Maximal violation of the I3322 inequality using infinite dimensional quantum systems

TL;DR

This paper demonstrates that the maximal quantum violation of the I3322 Bell inequality requires infinite dimensional quantum systems, providing a new measurement construction and an algorithm for computing quantum maxima.

Contribution

It introduces a measurement and state construction that achieves the maximum quantum violation of I3322 in infinite dimensions and proposes an algorithm for quantum maximum computation.

Findings

Maximal violation achieved in infinite-dimensional systems

Finite-dimensional systems are insufficient for the true maximum

Algorithm improves bounds for multiple Bell inequalities

Abstract

The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the Clauser-Horne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.