Generalized Iterative Formula for Bell Inequalities

TL;DR

This paper introduces a generalized iterative method for constructing Bell inequalities in multipartite quantum systems, unifying existing families and enabling detection of nonlocality across entire entangled regions.

Contribution

It presents a new iterative formula for generating multipartite Bell inequalities, including generalizations of known inequalities and a family of dual-use inequalities effective for GHZ states.

Findings

Recovers MABK and other known families as special cases

Proposes dual-use inequalities for GHZ states with full nonlocality detection

Generalizes I3322 and Sliwa inequalities to multipartite scenarios

Abstract

Bell inequalities are a vital tool to detect the nonlocal correlations, but the construction of them for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of $(n+1)$-partite Bell inequalities into $n$-partite ones, we present a generalized iterative formula to construct nontrivial $(n+1)$-partite ones from the $n$-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinski{\u{\i}}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the non-locality in the entire entangled region. Furthermore, we present generalizations of the the I3322 inequality to any $n$-partite case which are still tight, and of the $46$ \'{S}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.