Non-homogeneous Bell-type Inequalities for Two- and Three-qubit States

TL;DR

This paper introduces a systematic method to construct non-homogeneous Bell inequalities for two- and three-qubit systems, revealing new insights into quantum violations and detection efficiency thresholds.

Contribution

It presents a novel systematic approach to derive non-homogeneous Bell inequalities, unifies existing inequalities in the three-qubit case, and analyzes their behavior in loophole-free tests.

Findings

Maximal quantum violation approaches a constant with large subtracted terms.

Most significant three-qubit inequalities are recoverable within this framework.

Thresholds of detection efficiency are obtained for loophole-free Bell tests.

Abstract

A systematic approach is presented to construct non-homogeneous two- and three-qubit Bell-type inequalities. When projector-like terms are subtracted from homogeneous two-qubit CHSH polynomial, non-homogeneous inequalities are attained and the maximal quantum mechanical violation asymptotically equals a constant with the subtracted terms becoming sufficiently large. In the case of three-qubit system, it is found that most significant three-qubit inequalities presented in literature can be recovered in our framework. We aslo discuss the behavior of such inequalities in the loophole-free Bell test and obtain corresponding thresholds of detection efficiency.