# Greenberger–Horne–Zeilinger state

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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

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

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:

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

This state is non-biseparable[2] and is the representative of one of the two non-biseparable classes of three-qubit states (the other being the W state) which cannot be transformed (not even probabilistically) into each other by local quantum operations.[3] Thus ${\displaystyle |\mathrm {GHZ} \rangle }$ and ${\displaystyle |W\rangle }$ represent two very different kinds of tripartite entanglement. The W state is, in a certain sense "less entangled" than the GHZ state; however, that entanglement is, in a sense, more robust against single-particle measurements, in that, for an N-qubit W state, an entangled (N-1) qubit state remains after a single particle measurement. By contrast, certain measurements on the GHZ state collapse it into a mixture or a pure state.

## 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

${\displaystyle \mathrm {Tr} _{3}{\big (}(|000\rangle +|111\rangle )(\langle 000|+\langle 111|){\big )}={\frac {(|00\rangle \langle 00|+|11\rangle \langle 11|)}{2}}}$

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 ${\displaystyle |00\rangle }$ or ${\displaystyle |11\rangle }$ 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).

## Pairwise entanglement

Although a naive measurement of the third particle of the GHZ state results in an unentangled pair, a more clever measurement, along an orthogonal direction, can leave behind a maximally entangled Bell state. This is illustrated below. The lesson to be drawn from this is that pairwise entanglement in the GHZ is more subtle than it naively appears: measurements along the privileged Z-direction destroy pair-wise entanglement, but other measurements (along different axes) do not.

The GHZ state can be written as

${\displaystyle |000\rangle +|111\rangle =(|00\rangle +|11\rangle )\otimes |L\rangle +(|00\rangle -|11\rangle )\otimes |R\rangle }$

where the third particle ("Wigner's friend"[4]) is written as a superposition in the X-basis (as opposed to the Z-basis) as ${\displaystyle |0\rangle =|L\rangle +|R\rangle }$ and ${\displaystyle |1\rangle =|L\rangle -|R\rangle }$.

A measurement of the GHZ state along the X-basis for the third particle then yields either ${\displaystyle |00\rangle +|11\rangle }$, if ${\displaystyle |L\rangle }$ was measured, or ${\displaystyle |00\rangle -|11\rangle }$, if ${\displaystyle |R\rangle }$ was measured. In the later case, the phase can be rotated by applying a Z-quantum gate, to give ${\displaystyle |00\rangle +|11\rangle }$; while, in the former case, no additional transformations are applied. In either case, the end result of the operations is a maximally entangled Bell state.

The point of this example is that it illustrates that the pairwise entanglement of the GHZ state is more subtle than it first appears: a judicious measurement along an orthogonal direction, along with the application of a quantum transform depending on the measurement outcome, can leave behind a maximally entangled state.

2. ^ A pure state ${\displaystyle |\psi \rangle }$ of ${\displaystyle N}$ parties is called biseparable, if one can find a partition of the parties in two disjoint subsets ${\displaystyle A}$ and ${\displaystyle B}$ with ${\displaystyle A\cup B=\{1,...,N\}}$ such that ${\displaystyle |\psi \rangle =|\phi \rangle _{A}\otimes |\gamma \rangle _{B}}$, i.e. ${\displaystyle |\psi \rangle }$ is a product state with respect to the partition ${\displaystyle A|B}$.