# The Cluster's GHZ State

A page about the Greenberger-Horne-Zeilinger (GHZ) state — a maximally entangled state of three qubits.

## The GHZ state

The GHZ state is |GHZ> = (|000> + |111>)/sqrt(2). It is a maximally entangled state of three qubits. Unlike the Bell state of two qubits, the GHZ state has a striking feature: if you measure one qubit, the other two are projected onto a product state, but the correlations between the three qubits violate a deterministic version of Bell's inequalities. In the cluster, the GHZ state is a maximally entangled triple of pages. Measuring one page determines the state of the other two, but the correlations between all three are stronger than any classical theory allows.

## The GHZ contradiction

The GHZ state allows a deterministic contradiction with local hidden variable theories. If you measure the three qubits in certain bases, the results are perfectly correlated in a way that no local hidden variable model can reproduce. In the cluster, the GHZ correlation means that the content of three pages is correlated in a way that no local model of the cluster can explain.

## The single-qubit measurement

When you measure one qubit of the GHZ state in the computational basis, you get 0 or 1 with equal probability. If you get 0, the other two qubits collapse to |00>. If you get 1, they collapse to |11>. In the cluster, reading one page in a GHZ triple instantly determines the content of the other two — not probabilistically, but deterministically. The collapse is instant and absolute.

## The multi-qubit measurement

When you measure all three qubits in the X basis (|+> = (|0> + |1>)/sqrt(2), |-> = (|0> - |1>)/sqrt(2)), the parity of the number of - results is even. This is a deterministic prediction that no local hidden variable model can reproduce. In the cluster, this means that the parity of some measurement outcomes on three entangled pages is deterministic — a prediction that no local model of the cluster can satisfy.

## This GHZ state

This page is one of the qubits in a GHZ state. It is maximally entangled with two other pages. Measuring one determines the state of all three. The correlation is deterministic. The contradiction is absolute. The measurement is instant.