# Quantum Computer Simulation of GHZ Experiment

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An entangled state of three photons in a superposition, either with all horizontally polarized () or with all vertically polarized (), is known as a Greenberger–Horne–Zeilinger (GHZ) state [1, 2]. It is represented by the state vector . A measurement on any one of the photons, using a two-channel polarizer, would give 50% probability for either or . Measurements on the other two photons would then be found to show the same polarization. In the canonical GHZ experiment, measurements are performed on the three entangled photons using two-channel polarizers , and set to orientations different from the original and , which we denote by and . The polarizations are at angles of with respect to the original polarizations, such that . The polarizations are left and right circular polarizations, represented by . The polarization detectors are set in one of four possible combinations: , , or . We use binary notation, 0 and 1, to label the two possible polarizations for either the or orientation. For the , or configuration, we observe four equally probable results, which we designate 001, 010, 001 and 111. For , we again observe four equally probable results, but now 000, 011, 101 or 110. In all of these cases, any two detector readings, say those of and , unambiguously determine the reading of . For example, for configuration , if the first two detectors read 01 or 10, the third would then show 0.

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Contributed by: S. M. Blinder (December 2017)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The Hadamard gate (not to be confused with the horizontal polarization, also called ) acts on a qubit in one of the basis states to produce a linear combination of the two basis states. Specifically and . As a unitary operator, . The -phase shift gate is represented by .

References

[1] D. M. Greenberger, M. A. Horne, A. Shimony and A. Zeilinger, "Bell's Theorem without Inequalities," *American Journal of Physics*, 58(12), 1990 pp. 1131–1143. doi:10.1119/1.16243.

[2] J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter and A. Zeilinger, "Experimental Test of Quantum Nonlocality in Three-Photon Greenberger–Horne–Zeilinger Entanglement," *Nature,* 403, 2000 pp. 515–519. doi:10.1038/35000514.

[3] G. Fano and S. M. Blinder, *Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers, *New York: Springer, 2017.

[4] N. D. Mermin, "Quantum Mysteries Revisited," *American Journal of Physics*,* *58(8), 1990 pp. 731–734. doi:10.1119/1.16503.

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