Characterizing Nonlocality of Pure Symmetric Three-Qubit States

Abstract

We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser–Horne–Shimony–Holt (CHSH) inequality. We make use of the elegant parametrization in the canonical form of these states, proposed by Meill and Meyer (Phys Rev A 96:062310, 2017) based on Majorana geometric representation. The reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state, do not violate the CHSH inequality, and hence, they are CHSH-local. However, when Alice and Bob perform a CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. We have also shown that two different families of three-qubit pure symmetric states, consisting of two and three distinct spinors (qubits), respectively, can be distinguished based on the strength of violation in the conditional CHSH nonlocality test. Furthermore, we identify six of the 46 classes of tight Bell inequalities in the three-party, two-setting, two-outcome, i.e., (3,2,2) scenario (López-Rosa et al. in Phys Rev A 94:062121, 2016). Among the two inequivalent families of three-qubit pure symmetric states, only the states belonging to three distinct spinor class show maximum violations of these six tight Bell inequalities