The Greenberger–Horne–Zeilinger Argument & W states

A 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 $\vert\Psi^{\otimes M}_{\mathrm {GHZ}}\rangle=1/\sqrt 2\big(\vert 0\rangle^{\otimes M}+\vert 1\rangle^{\otimes M}\big)$. 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 $\mathcal H_2^A\otimes\mathcal H_2^B\otimes\mathcal H_2^C$ and given by $\vert\Psi^{ABC}_{\mathrm{GHZ}}\rangle=1/\sqrt 2\big(\vert 1^A\rangle \vert 1^B\rangle \vert 1^C\rangle+\vert 0^A\rangle \vert0^B\rangle \vert 0^C\rangle\big)$. Here $\vert0^i\rangle$ and $\vert1^i\rangle$ (or correspondingly $\vert H^i \rangle$ and $\vert V^i\rangle$) are the eigenstates of $\sigma_z$ with the corresponding eigenvalues +1 and -1, and $i=A,B,C$. Next local spin measurements, for different orientations, are performed. For example for a $yyx$-measurement, the observables $\sigma_y^A$, $\sigma_y^B$ and $\sigma_x^C$ 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 $\sigma_y^A$$\sigma_y^B$$\sigma_x^C$ 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° $\vert 0_x\rangle=(\vert +45^\circ\rangle=\vert +\rangle)=\frac{1}{\sqrt 2}\big(\vert 0\rangle+\vert 1\rangle\big)$, $\vert 1_x\rangle=(\vert -45^\circ\rangle=\vert -\rangle)=\frac{1}{\sqrt 2}\big(\vert 0\rangle-\vert 1\rangle\big)$. Right and left circularly polarized photons take the form $\vert 0_y\rangle=(\vert R\rangle)=\frac{1}{\sqrt 2}\big(\vert 0\rangle+{\mathrm i}\vert 1\rangle\big)$, and $\vert 1_y\rangle=(\vert L\rangle)=\frac{1}{\sqrt 2}\big(\vert 0\rangle-{\mathrm i}\vert 1\rangle\big)$. This finally yields: The quantum-mechanical expectation value can be calculated easily (+1 for every $\vert 1^i_j\rangle$ and -1 for every  $\vert 1^i_j\rangle$, with i=A,B,C and j=x,y, which yields $E(\sigma_y^A\sigma_y^B\sigma_x^C)=\langle\Psi^{ABC}_{\mathrm {GHZ}}\vert\sigma_y^A\sigma_y^B\sigma_x^C\vert\Psi^{ABC}_{\mathrm {GHZ}}\rangle=\frac{-4}{4}=-1$. Due to the symmetry of the state the result remains the same for the other two measurements: $E(\sigma_y^A\sigma_x^B\sigma_y^C)=E(\sigma_x^A\sigma_y^B\sigma_y^C)=-1$ 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 $v(\sigma^A_x)$ be the predefined result of the  $\sigma^A_x$ measurement, which can only be +1 or -1. A simple combination of of values, reproducing the quantum mechanical results from above is given by $v(\sigma^A_x)=v(\sigma^B_x)=v(\sigma^C_y)=1$ and $v(\sigma^A_y)=v(\sigma^B_y)=v(\sigma^C_x)=-1$, which gives $v(\sigma^A_y)v(\sigma^B_y)v(\sigma^C_x)=-1=E(\sigma_y^A\sigma_y^B\sigma_x^C)$, $v(\sigma^A_y)v(\sigma^B_x)v(\sigma^C_x)=-1=E(\sigma_y^A\sigma_x^B\sigma_y^C)$, and $v(\sigma^A_x)v(\sigma^B_y)v(\sigma^C_y)=-1=E(\sigma_x^A\sigma_y^B\sigma_y^C)$, and finally $v(\sigma^A_x)v(\sigma^B_x)v(\sigma^C_x)=-1$. However the predictions of quantum mechanics are not only different, but the complete opposite: Now the GHZ state is expressed in a $xxx$–basis

which yields $E(\sigma_x^A\sigma_x^B\sigma_x^C)=\langle\Psi^{ABC}_{\mathrm {GHZ}}\vert\sigma_x^A\sigma_x^B\sigma_x^C\vert\Psi^{ABC}_{\mathrm {GHZ}}\rangle=\frac{+4}{4}=+1$. 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 $\sigma_z$-measurement, the system will be left behind in a unentangled state. Dependent on the result the state is given by $\vert 0^A\rangle\vert 0^B\rangle$ or  $\vert 1^A\rangle\vert 1^B\rangle$, 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 $\vert\Psi^{ABC}_{\mathrm {GHZ}}\rangle=1/\sqrt 2\big(\vert 1^A\rangle \vert 1^B\rangle \vert 1^C\rangle+\vert 0^A\rangle \vert0^B\rangle \vert 0^C\rangle\big)=\frac{1}{2}\Big(\big(\vert 0^A\rangle \vert 0^B\rangle+\vert 1^A\rangle \vert 1^B\rangle \big)\vert 1_x^C\rangle+\big(\vert 0^A\rangle \vert 0^B\rangle-\vert 1^A\rangle \vert 1^B\rangle \big)\vert 0_x^C\rangle\Big)$. 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  $\vert\psi\rangle=\vert\psi\rangle_A\otimes\vert\psi\rangle_B$, or in other words $\vert\psi\rangle$ is a product-state).  In its original form a W-state is defined for three qubits and given by $\vert \Psi_W^{ABC}\rangle=1/\sqrt 3\big(\vert 0^A\rangle\vert 0^B\rangle\vert 1^C\rangle+\vert 0^A\rangle\vert 1^B\rangle\vert 0^C\rangle+\vert 1^A\rangle\vert 0^B\rangle\vert 0^C\rangle\big)$. 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” ( $\vert 1\rangle$) which can be written as $\vert \Psi_W^{m}\rangle=1/\sqrt m\big(\vert 1^{(1)}\rangle\vert 0^{(2)}\rangle \vert 0^{(3)}\rangle ... \vert 0^{(m)} \rangle+\vert 0^{(1)}\rangle\vert 1^{(2)}\rangle \vert 0^{(3)}\rangle ... \vert 0^{(m)} \rangle + ... + \vert 0^{(1)}\rangle\vert 0^{(2)}\rangle ... \vert 0^{(m-1)}\rangle \vert 1^{(m)}\rangle\big)$.

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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