On Bell inequality violations with high-dimensional systems

TL;DR

This paper investigates Bell inequality violations in high-dimensional quantum systems, clarifying misconceptions about Tsirelson's bound, and proposes a feasible method to verify d-dimensional entanglement under fair sampling conditions.

Contribution

It explains the overstepping of Tsirelson's bound as a fair sampling violation and introduces a practical test for confirming high-dimensional entanglement.

Findings

Tsirelson's bound violation can be due to fair sampling issues.

A method to verify d-dimensional entanglement using Bell inequalities is proposed.

The test is feasible for high-dimensional quantum systems.

Abstract

Quantum correlations resulting in violations of Bell inequalities have generated a lot of interest in quantum information science and fundamental physics. In this paper, we address some questions that become relevant in Bell-type tests involving systems with local dimension greater than 2. For CHSH-Bell tests within 2-dimensional subspaces of such high-dimensional systems, it has been suggested that experimental violation of Tsirelson's bound indicates that more than 2-dimensional entanglement was present. We explain that the overstepping of Tsirelson's bound is due to violation of fair sampling, and can in general be reproduced by a separable state, if fair sampling is violated. For a class of Bell-type inequalities generalized to d-dimensional systems, we then consider what level of violation is required to guarantee d-dimensional entanglement of the tested state, when fair sampling is satisfied. We find that this can be used as an experimentally feasible test of d-dimensional entanglement for up to quite high values of d.