GHZ

A '''Greenberger-Horne-Zeilinger''' state is an [[Theory of entanglement|entangled]] [[states|quantum state]] having extremely non-classical properties. = Definition = For a system of $n$ [[qubits]] the '''GHZ state''' can be written as :$|GHZ\backslash rangle\; =\; \backslash frac\{|0\backslash rangle^\{\backslash otimes\; n\}\; +\; |1\backslash rangle^\{\backslash otimes\; n\}\}\{\backslash sqrt\{2\}\}.$ The simplest one is the 3-qubit GHZ state, which already exhibits non-trivial multipartite entanglement: :$|GHZ\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash left(\; |000\backslash rangle+|111\backslash rangle\backslash right).$ = Motivation and entanglement observation = In experiments with two entangled particles, where [[Bell's theorem|Bell's inequalities]] are tested, one comes to the conclusion that ''statistical predictions'' of quantum mechanics are in conflict with local realism. According to an observation of Greenberger, Horne and Zeilinger, made back in 1989, the entanglement of more than two particles will show that not only ''statistical'' but also ''nonstatistical'' predictions of quantum mechanics are in conflict with local realism. [[image:Observation.jpg|thumb|right|schematic drawing of the experimental setup]] It took about ten years to realize the experiment where the desired GHZ [[Quantum entanglement|correlations]] have been observed. A schematic drawing of the experimental setup of the experiment by Bouwmeester ''et. al.'' for the demonstration of the polarization entanglement for three spatially separated photons is presented in the figure. Conditioned on the registration of one photon at the trigger detector $T$, the three photons registered at $D\_1$, $D\_2$ and $D\_3$ show the desired GHZ correlations. = Properties and applications = One of the most remarkable properties of the GHZ states is the fact that tracing out only one party completely destroys entanglement in the state and one ends up in a fully mixed state, which is fully separable: :$Tr\_k\backslash left(|GHZ\backslash rangle\_n\backslash langle\; GHZ|\backslash right)\; =\; \backslash mathbb\{I\}\_\{n\backslash backslash\; k\}$ [[image:BildchenGHZ.jpg|thumb|right|type of entanglement in GHZ-states]] Beside this, GHZ states maximize [[Entanglement measure|entanglement monotones]] and therefore can be called ''maximally entangled in multipartite sense''. Moreover GHZ states belong to the states for which the [[normal form]] coincides with the particular state. For example in the case of three qubits there is only one such state and this is exactly the GHZ state. Hence all three-qubit states with non-zero normal form carry some GHZ-type of entanglement. However there are also entangled states that have a zero normal form but are still entangled: :$\backslash ;\; |\backslash psi\backslash rangle\; \backslash approx\; |100\backslash rangle\; +\; |010\backslash rangle\; +\; |001\backslash rangle$ where $\backslash ;\; |W\backslash rangle\; =\; |100\backslash rangle\; +\; |010\backslash rangle\; +\; |001\backslash rangle$ is the so-called ''[[W-state]]''. It must be noticed, though, that these states form a set of zero measure. According to state classification by means of stochastic [[LOCC operations|LOCC]] (SLOCC), there are indeed only '''two classes''' of $\backslash ;\; 3$-qubit states which are truly '''$\backslash ;\; 3$-partite entangled''' corresponding to the GHZ and W states respectively. Moreover, having defined the [[Bipartite states and Schmidt Decomposition|'''generalized Schmidt Decomposition''']] $\backslash ;\; |\backslash Psi\_\{A\_1\; \backslash ldots\; A\_n\}\backslash rangle\; :=\; \backslash sum\_\{i=1\}^\{min\backslash \{d\_\{A\_1\},\backslash ldots,d\_\{A\_n\}\backslash \}\}\; a\_i\; |e\_\{A\_1\}^i\backslash rangle\; \backslash otimes\; \backslash ldots\; \backslash otimes\; |e\_\{A\_n\}\backslash rangle$ for an $\backslash ;\; n$-particle state $\backslash ;\; |\backslash Psi\_\{A\_1\; \backslash ldots\; A\_n\}\backslash rangle$, it can be easily seen that the GHZ states admit generalized Schmidt decomposition. In general, a state admits Schmidt decomposition if, tracing out any subsystem, the rest is in a [[Multipartite entanglement| fully separable]] state. Indeed this is true for the GHZ states, while it does not hold for W-states, which in fact do not admit Schmidt decomposition. There is wide range of applications of GHZ states, as states with multipartite entanglement. Entanglement between the several parties is the most essential feature in [[Quantum communication|quantum communication]] and [[Basic concepts in quantum computation|computation]] protocols. = References and further reading = * Daniel M. Greenberger, Michael A. Horne, Anton Zeilinger: ''Bell's theorem, Quantum Theory, and Conceptions of the Universe'', pp. '''73-76''', Kluwer Academics, Dordrecht, The Netherlands (1989); * D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, A. Zeilinger, ''Phys. Rev. Lett.'', '''82''' 1345 (1999) * F. Verstraete, J. Dehaene, B. De Moor, ''Phys. Rev. A'', '''68''', 012103 (2003) [[Category:Handbook of Quantum Information]] [[Category:Quantum States]]