# GHZ

A Greenberger-Horne-Zeilinger state is an entangled quantum state having extremely non-classical properties.

## Definition

For a system of n qubits the GHZ state can be written as

$$|GHZ\rangle = \frac{|0\rangle^{\otimes n} + |1\rangle^{\otimes n}}{\sqrt{2}}.$$

The simplest one is the 3-qubit GHZ state, which already exhibits non-trivial multipartite entanglement:

$$|GHZ\rangle = \frac{1}{\sqrt{2}}\left( |000\rangle+|111\rangle\right).$$

## Motivation and entanglement observation

In experiments with two entangled particles, where 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.

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It took about ten years to realize the experiment where the desired GHZ 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 D1, D2 and D3 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:

Trk(∣GHZnGHZ∣) = In ∖ k

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Beside this, GHZ states maximize 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:

∣ψ⟩ ≈ ∣100⟩ + ∣010⟩ + ∣001⟩

where  ∣W⟩ = ∣100⟩ + ∣010⟩ + ∣001⟩ 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 (SLOCC), there are indeed only two classes of  3-qubit states which are truly  3-partite entangled corresponding to the GHZ and W states respectively.

Moreover, having defined the generalized Schmidt Decomposition  ∣ΨA1An⟩ :  = ∑i = 1min{dA1, …, dAn}aieA1i⟩ ⊗ … ⊗ ∣eAn for an n-particle state  ∣ΨA1An, 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 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 and 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)