Efficiency of higher dimensional Hilbert spaces for the violation of Bell inequalities

TL;DR

This study numerically investigates the maximum quantum violations of bipartite Bell inequalities using higher-dimensional Hilbert spaces, revealing that qubits are often insufficient and that maximum violations do not always involve maximally entangled states.

Contribution

It demonstrates that higher-dimensional systems can achieve greater Bell inequality violations and shows that real Hilbert spaces suffice for reproducing bipartite quantum correlations.

Findings

Higher-dimensional Hilbert spaces can surpass qubits in Bell violations.

Maximum violations often involve non-maximally entangled states.

Real Hilbert spaces are sufficient for reproducing bipartite correlations.

Abstract

We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two dimensional systems, i.e., qubits are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate 'measurements' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result, that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore, in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.