Darboux's theorem (analysis): Difference between revisions

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{{merge|Cat state|date=February 2013|discuss=talk: Cat state #Proposed merge}}
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.
 
==Definition==
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>
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).
 
The simplest one is the 3-qubit GHZ state:
<math>|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math>
 
==Properties==
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]].
 
Another important property of the GHZ state is that when we [[partial trace|trace]] over one of the three systems
we get
:<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]].
 
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.
 
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).
 
==See also==
* [[Bell's theorem]]
* [[Bell state]]
* [[GHZ experiment]]
* [[Local hidden variable theory]]
* [[Quantum entanglement]]
* [[Qubit]]
* [[Measurement in quantum mechanics]]
 
==References==
<references/>
 
{{DEFAULTSORT:Greenberger-Horne-Zeilinger state}}
[[Category:Quantum information theory]]
 
 
{{Physics-stub}}
 
[[de:GHZ-Experiment|Greenberger-Horne-Zeilinger]]
[[ja:Greenberger-Horne-Zeilinger 状態]]

Revision as of 15:20, 15 January 2014



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Definition

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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).

The simplest one is the 3-qubit GHZ state:

Properties

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.

Another important property of the GHZ state is that when we trace over one of the three systems we get

which is an unentangled mixed state. It has certain two-particle (qubit) correlations, but these are of a classical nature.

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 or which are unentangled pure states. This is unlike the W state which leaves bipartite entanglements even when we measure one of its subsystems.

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 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).

See also

References

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Greenberger-Horne-Zeilinger ja:Greenberger-Horne-Zeilinger 状態