Greenberger–Horne–Zeilinger state

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems (particles). It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989.[1] They have noticed the extremely non-classical properties of the state.

Definition

The GHZ state is an entangled quantum state of M > 2 subsystems. In the case of each of the subsystems being two-dimensional, that is for qubits, it reads

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

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: $|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.$

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

$\mathrm{Tr}_3\big((|000\rangle + |111\rangle)(\langle 000|+\langle 111|) \big) = \frac{(|00\rangle \langle 00| + |11\rangle \langle 11|)}{2}$

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 $|00\rangle$ or $|11\rangle$ 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).