In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger (GHZ) state is an entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). 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 state has applications in quantum computing.
History The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, who predicted that it would lead to striking non-classical correlations inconsistent with any local hidden-variable theory. The following year Abner Shimony joined in and they published a three-particle version based on suggestions by N. David Mermin. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger, who was awarded a share of the 2022 Nobel Prize in physics for this work.
Definition The GHZ state is an entangled quantum state for 3 qubits and it can be written
| G H Z ⟩ = | 000 ⟩ + | 111 ⟩ 2 {\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}}
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. In physics applications the state may be written
| G H Z ⟩ = | 1 , 1 , 1 ⟩ + | − 1 , − 1 , − 1 ⟩ 2 {\displaystyle |\mathrm {GHZ} \rangle ={\frac {|1,1,1\rangle +|{-1},{-1},{-1}\rangle }{\sqrt {2}}}}
where the numbering of the states represents spin eigenvalues. Another example 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
| G H Z ⟩ = 1 2 ( | H H H ⟩ + | V V V ⟩ ) . {\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {HHH} \rangle +|\mathrm {VVV} \rangle ).}
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. However the result of all three measurements on the state gives the same result: all three polarizations are observed along the same axis.
Generalization The generalized GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension d, i.e., the local Hilbert space is isomorphic to C d {\displaystyle \mathbb {C} ^{d}} , then the total Hilbert space of an M-partite system is H t o t = ( C d ) ⊗ M {\displaystyle {\mathcal {H}}_{\rm {tot}}=(\mathbb {C} ^{d})^{\otimes M}} . This GHZ state is also called an M-partite qudit GHZ state. Its formula as a tensor product is
| G H Z ⟩ = 1 d ∑ i = 0 d − 1 | i ⟩ ⊗ ⋯ ⊗ | i ⟩ = 1 d ( | 0 ⟩ ⊗ ⋯ ⊗ | 0 ⟩ + ⋯ + | d − 1 ⟩ ⊗ ⋯ ⊗ | d − 1 ⟩ ) {\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes \cdots \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes \cdots \otimes |0\rangle +\cdots +|d-1\rangle \otimes \cdots \otimes |d-1\rangle )} . In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads
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