Greenberger–Horne–Zeilinger state
Named for the three authors that first described this state, the GHZ state predicts outcomes from experiments that directly contradict predictions by every classical local hidden-variable theory.
The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989.
[1] The following year Abner Shimony joined in and they published a three-particle version[2] based on suggestions by N. David Mermin.
[3][4] Experimental measurements on such states contradict 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.
where the 0 or 1 values of the qubit correspond to any two physical states.
For example the two states may correspond to spin-down and spin up along some physical axis.
where the numbering of the states represents spin eigenvalues.
[3] Another example[6] of a GHZ state is three photons in an entangled state, with the photons being in a superposition of being all horizontally polarized (HHH) or all vertically polarized (VVV), with respect to some coordinate system.
The GHZ state can be written in bra–ket notation as Prior to any measurements being made, the polarizations of the photons are indeterminate.
If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, each orientation will be observed, with 50% probability.
Its formula as a tensor product is In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads The results of actual experiments agree with the predictions of quantum mechanics, not those of local realism.
[7] In the language of quantum computation, the polarization state of each photon is a qubit, the basis of which can be chosen to be With appropriately chosen phase factors for
The quantum mechanical predictions of the GHZ experiment can then be summarized as which is consistent in quantum mechanics because all these multi-qubit Paulis commute with each other, and due to the anticommutativity between
These results lead to a contradiction in any local hidden variable theory, where each measurement must have definite (classical) values
[3] There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist.
[citation needed] Another important property of the GHZ state is that taking the partial trace over one of the three systems yields which is an unentangled mixed state.
It has certain two-particle (qubit) correlations, but these are of a classical nature.
This is unlike the W state, which leaves bipartite entanglements even when we measure one of its subsystems.
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.
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.
A measurement of the GHZ state along the X basis for the third particle then yields either
In the later case, the phase can be rotated by applying a Z quantum gate to give
In either case, the result of the operations is a maximally entangled Bell state.
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[11] or in the quantum Byzantine agreement.
