We start showing the strategy with the help of Theorem 4 to derive correlation conditions for the generalized four-qubit GHZ state:
|Φ(θ, φ)i = cos(θ) |0000iz+ eiφsin(θ) |1111iz, (4.3)
for 0 < θ < π/4 and 0 ≤ φ < π/2, where |v1v2v3v4iz = ⊗4k=1|vikz for v ∈ {0, 1} and
|vikz corresponds to an eigenstate of σz with eigenvalue (−1)v for the party k. Firstly, to
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describe the correlation between a specific party and others of the four-qubit system, we give four sets of correlator operators:
Cˆ0,nz(z) = (ˆ0nz− ˆ1nz) ⊗ ˆ0mz⊗ ˆ0pz ⊗ ˆ0qz, ˆC1,nz(z) = (ˆ1nz− ˆ0nz) ⊗ ˆ1mz⊗ ˆ1pz ⊗ ˆ1qz, (4.4)
for n = 1, ..., 4, where ˆvnz = |vinznzhv| and n, m, p, and q denote four different parties un-der the local measurement setting, M4z = (Z, Z, Z, Z). In order to have compact forms, in what follows, symbols of tensor product will be omitted from correlator operators. Then, for some experimental output state, the expectation values of the Hermitian operators Cˆ0,n(z) and ˆC1,n(z) are expressed in the following correlators in terms of joint probabilities:
DCb0,n(z)E
= P (vn= 0, v = 0) − P (vn = 1, v = 0), DCb1,n(z)E
= P (vn= 1, v = 3) − P (vn = 0, v = 3), (4.5)
where v = P4
i=1,i6=nvi. By Theorem 4, we know that if results of measurements reveal that D
Cb0,n(z)E D Cb1,n(z)E
> 0, the outcomes of measurements performed on the nth party are correlated with the ones performed on the rest. If the nth party is independent of the rest, we have
For a pure generalized four-qubit GHZ state, |Φ(θ, φ)i, we have
C0,n,Φ(θ,φ)(z) = cos2(θ), C1,n,Φ(θ,φ)(z) = sin2(θ), (4.6)
and hence C0,n,Φ(θ,φ)(z) C1,n,Φ(θ,φ)(z) > 0, which describes the outcomes of measurements are cor-related. Then the condition, C0,n(z)C1,n(z) > 0, is a necessary condition of the pure generalized four-qubit GHZ state.
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Further, we construct the following correlator operators to identify correlations be-tween a specific group, which is composed of the nth and mth parties, and another :
Cˆ0,nm(z) = (ˆ0nzˆ0mz − ˆ1nzˆ1mz)ˆ0pzˆ0qz, ˆC1,nm(z) = (ˆ1nzˆ1mz− ˆ0nzˆ0mz)ˆ1pzˆ1qz, (4.7)
for n, m = 1, ..., 4 and n 6= m. Moreover, we can express the expectation values of the Hermitian operators ˆC0,nm(z) and ˆC1,nm(z) in terms of joint probabilities for some output state:
DCb0,nm(z) E
= P (vnm = 0, v′ = 0) − P (vnm= 2, v′ = 0), DCb1,nm(z) E
= P (vnm = 2, v′ = 2) − P (vnm= 0, v′ = 2), (4.8)
where vnm = vn+ vm and v′ = P4
i=1,i6=n6=mvi. Theorem 4 shows that if the subsystem composed of the nth and the mth parties is uncorrelated with another one, the measured outcomes must satisfy D
It is clear that, for a pure generalized four-qubit GHZ state, we have DCb0,nm(z) E
> 0. Thus we know that the subsystem composed of the nth and the mth parties are correlated with another. Therefore, the condition,D
Cb0,nm(z) E D
Cb1,nm(z) E
>
0, is also a necessary condition of the state |Φ(θ, φ)i.
After introducing two correlation conditions for the pure generalized GHZ state under M4z, let us progress towards the third one for correlation. Under the local measurement setting, M4x = (X, X, X, X), we formulate four sets of correlators which correspond to the following operators for identifying correlations between the nth party and others:
Cˆ0,n(x) = (ˆ0nx− ˆ1nx) ⊗ ˆE, ˆC1,n(x)= (ˆ1nx− ˆ0nx) ⊗ ˆO, (4.10)
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where
Eˆ = (ˆ0mxˆ0pxˆ0qx+ ˆ0mxˆ1pxˆ1qx+ ˆ1mxˆ0pxˆ1qx+ ˆ1mxˆ1pxˆ0qx), (4.11) Oˆ = (ˆ1mxˆ1pxˆ1qx+ ˆ1mxˆ0pxˆ0qx+ ˆ0mxˆ1pxˆ0qx+ ˆ0mxˆ0pxˆ1qx). (4.12)
From the expectation values of ˆC0,n(x) and ˆC1,n(x) for some state and Theorem 4, we could know the correlation behavior of the system, i.e., for a system in which the nth party is uncorrelated with the rest under M4x, the outcomes of measurements must satisfy the condition: C0,n(x)C1,n(x) ≤ 0.
