= Greenberger–Horne–Zeilinger state
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{wiki}
= GHZ state
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A <GHZ state> is a multipartite <entangled state> with coherent amplitudes linking the alternatives in which all <qubits> are $0$ and all are $1$.
The three-<qubit> <GHZ state> is
$$
|\operatorname{GHZ}\rangle
=\frac{|000\rangle+|111\rangle}{\sqrt2}.
$$
<Computational-basis measurements> give only $000$ and $111$, each with <probability> one half. A classical mixture of those strings gives the same <probabilities>, but lacks the off-diagonal <quantum coherence>. For the displayed state, $\langle X\otimes X\otimes X\rangle=1$, whereas it is zero for the mixture. Tracing out one <qubit> leaves a classically correlated mixture of $|00\rangle$ and $|11\rangle$. <Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement by Dik Bouwmeester and others>[1999, three-photon entanglement; the state/mixture calculations here are direct algebra]{ref}
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