Large violation of Bell inequalities with low entanglement

TL;DR

This paper demonstrates large violations of bipartite Bell inequalities with low entanglement, providing explicit constructions and analyzing the role of entanglement and state choice in such violations.

Contribution

It constructs explicit Bell inequalities and quantum states achieving large violations with low entanglement, challenging the assumption that high entanglement is necessary.

Findings

Violations of order √n / log n with n inputs and outputs

Explicit construction of Bell inequalities and states for violations

Maximally entangled states are suboptimal for arbitrary coefficients

Abstract

In this paper we obtain violations of general bipartite Bell inequalities of order $\frac{\sqrt{n}}{\log n}$ with $n$ inputs, $n$ outputs and $n$-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a random choice of signs, all the elements involved in such violations: the coefficients of the Bell inequalities, POVMs measurements and quantum states. Analyzing this construction we find that, even though entanglement is necessary to obtain violation of Bell inequalities, the Entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.