General Greenberger-Horne-Zeilinger theorem of cluster states

TL;DR

This paper systematically constructs GHZ theorems for one-dimensional cluster states of N qubits, revealing multiple distinct forms and extending the approach to Bell inequalities.

Contribution

It introduces a comprehensive method to derive GHZ arguments for N-qubit cluster states, including multiple forms and Bell inequalities, advancing the understanding of quantum nonlocality.

Findings

Eight GHZ forms for four-qubit cluster states

Forty-eight GHZ forms for five-qubit cluster states

Extension of the method to N-qubit systems

Abstract

In this paper, we show that there are eight distinct forms of the Greenberger-Horne-Zeilinger (GHZ) argument for the four-qubit cluster state $|\phi_4>$ and forty eight distinct forms for the five-qubit cluster state $|\phi_5>$ in the case of the one-dimensional lattice. The proof is obtained by regarding the pair qubits as a single object and constructing the new Pauli-like operators. The method can be easily extended to the case of the N-qubit system and the associated Bell inequalities are also discussed. Consequently, we present a complete construction of the GHZ theorem for the cluster states of N-qubit in the case of the one-dimensional lattice.