ABSTRACT (original article): Violation of the Bell-type inequalities is very necessary to confirm the existence of the nonlocality in the nonclassical (entangled) states. We have designed a customized operator which is made of the sum of the identity and Pauli matrices (, , , and ). We theoretically evaluate the Bell-type violation for the two-qubit Bell state and a four-qubit Dicke state, which gives the Bell-CHSH parameter values and , respectively for our customized operator. For experimental implementation, IBM’s 127-qubitQuantum Processing Units (QPU) were utilized, where we have applied our customized operator to evaluate Bell-type inequalities for two-qubit Bell state and four-qubit Dicke state . We observed, for the two-qubit Bell state, the experimental Bell violation was . For Dicke state, we found the violation be to and respectively for two distinct methods of state preparation. All our results show clear violation of the local realism; however, we find that the experimental violation of the Bell state is close to the theoretical results due to lower circuit depth in state-preparation as well as fewer measurements, while the Dicke state shows greater errors (2.12 and 2.21 vs. 3.05) from higher depth and more measurements. CITATION (original article): Tomis, Harsh Mehta, Shreya Banerjee, Prasanta K. Panigrahi, V. Narayanan (2024), Experimental demonstration of the Bell-type inequalities for four qubit Dicke state using IBM Quantum Processing Unit, arXiv:2410.20241. https://doi.org/10.48550/arXiv.2410.20241
I
σ
x
σ
y
σ
z
2
2
3.05
(〉)
+
Φ
(2)
D
4
2.7507±0.0197
2.1239±0.0457
2.2175±0.0352
(2.75)
(2.82)
The goal here is to finding the Bell type inequality for the two qubit Bell state and four qubit Dicke state using our derived customized operator.
For the Two qubit
For the Two qubit
Defining the Parameters for all
In[]:=
Id={{1,0},{0,1}};Sigx={{0,1},{1,0}};Sigy={{0,-I},{I,0}};Sigz={{1,0},{0,-1}};H={{1},{0}};V={{0},{1}};
Defining the Bell state
In[]:=
ϕ=(KroneckerProduct[H,H]+KroneckerProduct[V,V]);
1
Sqrt[2]
Developing our customized operator for the violation of the Bell inequality for two qubit Bell state--
To achieve the maximum violation of any Bell-type inequalities, we assign a local arbitrary observable defined by a general operator X̂n = x · σ̂, where x = (x₀, x₁, x₂, x₃) ∈ ℝ⁴ with Σᵢ₌₀³ xᵢ² = 1, and the Pauli operator σ̂ = (Î, σ̂ₓ, σ̂ᵧ, σ̂𝓏). In a four-dimensional spherical space, X̂n can be represented as:
X̂n = cos(γ) Î − sin(γ) [ sin(θ) cos(ϕ) σ̂x + sin(θ) sin(ϕ) σ̂y + cos(θ) σ̂z ]
To achieve the maximum violation of any Bell-type inequalities, we assign a local arbitrary observable defined by a general operator X̂n = x · σ̂, where x = (x₀, x₁, x₂, x₃) ∈ ℝ⁴ with Σᵢ₌₀³ xᵢ² = 1, and the Pauli operator σ̂ = (Î, σ̂ₓ, σ̂ᵧ, σ̂𝓏). In a four-dimensional spherical space, X̂n can be represented as:
