The GHZ State

The GHZ state is a multiparty entangled state whose correlations refute local realism without any inequality, in a single run.

Definition

The three-qubit Greenberger-Horne-Zeilinger state is GHZ = (|000> + |111>)/sqrt(2), generalizing to n qubits as (|0...0> + |1...1>)/sqrt(2). It is genuinely multipartite entangled: all parties share correlation, yet the entanglement is fragile. Tracing out even one qubit leaves the remaining parties in a separable mixture, so GHZ entanglement lives only in the joint whole.

All-or-nothing nonlocality

GHZ delivers a deterministic contradiction with local hidden variables. Consider measuring products of Pauli X and Y operators across the three qubits. Quantum mechanics predicts, with certainty, XYY = +1, YXY = +1, YYX = +1, and XXX = -1. Any assignment of predetermined plus or minus one values to the local observables makes the product of the first three equal the fourth, forcing XXX = +1, a flat contradiction. Unlike CHSH, this needs no statistics: a single perfect run refutes local realism.

import numpy as np
ghz = np.zeros(8); ghz[0]=ghz[7]=1/np.sqrt(2)
X=np.array([[0,1],[1,0]]); Y=np.array([[0,-1j],[1j,0]])
XXX = np.kron(np.kron(X,X),X)
print(round((ghz@XXX@ghz).real,3))  # -1

Fragility and structure

GHZ entanglement is maximally connected but not robust: loss of any single qubit destroys all multipartite entanglement among the rest. This contrasts sharply with the W state, which retains pairwise entanglement under particle loss. The two represent inequivalent classes of tripartite entanglement that cannot be converted into each other by local operations.

Uses

GHZ states are the resource for quantum secret sharing, multiparty protocols, and certain metrology schemes where n-fold correlation sharpens phase sensitivity toward the Heisenberg limit. In error correction, GHZ-type entangled ancillas assist fault-tolerant syndrome extraction. Preparing high-fidelity GHZ states across many qubits is a standard benchmark for a processor's multi-qubit coherence.