Greenberger–Horne–Zeilinger state

<h2 id="history">History</h2>
The four-particle version was first studied by <a href="/facts/Daniel_Greenberger/THe2wFmd">Daniel Greenberger</a>, <a href="/facts/Michael_Horne_(physicist)/2l2RtuDh">Michael Horne</a> and <a href="/facts/Anton_Zeilinger/bJoKANHh">Anton Zeilinger</a> in 1989.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> The following year <a href="/facts/Abner_Shimony/x2NjV8p1">Abner Shimony</a> joined in and they published a three-particle version<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> based on suggestions
by <a href="/facts/N._David_Mermin/ALGkFowC">N. David Mermin</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="definition">Definition</h2>
The GHZ state is an <a href="/facts/Quantum_entanglement/afzslwN0">entangled</a> <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> for 3 <a href="/facts/Qubit/WveeEubw">qubits</a> 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.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Another example<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> of a GHZ state is three <a href="/facts/Photon/mFNhAEzA">photons</a> in an <a href="/facts/Quantum_entanglement/afzslwN0">entangled</a> state, with the photons being in a <a href="/facts/Quantum_superposition/4z1UJg9o">superposition</a> of being all horizontally <a href="/facts/Polarization_(waves)/fP33m0eD">polarized</a> (HHH) or all vertically polarized (VVV), with respect to some <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a>. The GHZ state can be written in <a href="/facts/Bra%25E2%2580%2593ket_notation/NDlKsaRh">bra–ket notation</a> 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 <a href="/facts/Polarizer/kPDwyboP">polarizer</a> 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.

<h3>Generalization</h3>
The generalized GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension 
 
 
 
 d
 
 
 {\displaystyle d}
 
, i.e., the local <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> is isomorphic to 
 
 
 
 
 
 C
 
 
 d
 
 
 
 
 {\displaystyle \mathbb {C} ^{d}}
 
, then the total Hilbert space of an 
 
 
 
 M
 
 
 {\displaystyle 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
 
 
 {\displaystyle 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

|
        
        
          G
          H
          Z
        
        ⟩
        =
        
          
            
              
                |
              
              0
              
                ⟩
                
                  ⊗
                  M
                
              
              +
              
                |
              
              1
              
                ⟩
                
                  ⊗
                  M
                
              
            
            
              2
            
          
        
        .
      
    
    {\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}}.}

<h2 id="ghz-experiment">GHZ experiment</h2>
In the language of <a href="/facts/Quantum_computation/nbbcgmvq">quantum computation</a>, the polarization state of each photon is a <a href="/facts/Qubit/WveeEubw">qubit</a>, the basis of which can be chosen to be

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

With appropriately chosen <a href="/facts/Phase_factor/UPfymTfP">phase factors</a> for 
 
 
 
 
 |
 
 
 H
 
 ⟩
 
 
 {\displaystyle |\mathrm {H} \rangle }
 
 and 
 
 
 
 
 |
 
 
 V
 
 ⟩
 
 
 {\displaystyle |\mathrm {V} \rangle }
 
, both types of measurements used in the experiment becomes <a href="/facts/Measurement_in_quantum_mechanics/VY7jPfP0">Pauli measurements</a>, with the two possible results represented as +1 and −1 respectively:

<ul><li>The 45° linear polarizer implements a Pauli 
 
 
 
 X
 
 
 {\displaystyle X}
 
 measurement, distinguishing between the eigenstates</li></ul>

|
        
        
          +
          X
        
        ⟩
        ≡
        
          |
        
        +
        ⟩
        =
        
          
            1
            
              2
            
          
        
        (
        
          |
        
        
          H
        
        ⟩
        +
        
          |
        
        
          V
        
        ⟩
        )
        ,
        
        
          |
        
        
          −
          X
        
        ⟩
        ≡
        
          |
        
        −
        ⟩
        =
        
          
            1
            
              2
            
          
        
        (
        
          |
        
        
          H
        
        ⟩
        −
        
          |
        
        
          V
        
        ⟩
        )
        .
      
    
    {\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 ).}

<ul><li>The circular polarizer implements a Pauli 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 measurement, distinguishing between the eigenstates</li></ul>

|
        
        
          +
          Y
        
        ⟩
        ≡
        
          |
        
        R
        ⟩
        =
        
          
            1
            
              2
            
          
        
        (
        
          |
        
        
          H
        
        ⟩
        +
        i
        
          |
        
        
          V
        
        ⟩
        )
        ,
        
        
          |
        
        
          −
          Y
        
        ⟩
        ≡
        
          |
        
        L
        ⟩
        =
        
          
            1
            
              2
            
          
        
        (
        
          |
        
        
          H
        
        ⟩
        −
        i
        
          |
        
        
          V
        
        ⟩
        )
        .
      
