Redshift Academy


Entangled States
----------------

Consider 2 electrons that are physically located 
at different points in space.  Because they are 
at different places, the Pauli exclusion principle 
is not violated. For each electron we can write
their spin wavefunctions as:

|ψ_A> = α|u_A> + β|d_B>

|ψ_B> = γ|u_A> + δ|d_B>

Each of these states describes a qubit.  

The states |u> and |d> are called PURE states.  
The superposition of these states is also a pure 
state.  Also, a superposition is said to be coherent 
if there is an operator that, if applied to one 
state, can turn it into another that is also 
present in the superposition.  In the above case 
σ_x(1/√2)|u> = (1/√2)|d> and vice versa so |ψ> is 
a coherent superposition.  If a system is a 
superposition of states it exists simultaneously 
in each of these states until a measurement is 
made. 

The joint state of the 2 qbits can be expressed 
as the tensor product of the respective individual 
qubits. Thus,

|ψ_AB> = (α|u_A> + β|d_A>) ⊗ (γ|u_B> + δ|d_B>) 

      = αγ|u_Au_B> + αδ|u_Ad_B> + βγ|d_Au_B> + βδ|d_Ad_B> 
                                ... (1)

Where α²γ² + α²δ² + β²γ² + β²δ² = 1

States belonging to a composite space that can 
be factored into individual states belonging 
to separate subspaces are also referred to as 
SEPARABLE states.  Separable states correspond 
to the situation in which each electron is prepared 
independently, and can be measured independently.  
This means that, for a separable state, there is 
always a direction along which you can measure 
the spin of the first electron to be ±1 with 100% 
certainty, and there is always a direction along 
which you will measure the spin of the second 
electron to be ±1 with 100% certainty. 

We can define a general state for 2 qubits as 
follows:

|ψ_AB> = a|u_Au_B> + b|u_Ad_B> + e|d_Au_B> + f|d_Ad_B> ... (2)

As it stands, this state cannot be factored into 
(1) unless a = αγ, b = αδ, e = βγ, f = βδ and the 
condition af - be = 0  is enforced.  In general,
however, this is not the case.  A state that cannot
be factored (is not separable) is referred to as 
an ENTANGLED state.  Entanglement requires coherence, 
but coherence does not imply entanglement.  Note 
that a qubit represents the basic state of a 
single spin 1/2 particle and is therefore neither 
separable nor entangled.

One other point to note is that (1) can be written
as αγ(|uu> + (δ/γ)|ud> + (β/α)|du> + (β/α)(δ/γ)|dd>  
indicating 3 degrees of freedom (αγ, β/α, δ/γ) to 
describe this state.  Whereas (2) has 4 degrees 
of freedom.  A consequence of this is that there 
are states where the 2 individual particles are 
not independent of one another.  

Now focus on the following state (the singlet): 

|ψ> = 1/√2{|u_Ad_B> - |d_Au_B>}

This state is not separable.  It is easy to see 
this since it would imply that αγ = 0, αδ = 1/√2, 
βγ = 1/√2 and βδ = 0, which is not possible.  

For any direction we can show that the expectation 
values(averages) of the associated spins are zero, 
meaning that measurements along this direction are 
equally likely to be +1 or -1.  First we need to 
discuss basis states.

Basis States
------------

The eigenvectors of the σ matrices are:

σ_x:

 -   -  - -       - -
| 0 1 || 1 | = +1| 1 |
| 1 0 || 1 |     | 1 |
 -   -  - -       - -

 -   -  -  -       -  -
| 0 1 ||  1 | = -1|  1 |
| 1 0 || -1 |     | -1 |
 -   -  -  -       -  -

σ_y:

 -    -  -  -       -  -
| 0 -i || -i | = +1| -i |
| i  0 ||  1 |     |  1 |
 -    -  -  -       -  -

 -    -  - -       - -
| 0 -i || i | = +1| i |
| i  0 || 1 |     | 1 |
 -    -  - -       - -

σ_z:

 -    -  - -       - -
| 1  0 || 1 | = +1| 1 | 
| 0 -1 || 0 |     | 0 |
 -    -  - -       - -

 -    -  - -       - -
| 1  0 || 0 | = -1| 0 |
| 0 -1 || 1 |     | 1 |
 -    -  - -       - -

The basis states |u> and |d> are chosen to be
the eigenvectors of σ_z. 

