In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990.[1] [2] [3] Extremely non-classical properties of the state have been observed, contradicting intuitive notions of locality and causality. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states.[4]
The GHZ state is an entangled quantum state for 3 qubits and its state is
|GHZ\rangle=
| |000\rangle+|111\rangle | |
| \sqrt{2 |
The generalized GHZ state is an entangled quantum state of subsystems. If each system has dimension
d
Cd
M
l{H}\rm=(Cd) ⊗
M
| |GHZ\rangle= | 1 |
| \sqrt{d |
In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads
|GHZ\rangle=
| |0\rangle ⊗ +|1\rangle ⊗ | |
| \sqrt{2 |
There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist. Nonetheless, many measures define the GHZ state to be maximally entangled state.
Another important property of the GHZ state is that taking the partial trace over one of the three systems yields
| \operatorname{Tr} | ||||
|
|00\rangle
|11\rangle
The GHZ state is non-biseparable[5] and is the representative of one of the two non-biseparable classes of 3-qubit states which cannot be transformed (not even probabilistically) into each other by local quantum operations, the other being the W state,
|W\rangle=(|001\rangle+|010\rangle+|100\rangle)/\sqrt{3}
|GHZ\rangle
|W\rangle
Experiments on the GHZ state lead 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 article. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger (1998), who was awarded a share of the 2022 Nobel Prize in physics for this work.[7] 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).
Although a measurement of the third particle of the GHZ state that distinguishes the two states results in an unentangled pair, a measurement along an orthogonal direction can leave behind a maximally entangled Bell state. This is illustrated below.
The 3-qubit GHZ state can be written as
| |GHZ\rangle= | 1 |
| \sqrt{2 |
|0\rangle=(|+\rangle+|-\rangle)/\sqrt{2}
|1\rangle=(|+\rangle-|-\rangle)/\sqrt{2}
A measurement of the GHZ state along the X basis for the third particle then yields either
|\Phi+\rangle=(|00\rangle+|11\rangle)/\sqrt{2}
|+\rangle
|\Phi-\rangle=(|00\rangle-|11\rangle)/\sqrt{2}
|-\rangle
|\Phi+\rangle
This example illustrates that, depending on which measurement is made of the GHZ state is more subtle than it first appears: a measurement along an orthogonal direction, followed by a quantum transform that depends on the measurement outcome, can leave behind a maximally entangled state.
GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing or in the quantum Byzantine agreement.
|\psi\rangle
N
A
B
A\cupB=\{1,...,N\}
|\psi\rangle=|\phi\rangleA ⊗ |\gamma\rangleB
|\psi\rangle
A|B