    
    {\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 <a href="/facts/Generalizations_of_Pauli_matrices/SjqqBE28">multi-qubit Pauli</a> 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 
 
 
 
 
 Y
 
 1
 
 
 
 Y
 
 2
 
 
 
 X
 
 3
 
 
 
 
 {\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

⟨
        
          G
          H
          Z
        
        
          |
        
        
          Y
          
            1
          
        
        
          Y
          
            2
          
        
        
          X
          
            3
          
        
        
          |
        
        
          G
          H
          Z
        
        ⟩
        =
        ⟨
        
          G
          H
          Z
        
        
          |
        
        
          Y
          
            1
          
        
        
          X
          
            2
          
        
        
          Y
          
            3
          
        
        
          |
        
        
          G
          H
          Z
        
        ⟩
        =
        ⟨
        
          G
          H
          Z
        
        
          |
        
        
          X
          
            1
          
        
        
          Y
          
            2
          
        
        
          Y
          
            3
          
        
        
          |
        
        
          G
          H
          Z
        
        ⟩
        =
        −
        1
        ,
      
    
    {\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,}

⟨
        
          G
          H
          Z
        
        
          |
        
        
          X
          
            1
          
        
        
          X
          
            2
          
        
        
          X
          
            3
          
        
        
          |
        
        
          G
          H
          Z
        
        ⟩
        =
        +
        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

Y
          
            1
          
        
        
          Y
          
            2
          
        
        
          X
          
            3
          
        
        ⋅
        
          Y
          
            1
          
        
        
          X
          
            2
          
        
        
          Y
          
            3
          
        
        ⋅
        
          X
          
            1
          
        
        
          Y
          
            2
          
        
        
          Y
          
            3
          
        
        ⋅
        
          X
          
            1
          
        
        
          X
          
            2
          
        
        
          X
          
            3
          
        
        =
        −
        1
      
    
    {\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 <a href="/facts/Anticommutativity/N2xrAKH5">anticommutativity</a> between 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.
These results lead to a contradiction in any local hidden variable theory, where each measurement must have definite (classical) values 
 
 
 
 
 x
 
 i
 
 
 ,
 
 y
 
 i
 
 
 =
 ±
 1
 
 
 {\displaystyle x_{i},y_{i}=\pm 1}
 
 determined by hidden variables, because

y
          
            1
          
        
        
          y
          
            2
          
        
        
          x
          
            3
          
        
        ⋅
        
          y
          
            1
          
        
        
          x
          
            2
          
        
        
          y
          
            3
          
        
        ⋅
        
          x
          
            1
          
        
        
          y
          
            2
          
        
        
          y
          
            3
          
        
        ⋅
        
          x
          
            1
          
        
        
          x
          
            2
          
        
        
          x
          
            3
          
        
        =
        
          x
          
            1
          
          
            2
          
        
        
          x
          
            2
          
          
            2
          
        
        
          x
          
            3
          
          
            2
          
        
        
          y
          
            1
          
          
            2
          
        
        
          y
          
            2
          
          
            2
          
        
        
          y
          
            3
          
          
            2
          
        
      
    
    {\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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
The results of actual experiments agree with the predictions of quantum mechanics, not those of local realism.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="properties">Properties</h2>
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 <a href="/facts/Maximally_entangled_state/afzslwN0">maximally entangled state</a>.
Another important property of the GHZ state is that taking the <a href="/facts/Partial_trace/ra7RhkkT">partial trace</a> over one of the three systems yields

Tr
          
            3
          
        
        ⁡
        
          [
          
            
              (
              
                
                  
                    
                      |
                    
                    000
                    ⟩
                    +
                    
                      |
                    
                    111
                    ⟩
                  
                  
                    2
                  
                
              
              )
            
            
              (
              
                
                  
                    ⟨
                    000
                    
                      |
                    
                    +
                    ⟨
                    111
                    
                      |
                    
                  
                  
                    2
                  
                
              
              )
            
          
          ]
        
        =
        
          
            
              (
              
                |
              
              00
              ⟩
              ⟨
              00
              
                |
              
              +
              
                |
              
              11
              ⟩
              ⟨
              11
              
                |
              
              )
            
