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.