       - -             - -
|u> = | 1 | and |d> = | 0 | 
      | 0 |           | 1 |
       - -             - -

   - -     - -      - -     - -
σ_x| 1 | = | 0 |  σ_x| 0 | = | 1 |
 | 0 |   | 1 |   | 1 |   | 0 |
   - -     - -      - -     - -

   - -     - -      - -     -  -
σ_y| 1 | = | 0 |  σ_y| 0 | = | -i |
 | 0 |   | i |   | 1 |   |  0 |
   - -     - -      - -     -  -

   - -     - -      - -     -  -
σ_z| 1 | = | 1 |  σ_z| 0 | = |  0 |
 | 0 |   | 0 |   | 1 |   | -1 |
   - -     - -      - -     -  -

Consider 2 qubits and different spin matrices, 
σ and τ.  σ acts on the first electron and does 
nothing to the second electron.  τ does the 
opposite.

For electron A we can write:

σ_x|u_A> =   |d_A>
σ_x|d_A> =   |u_A>
σ_y|u_A> =  i|d_A>
σ_y|d_A> = -i|u_A>
σ_z|u_A> =   |u_A>
σ_z|d_A> =  -|d_A>

              - -             - -
Where, |u> = | 1 | and |d> = | 0 |
             | 0 |           | 1 |
              - -             - -

Similarly, for electron B we can write:

τ_x|u_B> =   |d_B>
τ_x|d_B> =   |u_B>
τ_y|u_B> =  i|d_B>
τ_y|d_B> = -i|u_B>
τ_z|u_B> =   |u_B>
τ_z|d_B> =  -|d_B>

Recall, that the expectation value, <>, of an operator 
is given by <ψ_AB|σ|ψ_AB>.  For the singlet:

<σ_x> = <(<u_Ad_B| - <d_Au_B|)(1/√2)|σ_x|(1/√2)(|u_Ad_B - |d_Au_B>)>

     = (1/2)(<u_Ad_B| - <d_Au_B|)(|d_Ad_B> - |u_Au_B>) 

     = 0

<σ_y> = <(<u_Ad_B| - <d_Au_B|)(1/√2)|σ_y|(1/√2)(|u_Ad_b - |d_Au_B>)>

     = (1/2)(<u_Ad_B| - <d_Au_B|)(i|d_Ad_B> + i|u_Au_B>) 

     = 0

<σ_z> = <(<u_Ad_B| - <d_Au_B|)(1/√2)|σ_z|(1/√2)(|u_Ad_b - |d_Au_B>)> 

     = (1/2)(<u_Ad_B| - <d_Au_B|)(|u_Ad_B> + |d_Au_B>) 

     = 0

Likewise, if we did the same analysis for τ we 
would get the same result.  

Therefore, in the entangled situation there is no 
direction of definite spin for either A's or B's 
electron (i.e. <σ_x> = <τ_x> = <σ_y> = <τ_y> = <σ_z> = 
<τ_z> = 0).  For example, if the system is prepared 
in the +z direction, A will never be able measure 
the spin of her electron in the +z direction and 
obtain +1 with 100% certainty.  Instead, she will 
measure +1 50% of the time and -1 50% of the time. 
B will encounter exactly the same situation with 
his electron.  Because of this they both conclude 
that their electrons must be in an entangled state.

A’s measurement will cause the wave function to
collapse.  If A finds her electron to be spin-up
then B will find his electron to be spin down i.e.

For the singlet state prior to measurement, 
P(|u_A>) = P(|u_B>) = (1/√2)² = 1/2

After measurement the system collapses into the 
normalized eigenstate: 

1|u_Ad_B> + 0|d_Au_B> 

With P(|u_A>) = 1² = 1 and P(|u_B>) = 0² = 0.