X̂n = cos(γ) Î − sin(γ) [ sin(θ) cos(ϕ) σ̂x + sin(θ) sin(ϕ) σ̂y + cos(θ) σ̂z ]
In[]:=
A=Cos[γ1]Id-*Sin[γ1](Sin[θ1]Cos[ϕ1]Sigx+Sin[θ1]Sin[ϕ1]Sigy+Cos[θ1]Sigz);B=Cos[γ2]Id-*Sin[γ2](Sin[θ2]Cos[ϕ2]Sigx+Sin[θ2]Sin[ϕ2]Sigy+Cos[θ2]Sigz);a=Cos[γ1p]Id-*Sin[γ1p](Sin[θ1p]Cos[ϕ1p]Sigx+Sin[θ1p]Sin[ϕ1p]Sigy+Cos[θ1p]Sigz);b=Cos[γ2p]Id-*Sin[γ2p](Sin[θ2p]Cos[ϕ2p]Sigx+Sin[θ2p]Sin[ϕ2p]Sigy+Cos[θ2p]Sigz);
Finding the Tensor product of the operator necessary for the violation of the Bell inequality. Bell inequality can be defined as-
|S|Φ+⟩ =| ⟨Â B̂⟩ + ⟨Â B̂′⟩ − ⟨Â′ B̂⟩ + ⟨Â′ B̂′⟩ | ≤ 2
|S|Φ+⟩ =| ⟨Â B̂⟩ + ⟨Â B̂′⟩ − ⟨Â′ B̂⟩ + ⟨Â′ B̂′⟩ | ≤ 2
In[]:=
c1=KroneckerProduct[A,B];c2=KroneckerProduct[A,b];c3=KroneckerProduct[a,B];c4=KroneckerProduct[a,b];
The quantum mechanical expectation value specified in the first term of Eq.(5) can be calculated for the |Φ+⟩ Bell state as:
⟨Â B̂⟩ = ⟨Φ+| X̂nA ⊗ X̂nB |Φ+⟩, and all the remaining terms
⟨Â B̂⟩ = ⟨Φ+| X̂nA ⊗ X̂nB |Φ+⟩, and all the remaining terms
In[]:=
CH1=Dot[ϕ,c1,ϕ];CH2=Dot[ϕ,c2,ϕ];CH3=Dot[ϕ,c3,ϕ];CH4=Dot[ϕ,c4,ϕ];
Defining the Bell inequality
In[]:=
CHSH=CH1+CH2-CH3+CH4;Observable=CHSH[[1,1]];
The violation of the Bell inequality can be found as--
In[]:=
FindMaximum[{Re[Observable],0<θ1<π&&0<θ2<π&&0<θ1p<π&&0<θ2p<π&&0<ϕ1<2π&&0<ϕ2<2π&&0<ϕ1p<2π&&0<ϕ2p<2π&&0<γ1<π&&0<γ2<π&&0<γ1p<π&&0<γ2p<π},θ1,θ2,θ3,θ1p,θ2p,θ3p,ϕ1,ϕ2,ϕ3,ϕ1p,ϕ2p,ϕ3p,γ1,γ2,γ3,γ1p,γ2p,γ3p]
This is the final result for the two qubit Bell state
{{2.82843,{θ12.0315,θ20.299561,θ31.,θ1p0.357995,θ2p2.21797,θ3p1.,ϕ11.5978,ϕ20.89895,ϕ31.,ϕ1p0.895891,ϕ2p1.8081,ϕ3p1.,γ10.679214,γ21.1037,γ31.,γ1p1.42907,γ2p0.863141,γ3p1.}}}
Bell type inequality violation for the four qubit Dicke State
Bell type inequality violation for the four qubit Dicke State
Defining the Dicke state
In[]:=
D4=(KroneckerProduct[H,H,V,V]+KroneckerProduct[H,V,H,V]+KroneckerProduct[V,H,H,V]+KroneckerProduct[H,V,V,H]+KroneckerProduct[V,H,V,H]+KroneckerProduct[V,V,H,H]);
1
6
Defining the customized operator in the similar manner
In[]:=
A=Cos[γ1]Id-*Sin[γ1](Sin[θ1]Cos[ϕ1]Sigx+Sin[θ1]Sin[ϕ1]Sigy+Cos[θ1]Sigz);B=Cos[γ2]Id-*Sin[γ2](Sin[θ2]Cos[ϕ2]Sigx+Sin[θ2]Sin[ϕ2]Sigy+Cos[θ2]Sigz);S=Cos[γ3]Id-*Sin[γ3](Sin[θ3]Cos[ϕ3]Sigx+Sin[θ3]Sin[ϕ3]Sigy+Cos[θ3]Sigz);T=Cos[γ4]Id-*Sin[γ4](Sin[θ4]Cos[ϕ4]Sigx+Sin[θ4]Sin[ϕ4]Sigy+Cos[θ4]Sigz);a=Cos[γ1p]Id-*Sin[γ1p](Sin[θ1p]Cos[ϕ1p]Sigx+Sin[θ1p]Sin[ϕ1p]Sigy+Cos[θ1p]Sigz);b=Cos[γ2p]Id-*Sin[γ2p](Sin[θ2p]Cos[ϕ2p]Sigx+Sin[θ2p]Sin[ϕ2p]Sigy+Cos[θ2p]Sigz);s=Cos[γ3p]Id-*Sin[γ3p](Sin[θ3p]Cos[ϕ3p]Sigx+Sin[θ3p]Sin[ϕ3p]Sigy+Cos[θ3p]Sigz);t=Cos[γ4p]Id-*Sin[γ4p](Sin[θ4p]Cos[ϕ4p]Sigx+Sin[θ4p]Sin[ϕ4p]Sigy+Cos[θ4p]Sigz);
For the four qubit Dicke state, the inequality is given by:
| Â B̂ Ĉ D̂ + Â B̂′ Ĉ′ D̂′ + Â′ B̂ Ĉ′ D̂ − Â′ B̂′ Ĉ D̂′ | ≤ 2.
| Â B̂ Ĉ D̂ + Â B̂′ Ĉ′ D̂′ + Â′ B̂ Ĉ′ D̂ − Â′ B̂′ Ĉ D̂′ | ≤ 2.
