Greenberger–Horne–Zeilinger state

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. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states.^[4]

Generation of the 3-qubit GHZ state using quantum logic gates.

Definition

The GHZ state is an entangled quantum state for 3 qubits and its state is

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

Generalization

The generalized GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension ${\displaystyle d}$ , i.e., the local Hilbert space is isomorphic to ${\displaystyle \mathbb {C} ^{d}}$ , then the total Hilbert space of an ${\displaystyle M}$ -partite system is ${\displaystyle {\mathcal {H}}_{\rm {tot}}=(\mathbb {C} ^{d})^{\otimes M}}$ . This GHZ state is also called an ${\displaystyle M}$ -partite qudit GHZ state. Its formula as a tensor product is

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes \cdots \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes \cdots \otimes |0\rangle +\cdots +|d-1\rangle \otimes \cdots \otimes |d-1\rangle )}$ .

In the case of each of the subsystems being two-dimensional, that is for a collection of M qubits, it reads

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

Properties

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.^{[citation needed]}

Another important property of the GHZ state is that taking the partial trace over one of the three systems yields

${\displaystyle \operatorname {Tr} _{3}\left[\left({\frac {|000\rangle +|111\rangle }{\sqrt {2}}}\right)\left({\frac {\langle 000|+\langle 111|}{\sqrt {2}}}\right)\right]={\frac {(|00\rangle \langle 00|+|11\rangle \langle 11|)}{2}},}$ 

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.^{[citation needed]}

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, ${\displaystyle |\mathrm {W} \rangle =(|001\rangle +|010\rangle +|100\rangle )/{\sqrt {3}}}$ .^[6] Thus ${\displaystyle |\mathrm {GHZ} \rangle }$  and ${\displaystyle |\mathrm {W} \rangle }$  represent two very different kinds of entanglement for three or more particles.^[7] 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.

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 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.^[8] 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 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

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {2}}}\left(|000\rangle +|111\rangle \right)={\frac {1}{2}}\left(|00\rangle +|11\rangle \right)\otimes |+\rangle +{\frac {1}{2}}\left(|00\rangle -|11\rangle \right)\otimes |-\rangle ,}$ 

where the third particle is written as a superposition in the X basis (as opposed to the Z basis) as ${\displaystyle |0\rangle =(|+\rangle +|-\rangle )/{\sqrt {2}}}$  and ${\displaystyle |1\rangle =(|+\rangle -|-\rangle )/{\sqrt {2}}}$ .

A measurement of the GHZ state along the X basis for the third particle then yields either ${\displaystyle |\Phi ^{+}\rangle =(|00\rangle +|11\rangle )/{\sqrt {2}}}$ , if ${\displaystyle |+\rangle }$  was measured, or ${\displaystyle |\Phi ^{-}\rangle =(|00\rangle -|11\rangle )/{\sqrt {2}}}$ , if ${\displaystyle |-\rangle }$  was measured. In the later case, the phase can be rotated by applying a Z quantum gate to give ${\displaystyle |\Phi ^{+}\rangle }$ , while in the former case, no additional transformations are applied. 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.

Applications

GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing^[9] or in the quantum Byzantine agreement.

See also

• Bell's theorem
• Local hidden-variable theory
• NOON state
• Quantum pseudo-telepathy

References

1. Greenberger, Daniel M.; Horne, Michael A.; Zeilinger, Anton (1989). "Going beyond Bell's Theorem". In Kafatos, M. (ed.). Bell's Theorem, Quantum Theory and Conceptions of the Universe. Dordrecht: Kluwer. p. 69. arXiv:0712.0921. Bibcode:2007arXiv0712.0921G.
2. Mermin, N. David (August 1, 1990). "Quantum mysteries revisited". American Journal of Physics. 58 (8): 731–734. Bibcode:1990AmJPh..58..731M. doi:10.1119/1.16503. ISSN 0002-9505. S2CID 119911419.
3. Caves, Carlton M.; Fuchs, Christopher A.; Schack, Rüdiger (August 20, 2002). "Unknown quantum states: The quantum de Finetti representation". Journal of Mathematical Physics. 43 (9): 4537–4559. arXiv:quant-ph/0104088. Bibcode:2002JMP....43.4537C. doi:10.1063/1.1494475. ISSN 0022-2488. S2CID 17416262. Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger [...] where a similar four-system state was proposed.
4. Eldredge, Zachary; Foss-Feig, Michael; Gross, Jonathan A.; Rolston, S. L.; Gorshkov, Alexey V. (April 23, 2018). "Optimal and secure measurement protocols for quantum sensor networks". Physical Review A. 97 (4): 042337. arXiv:1607.04646. Bibcode:2018PhRvA..97d2337E. doi:10.1103/PhysRevA.97.042337. PMC 6513338. PMID 31093589.
5. A pure state ${\displaystyle |\psi \rangle }$  of ${\displaystyle N}$  parties is called biseparable, if one can find a partition of the parties in two nonempty disjoint subsets ${\displaystyle A}$  and ${\displaystyle B}$  with ${\displaystyle A\cup B=\{1,\dots ,N\}}$  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 ${\displaystyle A|B}$ .
6. W. Dür; G. Vidal & J. I. Cirac (2000). "Three qubits can be entangled in two inequivalent ways". Phys. Rev. A. 62 (6): 062314. arXiv:quant-ph/0005115. Bibcode:2000PhRvA..62f2314D. doi:10.1103/PhysRevA.62.062314. S2CID 16636159.
7. Piotr Migdał; Javier Rodriguez-Laguna; Maciej Lewenstein (2013), "Entanglement classes of permutation-symmetric qudit states: Symmetric operations suffice", Physical Review A, 88 (1): 012335, arXiv:1305.1506, Bibcode:2013PhRvA..88a2335M, doi:10.1103/PhysRevA.88.012335, S2CID 119536491
8. "Scientific Background on the Nobel Prize in Physics 2022" (PDF). The Nobel Prize. October 4, 2022.
9. Mark Hillery; Vladimír Bužek; André Berthiaume (1998), "Quantum secret sharing", Physical Review A, 59 (3): 1829–1834, arXiv:quant-ph/9806063, Bibcode:1999PhRvA..59.1829H, doi:10.1103/PhysRevA.59.1829, S2CID 55165469