Bell's  Inequalities  I

Introduction

We have seen in the previous notebook that the so-called Bell (or entangled) 2-Qubit states have some interesting properties and played a very important role in the the quantum teleportation protocol.

The unusual fetaures of Bell states was already recognized by Schrodinger (who invented the terminology of entangled state) and Einstein.

In order to fully appreciate their significance and implications we need to review a bit more the basics of the quantum theory.

Eigenstates, Eigenvalues and Observables.

We have already been introduced to the computational basis states |0⟩ and |1⟩. We know that any
quantum state |ψ⟩ is a linear combination of them,  and the Born rule tells us, if the system is in state |ψ⟩,   the probablities that one observes the value 0,1.  It turns out,  that there is more than one way to measure a quantum state.

Quantum postulate I. In the quantum theory an observable is associated with an Hermitian operator.  For example the states |0⟩, |1⟩ are eigenstates of the Hermitian operator

Matrix Representation.

In the discussion above we used the Dirac bra-ket language to describe operators, both Hermitian (observables and untary operators), in terms of the ket states  |0⟩,|1⟩ and their adjoints. As we know from our previous work that we can also express the former in terms of matrices.

We know that

|0⟩  ⟶ (({{1}, {0}}))

and

|1⟩  ⟶ (({{0}, {1}}))


and  so the matrix representation for hermitian operator (Subscript[S, 3]) is

(({{⟨0|(Subscript[S, 3])|0⟩, ⟨0|(Subscript[S, 3])|1⟩}, {⟨1|(Subscript[S, 3])|0⟩, ⟨1|(Subscript[S, 3])|1⟩}}))   = (({{1, 0}, {0, -1}}))

Note that

S3={{1,0},{0,-1}};
zeroket={1,0};
oneket={0,1};

S3.zeroket == 1 zeroket

S3.oneket == -1 oneket

So the matrix representation of our computational basis states satisfy the eigenvalue equation for operator  (Subscript[S, 3])
Can you find the matrix representation of Hermitian operator  (Subscript[S, 1]) ?

S1={{0,1},{1,0}};
S1.zeroket == 1 zeroket
S1.oneket == -1 oneket

Eigenvalues[S1]

Eigensystem[S1]

upket={1,1};
downket={-1,1};

S1.upket == 1 upket
S1.downket == -1 downket

Note the two state upket and downket are just linear combinations of the computational basis states. (We can normalize them by simply multiplying a normalization factor of 1/(Sqrt[2]))

So we have two Hermitian operators (Subscript[S, 3]), and (Subscript[S, 1])both of which have eigenvalues ± 1, but different
eigenstates.

Class question: Alice makes a measurement with (Subscript[S, 3]) and she obtains the value 1.
(i)  what state is the system in immediatly after she makes that measurement ?
(ii) She makes another measurement with (Subscript[S, 3])what eigenvalues will she get? What is
the state immediatly after that measurement ?
(iii) Now Alice makes a measurement with (Subscript[S, 1]) what are the possible value of her measurement?
(iv) Suppose she obtains the value -1 what state is the system in now?
(v) She now measures with (Subscript[S, 3])again what is the probability that she obtains the value 1/2?

Consider now the following operator

S2={{0,-I},{I,0}};
MatrixForm[S2]

Is (Subscript[S, 2]) Hermitian ?
What are the eigenvalues and eigenvectors of (Subscript[S, 2])?

Eigensystem[S2]

| + ⟩ = (1/(Sqrt[2]))(({{-i}, {1}}))

| - ⟩ = (1/(Sqrt[2]))(({{i}, {1}}))

These three Hermitian operators are called the Pauli spin operators (because they are also
unitary, in addition to being Hermitian, they also serve as possible gates) and their eigenvalues
are usually interpreted as the component of spin angular momentum along the three cartesian
coordinate axis. Many particles, including electrons, atoms, protons, molecules have this property of
spin. Because there are three components, Alice can measure the spin of the system along her axis
of choice.

Question: An atom has total spin of "1/2"  (in our notebooks we ignored the 1/2 factor.  The spin operators are actually the Pauli operators multiplied by a factor of 1/2, but to keep the discussion simple we ignore the factor of 1/2 here ). Alice makes a measurement for the spin along the z-direction with (Subscript[S, 3])and she obtains the value +1 . After that she measures the y-component with (Subscript[S, 2]) followed again by a measurement of the z-component with (Subscript[S, 3]).  What are the probablities for her possible measurements as well as their probability.

Consider the following operator

Sn=FullSimplify[Sin[θ]Cos[ϕ] S1 + Sin[θ]Sin[ϕ] S2  + Cos[θ] S3];
MatrixForm[Sn]

Is (Subscript[S, n])  Hermitian ?

Hermitianconjugate=FullSimplify[ComplexExpand[Conjugate[Transpose[Sn]]]]
MatrixForm[%]

Sn==Hermitianconjugate

Eigenvalues[Sn]

FullSimplify[Eigensystem[Sn]]

Do you have any physical interpretation for what measurement  (Subscript[S, n]) represents ?

up={(E^(-I ϕ)) (Cot[θ]+Csc[θ]),1};
nup[θ_,ϕ_]={Cos[θ/2],Exp[I ϕ] Sin[θ/2]};

Simplify[Sn.nup[θ,ϕ]]

Simplify[ComplexExpand[Conjugate[nup[θ,ϕ]]].nup[θ,ϕ]]

ndown[θ_,ϕ_]={-Sin[(θ/2)],(E^(I ϕ)) Cos[(θ/2)]};

Simplify[Sn.ndown[θ,ϕ]]

Simplify[ComplexExpand[Conjugate[ndown[θ,ϕ]]].ndown[θ,ϕ]]

Simplify[ComplexExpand[Conjugate[ndown[θ,ϕ]]].nup[θ,ϕ]]

So according to the Born rule, if the quantum system is in state |ψ⟩  and Alice makes a measurement with
device  S(θ,ϕ)  the she will either get the result (eigenvalue) + 1

with probability    | ⟨θϕ up | ψ⟩ ((|)^2). Immediatly after that measurement the system collapse to |θϕ up ⟩

or the result  (eigenvalue)  -1 with probability   | ⟨θϕ down | ψ⟩ ((|)^2). 
Immediatly after that measurement the system collapse to |θϕ down ⟩

Examples.  Suppose |ψ⟩ is simply our computational state |0⟩ = (({{1}, {0}})) we note this is the state
|θ=0 ϕ=0 up ⟩

nup[0,0]

Alice decides to make a measurement with the Pauli X instrument (Stern-Gerlach device measuring spin along x-direction)
According to the Born rule the probability of her getting the result +1 is

|⟨θ=(π/2) ϕ=0 up | 0 ⟩((|)^2)

ComplexExpand[Conjugate[nup[π/2,0]]].nup[0,0]
%^2

Suppose Alice does get the value + 1, she now makes another measuremt with the Pauli Z gate, what is the probablity that
she will get -1 ?

Question II   Alice gets the state |ψ⟩=|1⟩, she makes a measurement with a Sten-Gerlach device that is pointed
at the angle  θ=π/3  and ϕ=π. She does this measurement 1000 times (each time with the same state  |ψ⟩=|1⟩) how many
times (approximatley) will she get the result + 1 ?