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Greenberger-Horne-Zeilinger State

A Greenberger-Horne-Zeilinger state, or GHZ state, is a multipartite entangled state of n>=3 qubits of the form

|GHZ_n>=(|0>^( tensor n)+|1>^( tensor n))/(sqrt(2)).

The two terms are orthogonal unit vectors in the tensor product space (C^2)^( tensor n), so the state is normalized. Across any division into two nonempty groups of qubits, this state has Schmidt rank two and hence exhibits quantum entanglement.

For three qubits,

|GHZ_3>=(|000>+|111>)/(sqrt(2)).

Its perfect correlations under products of Pauli matrices yield the Greenberger-Horne-Zeilinger theorem, a contradiction with predetermined local outcomes that does not require an inequality (Greenberger et al. 1990).

See also

Greenberger-Horne-Zeilinger Theorem, Pauli Matrices, Quantum Entanglement, Qubit, Schmidt Decomposition, Tensor Product

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References

Greenberger, D. M.; Horne, M. A.; Shimony, A.; and Zeilinger, A. "Bell's Theorem without Inequalities." Amer. J. Phys. 58, 1131-1143, 1990. https://doi.org/10.1119/1.16243.Mermin, N. D. "Quantum Mysteries Revisited." Amer. J. Phys. 58, 731-734, 1990. https://doi.org/10.1119/1.16503.Zeng, H.-S. and Kuang, L.-M. "Preparation of GHZ States via Grover's Quantum Searching Algorithm." Chin. Phys. Lett. 17, 410-412, 2000. https://doi.org/10.1088/0256-307X/17/6/009.

Cite this as:

Weisstein, Eric W. "Greenberger-Horne-Zeilinger State." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Greenberger-Horne-ZeilingerState.html

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