Quantum Bounds on Bell inequalities

TL;DR

This paper determines the maximum quantum violations of 241 bipartite Bell inequalities with multiple measurement settings, revealing the role of higher-dimensional spaces and providing bounds and detection efficiencies.

Contribution

It introduces 129 new Bell inequalities and explores the maximum quantum violations across different Hilbert space dimensions using numerical optimization.

Findings

Higher-dimensional spaces can yield larger violations than qubits.

Maximum violations often match the upper bounds, validating the optimization method.

Detection efficiencies can be lowered with degenerate measurements.

Abstract

We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real component Hilbert spaces using numerical optimization. Out of these inequalities 129 has been introduced here. In 43 cases higher dimensional component spaces gave larger violation than qubits, and in 3 occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.