            2
          
        
        ,
      
    
    {\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 <a href="/facts/Mixed_state_(physics)/7jbxd7Dx">mixed state</a>. It has certain two-particle (qubit) correlations, but these are <a href="/facts/Covariation/hILocGNH">of a classical nature</a>. 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 
 
 
 
 
 |
 
 00
 ⟩
 
 
 {\displaystyle |00\rangle }
 
 or 
 
 
 
 
 |
 
 11
 ⟩
 
 
 {\displaystyle |11\rangle }
 
, which are unentangled pure states. This is unlike the <a href="/facts/W_state/TTmHz55g">W state</a>, which leaves bipartite entanglements even when we measure one of its subsystems.
A pure state 
 
 
 
 
 |
 
 ψ
 ⟩
 
 
 {\displaystyle |\psi \rangle }
 
 of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 parties is called biseparable, if one can find a partition of the parties in two nonempty disjoint subsets 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 with 
 
 
 
 A
 ∪
 B
 =
 {
 1
 ,
 …
 ,
 N
 }
 
 
 {\displaystyle A\cup B=\{1,\dots ,N\}}
 
 such that 
 
 
 
 
 |
 
 ψ
 ⟩
 =
 
 |
 
 ϕ
 
 ⟩
 
 A
 
 
 ⊗
 
 |
 
 γ
 
 ⟩
 
 B
 
 
 
 
 {\displaystyle |\psi \rangle =|\phi \rangle _{A}\otimes |\gamma \rangle _{B}}
 
, i.e. 
 
 
 
 
 |
 
 ψ
 ⟩
 
 
 {\displaystyle |\psi \rangle }
 
 is a <a href="/facts/Product_state/uVfzaRUq">product state</a> with respect to the partition 
 
 
 
 A
 
 |
 
 B
 
 
 {\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 <a href="/facts/LOCC/m4gfHgUC">local quantum operations</a>, the other being the <a href="/facts/W_state/TTmHz55g">W state</a>, 
 
 
 
 
 |
 
 
 W
 
 ⟩
 =
 (
 
 |
 
 001
 ⟩
 +
 
 |
 
 010
 ⟩
 +
 
 |
 
 100
 ⟩
 )
 
 /
 
 
 
 3
 
 
 
 
 {\displaystyle |\mathrm {W} \rangle =(|001\rangle +|010\rangle +|100\rangle )/{\sqrt {3}}}
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>: 903 
Thus 
 
 
 
 
 |
 
 
 G
 H
 Z
 
 ⟩
 
 
 {\displaystyle |\mathrm {GHZ} \rangle }
 
 and 
 
 
 
 
 |
 
 
 W
 
 ⟩
 
 
 {\displaystyle |\mathrm {W} \rangle }
 
 represent two very different kinds of entanglement for three or more particles.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
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 <a href="/facts/Bell%2527s_theorem/lg1V6kpk">Bell's theorem</a>, which shows the internal inconsistency with the notion of "elements of reality" introduced in the famous <a href="/facts/Einstein%25E2%2580%2593Podolsky%25E2%2580%2593Rosen_paradox/rulT1oGp">Einstein–Podolsky–Rosen</a> article. The first laboratory observation of GHZ correlations was by the group of <a href="/facts/Anton_Zeilinger/bJoKANHh">Anton Zeilinger</a> (1998), who was awarded a share of the 2022 Nobel Prize in physics for this work.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> Many more accurate observations followed. The correlations can be utilized in some <a href="/facts/Quantum_information/Bl9QIvHj">quantum information</a> tasks. These include multipartner <a href="/facts/Quantum_cryptography/FPqPdNXT">quantum cryptography</a> (1998) and <a href="/facts/Communication_complexity/TXLhEfQW">communication complexity</a> tasks (1997, 2004).

<h2 id="pairwise-entanglement">Pairwise entanglement</h2>

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 <a href="/facts/Bell_state/8ubaX7xG">Bell state</a>. This is illustrated below.
The 3-qubit GHZ state can be written as

|
        
        
          G
          H
          Z
        
        ⟩
        =
        
          
            1
            
              2
            
          
        
        
          (
          
            
              |
            
            000
            ⟩
            +
            
              |
            
            111
            ⟩
          
          )
        
        =
        
          
            1
            2
          
        
        
          (
          
            
              |
            
            00
            ⟩
            +
            
              |
            
            11
            ⟩
          
          )
        
        ⊗
        
          |
        
        +
        ⟩
        +
        
          
            1
            2
          
        
        
          (
          
            
              |
            
            00
            ⟩
            −
            
              |
            
            11
            ⟩
          
          )
        