In[]:=
e1=KroneckerProduct[A,B,S,T];e2=KroneckerProduct[A,b,s,t];e3=KroneckerProduct[a,B,s,T];e4=KroneckerProduct[a,b,S,t];
The evaluation of the first term of the inequality and as well remaining terms
In[]:=
E1=Dot[D4,e1,D4]//FullSimplify
Out[]=
(Cos[γ1](Sin[γ2](Cos[γ4]Sin[γ3](Cos[θ2]Cos[θ3]-2Cos[ϕ2-ϕ3]Sin[θ2]Sin[θ3])+Cos[γ3]Sin[γ4](Cos[θ2]Cos[θ4]-2Cos[ϕ2-ϕ4]Sin[θ2]Sin[θ4]))+Cos[γ2](3Cos[γ3]Cos[γ4]+Sin[γ3]Sin[γ4](Cos[θ3]Cos[θ4]-2Cos[ϕ3-ϕ4]Sin[θ3]Sin[θ4])))+Sin[γ1](Cos[γ3](Cos[γ4]Sin[γ2](Cos[θ1]Cos[θ2]-2Cos[ϕ1-ϕ2]Sin[θ1]Sin[θ2])+Cos[γ2]Sin[γ4](Cos[θ1]Cos[θ4]-2Cos[ϕ1-ϕ4]Sin[θ1]Sin[θ4]))+Sin[γ3](Cos[γ2]Cos[γ4](Cos[θ1]Cos[θ3]-2Cos[ϕ1-ϕ3]Sin[θ1]Sin[θ3])+Sin[γ2]Sin[γ4](Cos[θ4](-2Sin[θ1](Cos[θ3]Cos[ϕ1-ϕ2]Sin[θ2]+Cos[θ2]Cos[ϕ1-ϕ3]Sin[θ3])+Cos[θ1](3Cos[θ2]Cos[θ3]-2Cos[ϕ2-ϕ3]Sin[θ2]Sin[θ3]))+(-2Cos[θ2](Cos[θ3]Cos[ϕ1-ϕ4]Sin[θ1]+Cos[θ1]Cos[ϕ3-ϕ4]Sin[θ3])+Sin[θ2](-2Cos[θ1]Cos[θ3]Cos[ϕ2-ϕ4]+(Cos[ϕ1+ϕ2-ϕ3-ϕ4]+2Cos[ϕ1-ϕ2]Cos[ϕ3-ϕ4])Sin[θ1]Sin[θ3]))Sin[θ4]))))
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3
E1=Dot[D4,e1,D4]//FullSimplify;E2=Dot[D4,e2,D4]//FullSimplify;E3=Dot[D4,e3,D4]//FullSimplify;E4=Dot[D4,e4,D4]//FullSimplify;
Here we define the Bell inequality for the four qubit Dicke state
In[]:=
B=E1+E2+E3-E4//FullSimplify
Out[]=
Using this code we found the maximal violation of the Bell inequality for four qubit Dicke state
In[]:=
Fi=B[[1,1]];
In[]:=
FindMaximumFi,-<θ1<&&-<θ2<&&-<θ3<&&-<θ4<&&-<θ1p<&&-<θ2p<&&-<θ3p<&&-<θ4p<&&-π<ϕ1<π&&-π<ϕ2<π&&-π<ϕ3<π&&-π<ϕ4<π&&-π<ϕ1p<π&&-π<ϕ2p<π&&-π<ϕ3p<π&&-π<ϕ4p<π&&-<γ1<&&-<γ2<&&-<γ3<&&-<γ4<&&-<γ1p<&&-<γ2p<&&-<γ3p<&&-<γ4p<,θ1,θ2,θ3,θ4,θ1p,θ2p,θ3p,θ4p,ϕ1,ϕ2,ϕ3,ϕ4,ϕ1p,ϕ2p,ϕ3p,ϕ4p,γ1,γ2,γ3,γ4,γ1p,γ2p,γ3p,γ4p
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This is the final result
{3.05505,{θ1-0.670099,θ2-0.670899,θ31.33755,θ4-0.670899,θ1p1.33826,θ2p0.756904,θ3p-0.671585,θ4p0.756912,ϕ10.993418,ϕ20.993418,ϕ30.993418,ϕ40.993418,ϕ1p0.993418,ϕ2p0.993418,ϕ3p0.993418,ϕ4p0.993418,γ10.513594,γ20.513863,γ31.00891,γ40.513863,γ1p1.0903,γ2p0.546396,γ3p0.514098,γ4p0.546399}}
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Experimental demonstration of the Bell-type inequalities for four qubit Dicke state using IBM quantum processing unit
by Tomis Prajapati, Harsh Mehta, Shreya Banerjee, Prasanta K. Panigrahi, V. Narayanan
Wolfram Community, STAFF PICKS, November 12, 2024
https://community.wolfram.com/groups/-/m/t/3318770
by Tomis Prajapati, Harsh Mehta, Shreya Banerjee, Prasanta K. Panigrahi, V. Narayanan
Wolfram Community, STAFF PICKS, November 12, 2024
https://community.wolfram.com/groups/-/m/t/3318770