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]

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