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| In [[physics]], in the area of [[quantum information theory]], a '''Greenberger–Horne–Zeilinger state''' is a certain type of [[quantum entanglement|entangled]] [[quantum state]] which involves at least three subsystems (particles). It was first studied by D. Greenberger, M.A. Horne and [[Anton Zeilinger]] in 1989.<ref>{{citation |author=Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger |year=2007 |title=Going beyond Bell's Theorem |arxiv=0712.0921|bibcode = 2007arXiv0712.0921G }}</ref> They have noticed the extremely non-classical properties of the state.
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| ==Definition==
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| The '''GHZ state''' is an [[quantum entanglement|entangled]] [[quantum state]] of {{math|''M'' > 2}} subsystems. In the case of each of the subsystems being two-dimensional, that is for [[qubit]]s, it reads | |
| :<math>|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}.</math>
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| In simple words it is a quantum superposition of all subsystems being in state 0 with all of them being in state 1 (states 0 and 1 of a single subsystem are fully distinguishable).
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| The simplest one is the 3-qubit GHZ state:
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| <math>|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math> | |
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| ==Properties==
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| There is no standard measure of multi-partite entanglement because different types of multi-partite entanglement exist which are not mutually convertible. Nonetheless, many measures define the GHZ to be [[Maximally entangled state|maximally entangled]].
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| Another important property of the GHZ state is that when we [[partial trace|trace]] over one of the three systems
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| we get
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| :<math>\mathrm{Tr}_3\big((|000\rangle + |111\rangle)(\langle 000|+\langle 111|) \big) = |00\rangle \langle 00| + |11\rangle \langle 11|</math> | |
| which is an unentangled [[mixed state (physics)|mixed state]]. It has certain two-particle (qubit) correlations, but these are [[covariation|of a classical nature]].
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| On the other hand, if we were to measure one of the subsystems, in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either <math>|00\rangle</math> or <math>|11\rangle</math> which are unentangled pure states. This is unlike the [[W state]] which leaves bipartite entanglements even when we measure one of its subsystems.
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| The GHZ state leads to striking non-classical correlations (1989). Particles prepared in this state lead to a version of [[Bell's theorem]], which shows the internal inconsistency of the notion of elements-of-reality introduced in the famous [[Einstein–Podolsky–Rosen paradox|Einstein–Podolsky–Rosen]] paper. The first laboratory observation of GHZ correlations was by the group of [[Anton Zeilinger]] (1998). Many, more accurate observations followed. The correlations can be utilized in some [[quantum information]] tasks. These include multipartner [[quantum cryptography]] (1998) and [[communication complexity]] tasks (1997, 2004).
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| ==See also==
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| * [[Bell's theorem]]
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| * [[Bell state]]
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| * [[GHZ experiment]]
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| * [[Local hidden variable theory]]
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| * [[Quantum entanglement]]
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| * [[Qubit]]
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| * [[Measurement in quantum mechanics]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Greenberger-Horne-Zeilinger state}}
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| [[Category:Quantum information theory]]
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| {{Physics-stub}}
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| [[de:GHZ-Experiment|Greenberger-Horne-Zeilinger]]
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| [[ja:Greenberger-Horne-Zeilinger 状態]]
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