Undetectable Werner states using linear Bell inequalities

TL;DR

This paper investigates the limitations of linear Bell inequalities in detecting Werner states, revealing that most such states, including those related to GHZ states, are undetectable with these methods.

Contribution

It demonstrates that almost all generalized multipartite GHZ Werner states cannot be detected by homogeneous linear Bell inequalities, providing a method to identify undetectable Werner states.

Findings

Most Werner states of GHZ states are undetectable by linear Bell inequalities.

The method applies to all pure states, showing a broad undetectability.

Numeric algorithms reveal undetectable Werner states with few particles.

Abstract

Bell inequality serves as an important method to detect quantum entanglement, a problem which is generally known to be NP-hard. Our goal in this work is to detect Werner states using linear Bell inequality. Surprisingly, we show that Werner states of almost all generalized multipartite Greenberger-Horne-Zeilinger (GHZ) states cannot be detected by homogeneous linear Bell inequalities with dichotomic inputs and outputs of each local observer. The main idea is to estimate the largest violations of Werner states in the case of general linear Bell inequalities. The presented method is then applied to Werner states of all pure states to show a similar undetectable result. Moreover, we provide an accessible method to determine undetectable Werner states for general linear Bell inequality including sub-correlations. The numeric algorithm shows that there are nonzero measures of Werner states with small number of particles that cannot be detected by general linear Bell inequalities.