinism, and we supply a quantitative bound for it. We then apply it to provide an explicit {\it universal upper bound} for Bell inequalities with $2\times n$ inputs. As our main mathematical tool we introduce and prove a {\it 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.