For the pure state, |Φ(θ, φ)i, the expectation values of ˆCk,n(x) is given by DCb0,n(x)E
=D Cb1,n(x)E
= sin(2θ) cos(φ)/2, (4.13)
and ensure that there are correlations between measured outcomes under the local mea-surement setting, M4x. Thus the condition, C0,n(x)C1,n(x) > 0, is necessary for the pure generalized four-qubit GHZ state.
Entanglement imbedded in the pure generalized four-qubit GHZ state manifests itself via necessary conditions of correlations presented above under two local measurement settings. Therefore we combine all of the correlator operators involved in the necessary conditions:
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Table 4.1: Summaries of numerical results of αΦ(θ, φ) for WΦ(θ, φ), the parameters, γΦ, which are utilized to prove WΦ(θ, φ) and δnoise,Φ involved in robustness of the proposed witness operator for detecting truly multipartite entanglement. Three different cases for the state |Φ(θ, φ)i corresponding to WΦ(θ, φ) have been demonstrated.
(θ, φ) (π4,π6) (4.9π , 0) (3.7π ,π9) αΦ 9.01 9.21 8.92 γΦ 6.54 6.44 6.86 δΦ 0.139 0.150 0.169
and then utilize the operator ˆCΦ to construct witness operator for detections of truly multipartite entanglement. Three example are shown as follows. The witness operator:
WΦ(θ, φ) = αΦ(θ, φ)1 − ˆCΦ, (4.16)
where αΦ(θ, φ) is some constant, detects genuine multipartite entanglement for the cases, (θ, φ): (π/4, π/6) , (π/4.9, 0), and (π/3.7, π/9). Table 4.1 gives a summary of αΦ(θ, φ) for these cases.
In order to prove that WΦ(θ, φ) is a entanglement witness for detecting genuine mul-tipartite entanglement, we have to show the following comparison between
WΦp(θ, φ) = αΦp1− |Φ(θ, φ)i hΦ(θ, φ)| , (4.17)
and WΦ(θ, φ) [72]: if a state ρ satisfies Tr(WΦ(θ, φ)ρ) < 0, it also satisfies Tr(WΦp(θ, φ)ρ) <
0, i.e., WΦ(θ, φ) − γΦWΦp(θ, φ) ≥ 0, where γΦ(θ, φ) is some positive constant. Through the method given by Bourennane et al. [70], we derive the operator WΦp(θ, φ) and have αpΦ = cos2(θ) for 0 < θ ≤ π/4 and αpΦ = sin2(θ) for π/4 ≤ θ < π/2. Table 4.1 summarizes the parameters γΦ utilized to prove that the proposed operators are indeed entanglement witnesses for detecting truly multipartite entanglement.
In addition, we are concerned with the robustness to noise for the witness WΦ(θ, φ).
The robustness of WΦ(θ, φ) depends on the noise tolerance: pnoise < δnoise, is such that
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Table 4.2: Expectation values of three proposed entanglement witnesses including WΦ(π4,π6), WΦ(4.9π , 0), and WΦ(3.7π ,π9) for the pure states |Φi: en-tanglement. Three cases for the robustness to noise for the witness WΦ(θ, φ) have been summarized in Table 4.1.
Further, we show the expectation values of the proposed entanglement witnesses for different pure states by Table 4.2. From comparison with the results we know that a aim state, say |Φ(θ′, φ′)i, does not always give the smallest expectation value of the corresponding witness operator, WΦ(θ′, φ′). One can identify with the operator WΦ(θ′, φ′) that an experimental output ρ is truly multipartite entanglement if Tr(WΦ(θ′, φ′)ρ) < 0.
Further, if Tr(WΦ(θ′, φ′)ρ) < Tr(WΦ(θ′, φ′)|Φ(θ′, φ′)ihΦ(θ′, φ′)|), the state ρ is not in the state |Φ(θ′, φ′)i class.
The novel approach to derive ˆCΦ shown above can be applied to the cases for arbitrary number of qubits straightforwardly. One can formulate sets of correlator operators to identify correlations between two subsystems under two local measurement settings and then construct the witness operators further.