Greenberger–Horne–Zeilinger state

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.

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 an 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.^[5]

The GHZ state is an entangled quantum state for 3 qubits and it can be written $${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}.}$$ 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 $${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|1,1,1\rangle +|-1,-1,-1\rangle }{\sqrt {2}}}.}$$ 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

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {HHH} \rangle +|\mathrm {VVV} \rangle ).}$

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.

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

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

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

${\displaystyle |0\rangle \equiv |\mathrm {H} \rangle ,\qquad |1\rangle \equiv |\mathrm {V} \rangle .}$

With appropriately chosen phase factors for ${\displaystyle |\mathrm {H} \rangle }$ and ${\displaystyle |\mathrm {V} \rangle }$, both types of measurements used in the experiment becomes Pauli measurements, with the two possible results represented as +1 and −1 respectively:^{[citation needed]}

• The 45° linear polarizer implements a Pauli ${\displaystyle X}$ measurement, distinguishing between the eigenstates

${\displaystyle |{+X}\rangle \equiv |+\rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {H} \rangle +|\mathrm {V} \rangle ),\qquad |{-X}\rangle \equiv |-\rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {H} \rangle -|\mathrm {V} \rangle ).}$

• The circular polarizer implements a Pauli ${\displaystyle Y}$ measurement, distinguishing between the eigenstates

${\displaystyle |{+Y}\rangle \equiv |R\rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {H} \rangle +i|\mathrm {V} \rangle ),\qquad |{-Y}\rangle \equiv |L\rangle ={\frac {1}{\sqrt {2}}}(|\mathrm {H} \rangle -i|\mathrm {V} \rangle ).}$

A combination of those measurements on each of the three qubits can be regarded as a destructive multi-qubit Pauli measurement, the result of which being the product of each single-qubit Pauli measurement. For example, the combination "circular polarizer on photons 1 and 2, 45° linear polarizer on photon 3" corresponds to a ${\displaystyle Y_{1}Y_{2}X_{3}}$ measurement, and the four possible result combinations (RL+, LR+, RR−, LL−) are exactly the ones corresponding to an overall result of −1.

The quantum mechanical predictions of the GHZ experiment can then be summarized as

${\displaystyle \langle \mathrm {GHZ} |Y_{1}Y_{2}X_{3}|\mathrm {GHZ} \rangle =\langle \mathrm {GHZ} |Y_{1}X_{2}Y_{3}|\mathrm {GHZ} \rangle =\langle \mathrm {GHZ} |X_{1}Y_{2}Y_{3}|\mathrm {GHZ} \rangle =-1,}$

${\displaystyle \langle \mathrm {GHZ} |X_{1}X_{2}X_{3}|\mathrm {GHZ} \rangle =+1,}$

which is consistent in quantum mechanics because all these multi-qubit Paulis commute with each other, and

${\displaystyle Y_{1}Y_{2}X_{3}\cdot Y_{1}X_{2}Y_{3}\cdot X_{1}Y_{2}Y_{3}\cdot X_{1}X_{2}X_{3}=-1}$

due to the anticommutativity between ${\displaystyle X}$ and ${\displaystyle Y}$.

These results lead to a contradiction in any local hidden variable theory, where each measurement must have definite (classical) values ${\displaystyle x_{i},y_{i}=\pm 1}$ determined by hidden variables, because

${\displaystyle y_{1}y_{2}x_{3}\cdot y_{1}x_{2}y_{3}\cdot x_{1}y_{2}y_{3}\cdot x_{1}x_{2}x_{3}=x_{1}^{2}x_{2}^{2}x_{3}^{2}y_{1}^{2}y_{2}^{2}y_{3}^{2}}$

must equal +1, not −1.^[3]

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 a 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]}

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 ${\displaystyle A|B}$. The GHZ state is non-biseparable 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, ${\displaystyle |\mathrm {W} \rangle =(|001\rangle +|010\rangle +|100\rangle )/{\sqrt {3}}}$.^[8]: 903 Thus ${\displaystyle |\mathrm {GHZ} \rangle }$ and ${\displaystyle |\mathrm {W} \rangle }$ represent two very different kinds of entanglement for three or more particles.^[9] 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.

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.^[10] 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).

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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.

GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing^[11] or in the quantum Byzantine agreement.

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