We can compare this result to the one obtained for 
the separable state (1/2(|u_A> + |d_A>) ⊗ (|u_B> + |d_B>).  
We get:

<σ_x> = <(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          |σ_x|(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

     = (1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          (1/2)(|d_Au_B> + |d_Ad_B> + |u_Au_B> + |u_Ad_B>)

     = (1/4)(4) = 1

<σ_y> = <(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          |σ_y|(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

     = (1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          (1/2)(i|d_Au_B> + i|d_Ad_B> - i|u_Au_B> - i|u_Ad_B>)

     = 0

<σ_z> = <(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          |σ_z|(1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

     = (1/2)(|u_Au_B> + |u_Ad_B> + |d_Au_B> + |d_Ad_B>)

          (1/2)(|u_Au_B> + |u_Ad_B> - |d_Au_B> - |d_Ad_B>)

     = 0

Likewise, if we did the same analysis for τ we 
would get the same result.  

Therefore, if a system of qubits is prepared in 
the +z direction and A's electron is independent, 
she would measure the spin in the +z direction to 
be +1 (i.e. <σ_x> = 1, <σ_y> = <σ_z> = 0).  Likewise 
for Bob's electron (i.e. <τ_x> = 1, <τ_y> = <τ_z> = 0).

We could also show that if the system of qubits
is prepared and measured in the ±x direction we get:

<σ_x> = (1/2)((<ψ_x|σ_x|ψ_x) = +1

<σ_x> = (1/2)((<ψ_-x|σ_x|ψ_-x) = -1

<σ_y> = <σ_z> = 0

and so on.

For the separable state:

|ψ> = (1/√2)(|u_A> + |d_A>) ⊗ (1/√2)(|u_B> + |d_B>)

    = (1/2)|u_Au_B> + (1/2)|u_Ad_B> + (1/2)|d_Au_B> + (1/2)|d_Ad_B>

(Normalization:  (1/2)² + (1/2)² + (1/2)² + (1/2)² = 1)

For the separable state prior to measurement, 
P(|u_A>) = P(|u_B>) = (1/2)² + (1/2)² = 1/2

After measurement by A the system collapses into 
the normalized eigenstate: 

(1/√2)|u_Au_B> + (1/√2)|u_Ad_B> + 0|d_Au_B> + 0|d_Ad_B>

∴ P(|u_A>) = (1/√2)² = 1/2 

After measurement by B the system collapses into 
the normalized eigenstate: 

(1/√2)|u_Au_B> + 0|u_Ad_B> + (1/√2)|d_Au_B> + 0|d_Ad_B>

∴ P(|u_B>) = (1/√2)² = 1/2 

We have focussed on the singlet state in our analysis
but there are 3 other entangled states of particular 
importance.  These 3 states together with the singlet 
are called the BELL STATES: 

|Φ⁺> = 1/√2{|u_Au_B> + |d_Ad_B>}

|Φ^-> = 1/√2{|u_Au_B> - |d_Ad_B>}

|Ψ⁺> = 1/√2{|u_Ad_B> + |d_Au_B>}

|Ψ^-> = 1/√2{|u_Ad_B> - |d_Au_B>} - singlet

It should be pointed out that for the Φ states A 
and B's measurements will result in identical spins
whereas in the Ψ states, measurements will result 
in opposite spins.

Einstein - Podolsky - Rosen Correlation
---------------------------------------

The curious fact is that A and B's electrons will 
remain entangled regardless of the separation between 
them unless either is 'disturbed' by a measurement.  
Once a measurement has been made, the state of the
original entangled system collapses and the entanglement 
is destroyed.  The fact that the spin of B's electron
instantly "knows" what spin it is supposed to take 
on (and vice versa), seemingly involves communication 
between the two particles at speeds greater than the 
speed of light, which is in conflict with Einstein's 
special theory of relativity (violation of locality). 
Einstein referred to the phenomenon as 'spooky; action
at a distance and supported an alternative idea 
called 'hidden variables theory' which suggested 
that quantum mechanics was incomplete.  However,
later on Bell's theorem would suggest that the 
existance of local hidden variables is impossible.  

