# The Greenberger–Horne–Zeilinger Argument & W states

October 11, 2016 4:32 pm Leave your thoughtsA * Greenberger–Horne–Zeilinger* (GHZ) state

^{1}is an entangled quantum state involving at least three subsystems (particles) and was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. The GHZ state has extremely non-classical properties, which manifest not only a statistical violation, as it is the case for violations of

*Bell’s inequality*, but in a contradiction between

*quantum mechanics*and

*local hidden variable theories*. The GHZ argument is independent of the Bell approach, and shows in a non-statistic manner that quantum mechanics and local realism are mutually incompatible. The general form of a GHZ state of

*M*> 2 subsystems is given by . The simples form of a GHZ state is a tripartite entangled system, where the objects are spatially separated, being an element of the product Hilbert space and given by . Here and (or correspondingly and ) are the eigenstates of with the corresponding eigenvalues +1 and -1, and . Next local spin measurements, for different orientations, are performed. For example for a -measurement, the observables , and are measured on the corresponding composite system, which is illustrated below.

An easy way to calculate the expectation value is to decompose the GHZ-state in the eigenfunctions of the measurement operator: in the case of a measurement system *A* in *y*-basis, *B* in *y*-basis and *C* in *x*-basis. For photons we get in the in rotated basis with polarization along ±45° , . Right and left circularly polarized photons take the form , and . This finally yields: The quantum-mechanical expectation value can be calculated easily (+1 for every and -1 for every , with *i=A,B,C* and *j=x,y*, which yields . Due to the symmetry of the state the result remains the same for the other two measurements: The unique property of this system is that the result of the *x*-measurement of one system can be predicted with certain, when the results of the *y*-measurement of the other systems are known. Analogously, the result of one *y*-measurement can be predicted of the results of the other *y*-measurement and the *x*-measurement are known. From the point of view of a *local realistic theory* this behaviour can be reproduced simply by addressing predefined value to the individual spin measurements. Let for example be the predefined result of the measurement, which can only be +1 or -1. A simple combination of of values, reproducing the quantum mechanical results from above is given by and , which gives , , and , and finally . However the predictions of quantum mechanics are not only different, but the complete opposite: Now the GHZ state is expressed in a *–*basis

which yields . Here a rigorous contradiction between the predictions of local-realistic theories and quantum mechanics has been disclosed.

Another important property of the *maximally* entangles GHZ state is that if a measurement on one of the subsystems is performed in such way that it distinguishes between the states 0 and 1, i.e., a -measurement, the system will be left behind in a *unentangled* state. Dependent on the result the state is given by or , for results -1 and 1, respectively. On the other hand if the measurement on the third particle is carried out in another basis, for instance *x*, a completely different behavior is observed. The GHZ state can be written as . Independent of the result, in either case, the end result of the operations is a *maximally entangled Bell state*.

The second class of so called *non-biseparable *three qubit states is found by the* W-state *

^{2 }(biseparable means that one can find a partition of the parties in two disjoint subsets

*A*and

*B,*expressed as , or in other words is a product-state).

*In its original form a W-state is defined for three qubits and given by . It is impossible to transform the W-state into a GHZ state applying local quantum operations. An interesting property of the W-state is if one of the three qubits is lost, the state of the remaining 2-qubit system is still*

*entangled*. This

*robustness*of W-type entanglement contrasts strongly with the GHZ state, which is left in a

*fully separable*state after one of the three qubits is lost (see above). The generalized form of the W-state for

*m*qubits as a quantum superpostion with equal expansion coefficients of all possible pure states in which exactly one of the qubits in an “excited state” ( ) which can be written as .

1. D.M. Greenberger, M.A. Horne, A. Shimony and A. Zeilinger, *Am. J. Phys*. **58**, 1131 (1990). ↩

2. W. Dür, G. Vidal, and J.I. Cirac, *Phys. Rev. A* **62**, 062314 (1990). ↩

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