        ⊗
        
          |
        
        −
        ⟩
        ,
      
    
    {\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 
 
 
 
 
 |
 
 0
 ⟩
 =
 (
 
 |
 
 +
 ⟩
 +
 
 |
 
 −
 ⟩
 )
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle |0\rangle =(|+\rangle +|-\rangle )/{\sqrt {2}}}
 
 and 
 
 
 
 
 |
 
 1
 ⟩
 =
 (
 
 |
 
 +
 ⟩
 −
 
 |
 
 −
 ⟩
 )
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle |1\rangle =(|+\rangle -|-\rangle )/{\sqrt {2}}}
 
.
A measurement of the GHZ state along the X basis for the third particle then yields either 
 
 
 
 
 |
 
 
 Φ
 
 +
 
 
 ⟩
 =
 (
 
 |
 
 00
 ⟩
 +
 
 |
 
 11
 ⟩
 )
 
 /
 
 
 
 2
 
 
 
 
 {\displaystyle |\Phi ^{+}\rangle =(|00\rangle +|11\rangle )/{\sqrt {2}}}
 
, if 
 
 
 
 
 |
 
 +
 ⟩
 
 
 {\displaystyle |+\rangle }
 
 was measured, or 
 
 
 
 
 |
 
 
 Φ
 
 −
 
 
 ⟩
 =
 (
 
 |
 
 00
 ⟩
 −
 
 |
 
 11
 ⟩
 )
 
 /
 
 
 
 2
 
 
 
 
 {\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 <a href="/facts/Quantum_gate/EFV5sPCU">quantum gate</a> to give 
 
 
 
 
 |
 
 
 Φ
 
 +
 
 
 ⟩
 
 
 {\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 <a href="/facts/Maximally_entangled_state/afzslwN0">maximally entangled state</a>.

<h2 id="applications">Applications</h2>
GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> or in the <a href="/facts/Quantum_Byzantine_agreement/suau4UR3">quantum Byzantine agreement</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bell%2527s_theorem/lg1V6kpk">Bell's theorem</a></li>
<li><a href="/facts/Local_hidden-variable_theory/wXo2mmvp">Local hidden-variable theory</a></li>
<li><a href="/facts/NOON_state/CmYRdevv">NOON state</a></li>
<li><a href="/facts/Quantum_pseudo-telepathy/EA4Nn93J">Quantum pseudo-telepathy</a></li>
<li><a href="/facts/Dicke_state/jX9d4hzf">Dicke state</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">D. Greenberger; M. Horne; A. Shimony; A. Zeilinger (1990). "Bell's theorem without inequalities". Am. J. Phys. 58 (12): 1131. Bibcode:1990AmJPh..58.1131G. doi:10.1119/1.16243. <a href="https://doi.org/10.1119%2F1.16243" target="_blank">https://doi.org/10.1119%2F1.16243</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="/wiki/American_Journal_of_Physics" target="_blank">/wiki/American_Journal_of_Physics</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. <a href="/wiki/Carlton_M._Caves" target="_blank">/wiki/Carlton_M._Caves</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">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. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6513338" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6513338</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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. <a href="/wiki/American_Journal_of_Physics" target="_blank">/wiki/American_Journal_of_Physics</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">A. Zeilinger, Dance of the Photons, Farrar, Straus and Giroux, New York, 2010, pp. 218–223. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">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. <a href="/wiki/American_Journal_of_Physics" target="_blank">/wiki/American_Journal_of_Physics</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Jian-Wei Pan; D. Bouwmeester; M. Daniell; H. Weinfurter; A. Zeilinger (2000). "Experimental test of quantum nonlocality in three-photon GHZ entanglement". Nature. 403 (6769): 515–519. Bibcode:2000Natur.403..515P. doi:10.1038/35000514. PMID 10676953. S2CID 4309261. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Horodecki, Ryszard; Horodecki, Pawel; Horodecki, Michal; Horodecki, Karol (2009). "Quantum entanglement". Reviews of Modern Physics. 81 (2): 865–942. arXiv:quant-ph/0702225. Bibcode:2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865. S2CID 59577352. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">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 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Scientific Background on the Nobel Prize in Physics 2022" (PDF). The Nobel Prize. October 4, 2022. <a href="https://www.nobelprize.org/uploads/2023/10/advanced-physicsprize2022-4.pdf" target="_blank">https://www.nobelprize.org/uploads/2023/10/advanced-physicsprize2022-4.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">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 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
</ol>

Greenberger–Horne–Zeilinger state open-in-new

Greenberger–Horne–Zeilinger state