Bell's Theorem

Alice and Bob have instrument S(θ,ϕ) that can measure the spin along an axis defined by the unit vector 
(OverVector[n ])  = Sin(θ)Cos(ϕ) (OverVector[i]) +  Sin(θ)Cos(ϕ) (OverVector[j])  + Cos(θ)(OverVector[ k])

S(θ,ϕ) ≡ S·(OverVector[n ])

S[θ_,ϕ_]={{Cos[θ],Exp[-I ϕ]Sin[θ]},{Exp[I ϕ]Sin[θ],-Cos[θ]}};
up[θ_,ϕ_]={Cos[θ/2],Exp[I ϕ] Sin[θ/2]};
down[θ_,ϕ_]={-Sin[(θ/2)],(E^(I ϕ)) Cos[(θ/2)]};
unit={{1,0},{0,1}};

They choose three possible angles A, B, C, so that S·(OverVector[n ]) is alinged in the x-z plane to make their measurements. We have
θ=(Subscript[θ, A]), (Subscript[ϕ, A])=0   θ=(Subscript[θ, B]), (Subscript[ϕ, B])=0  θ=(Subscript[θ, C])   (Subscript[ϕ, B])=π  (i.e. A and B are in the first and second quadrant of the z-x plane, and C is in the 4'th quadrant, going around counterclockwise in that plane)

Alice and Bob measure the the spin of the Qubits they each receive. The two Qubits are in the Bell state

|ψ⟩ = (1/(Sqrt[2])) |01⟩ - (1/(Sqrt[2]))|1 0⟩

If Alice chooses the instrument A and Bob the instrument B, their measurement setup would look  like

After a repeated number of mesurements we can calculate the expectation value (correlation) of the product A B for those measurements. According to the Born rule it is given by the eqpression  ⟨ψ| A B |ψ⟩ which we evaluate below.

psi=1/Sqrt[2](Flatten[KroneckerProduct[{1,0},{0,1}]]-Flatten[KroneckerProduct[{0,1},{1,0}]]);
MatrixForm[psi]

A=KroneckerProduct[S[θA,0],unit];
B=KroneckerProduct[unit,S[θB,0]];

EAB=FullSimplify[Conjugate[psi].A.B.psi]

So if A and B have the same orientation, we find that EAB=-1.  That is perfect anti-correlation, similar to a Bertlemann's socks scenario. Consider the case where Alice has orientation A and Bob orientation C

c=KroneckerProduct[unit,S[θC,Pi]];
EAC=FullSimplify[Conjugate[psi].A.c.psi]

Finally we choose Alice to have orientation B and Bob orientation C, then

B=KroneckerProduct[S[θB,0],unit];
EBC=FullSimplify[Conjugate[psi].B.c.psi]

Bell's proof,  incorporating local realism.

John Bell asked the question, what would the correlation values be if we strictly enforced local realism? Under this
restriction the random variable A=±1 obtained with measurements of Alice's device must only depend on the orientation of Alice's device  A = A((Subscript[(OverHat[n]), A])) , it should not depend on (Subscript[(OverHat[n]), B]) the orientation of Bob's device. However it may depend on other local variables, which we collectively call λ, and are hidden from us. These hidden parameters determine the properties of both Alice's and Bob's observed particles . Therefore we let  A = A((Subscript[(OverHat[n]), A]),λ). In the same manner, for the random variable measured by Bob we assume B = B((Subscript[(OverHat[n]), B]),λ).   so the expectation value  

⟨A B⟩  =  ∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) B((Subscript[(OverHat[n]), B]),λ) 

is the central assumption of Bell's local realism requirement. Here ρ(λ) is a probabilty for the system to be in a state for a given value of λ, the hidden variables.  Now if  ρ(λ) is a probability distribution

 ∫dλ ρ(λ)=1

If  (Subscript[(OverHat[n]), A])=(Subscript[(OverHat[n]), B]) we know that we get perfect anti-correlation, i.e. ⟨A B⟩  = -1, therefore we require B((Subscript[(OverHat[n]), B]),λ) = -A((Subscript[(OverHat[n]), A]),λ) since 


- ∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) A((Subscript[(OverHat[n]), B]),λ)  = -1  if (Subscript[(OverHat[n]), B])=(Subscript[(OverHat[n]), A])   (since (A^2)=1).

and the former relation must  hold  for all possible orientations.  Therefore

⟨A B⟩  =  -∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) A((Subscript[(OverHat[n]), B]),λ) 

 Thus

⟨A C⟩  =  -∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) A((Subscript[(OverHat[n]), C]),λ)  

⟨B C⟩  =  -∫dλ ρ(λ) A((Subscript[(OverHat[n]), B]),λ) A((Subscript[(OverHat[n]), C]),λ)

Now 

⟨AB⟩ - ⟨AC⟩  =  -∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) [  A((Subscript[(OverHat[n]), B]),λ) - A((Subscript[(OverHat[n]), C]),λ) ] =   -∫dλ ρ(λ) A((Subscript[(OverHat[n]), A]),λ) A((Subscript[(OverHat[n]), B]),λ)[ 1 - A((Subscript[(OverHat[n]), B]),λ)  A((Subscript[(OverHat[n]), C]),λ)]

the second equality comes from the fact that  A((Subscript[(OverHat[n]), B]),λ) A((Subscript[(OverHat[n]), B]),λ) =1.

Now, since [ 1 - A((Subscript[(OverHat[n]), B]),λ)  A((Subscript[(OverHat[n]), C]),λ)] ⩾ 0 we have 

|⟨AB⟩ - ⟨AC⟩ |  ⩽   |∫dλ ρ(λ) [ 1 - A((Subscript[(OverHat[n]), B]),λ)  A((Subscript[(OverHat[n]), C]),λ) ] | = 1 + ⟨B C⟩

Thus we get the Bell inequality

|⟨AB⟩ - ⟨AC⟩ | ⩽ 1 + ⟨B C⟩

Any local realistic theory must obey this inequality !  What does QM predict?  Well, we use the results
obtained above

lefthandside=Abs[EAB-EAC]

righthandside=1+EBC

angles={θA ->Pi/3,θB->0,θC->Pi/2}

lefthandside /. angles
righthandside /. angles

lefthandside⩽ righthandside  /. angles

Implications

Despite the epistemological issues raised by the Bell inequalities, their introduction has allowed profound novel implications in communication and security protocals. The Ekert protocal (1991) for Quantum Key Distribution (QKY) is an example of the Bell inequalities at work.

Additional reading

Hyperlink["http://www.physics.unlv.edu/%7Ebernard/2014_QIC/week13/despagnet.pdf"]
Hyperlink["http://www.physics.unlv.edu/%7Ebernard/2014_QIC/week13/antirealists.pdf"]
Hyperlink["http://www.physics.unlv.edu/%7Ebernard/2014_QIC/week14/aspect.pdf"]
Hyperlink["http://www.physics.unlv.edu/%7Ebernard/2014_QIC/week14/epr_horizon.pdf"]