Hidden Variables and Bell's Theorem 
-----------------------------------

Bell's theorem showed that if local hidden variables 
existed, certain experiments could be performed where 
the result would satisfy a Bell inequality. If, on the 
other hand, if Bell's inequality was violated local 
hidden variables could not exist.

Bell's Theorem
--------------



Let N be the number of objects.

N(A not B) + N(B not C) ≥ N(A not C)

Therefore,

N(1 + 2) + N(4 + 7) ≥ N(1 + 4)

Therefore,

N(1 + 4) + N(2 + 7) ≥ N(1 + 4)

So that N(2 + 7) ≥ 0

This is generaly written as:

N(A,~B) + N(B,~C) ≥ N(A,~C)

Let ↑↗→ be possible measurement directions for A 
and B's electrons.  Interpret P(↑,↗) as the 
probability A finds her electron in the ↑ direction 
and B finds his electron in the ↗ direction which, 
for the singlet, is equivalent to the ~↗ = ↙ 
direction for A.  Returning to the above Venn diagram, 
we can define A = ↑, B = ↗ and C = →.

Substituting into Bell's theorem gives:

N(↑,↗) + N(↗,→) ≥ N(↑,→)

We can equate N with a probability (i.e. P = n/N_Total).  
To compute these probabilities it is helpful to 
use the projection operators for spin.  These can 
be written in terms of the σ matrices as follows:

                   -   -       -   -     -   -
ℙ_↑ = (I + σ_z)/2 = | 1  0 | + | 1  0 | = | 1 0 |
                | 0  1 |   | 0 -1 |   | 0 0 |
                   -   -       -   -     -   -

ℙ_↗ = (I/2 + (σ_x + σ₂)/2√2 

                   -   -       -   -     -       -
ℙ_→ = (I + σ_x)/2 = | 1  0 | + | 0  1 | = | 1/2 1/2 |
                | 0  1 |   | 1  0 |   | 1/2 1/2 |
                   -   -       -   -     -       -

These act on electron 1.  There is an equivalent set 
of operators that act on electron 2:

For ℙ_↑:

 -   -  - -     - -       -   -  - -     - -
| 1 0 || 1 | = | 1 | and | 1 0 || 0 | = | 0 | 
| 0 0 || 0 |   | 0 |     | 0 0 || 1 |   | 0 |
 -   -  - -     - -       -   -  - -     - -

Therefore, ℙ_↑|u> = |u> and ℙ_↑|d> = |0>

For ℙ_→:

 -       -  - -     -   -       -       -  - -     -   -
| 1/2 1/2 || 1 | = | 1/2 | and | 1/2 1/2 || 0 | = | 1/2 | 
| 1/2 1/2 || 0 |   | 1/2 |     | 1/2 1/2 || 1 |   | 1/2 |
 -       -  - -     -   -       -       -  - -     -   -

Therefore, ℙ_→|u> = |u> and ℙ_↑|d> = |0>

For ℙ_↗:

        -         -  - -            -    -       
(1/2√2)| √2+1   1  || 1 | = (1/2√2)| √2+1 | 
       |   1  √2-1 || 0 |          |   1  |     
        -         -  - -            -    -    

and,

        -         -  - -     -    -       
(1/2√2)| √2+1   1  || 0 | = |   1  | 
       |   1  √2-1 || 1 |   | √2-1 |     
        -         -  - -     -    -          

The probabilities can then be computed as:

P = <ψ|ℙ₁ℙ₂|ψ>

Return to the singlet state and compute the 
probability P(↑,↗).

<ψ|(I/2 + σ_z/2)(I/2 + (τ_x + τ_z)/2√2)|ψ> 

Breaking it down piece by piece:

ℙ_↑|(1/√2)(|ud> - |du>) = (1/√2)|ud>

ℙ_↗|(1/√2)|ud>)> = (1/√2){|u> ⊗ ℙ_↗|d>}

                = (1/√2){|u> ⊗ {1|u> + (√2-1)/2√2|d>}

                = (1/√2){|uu> ⊗ (√2-1)/2√2|ud>}

P(↑,↗) = <(<ud| - <du|)(1/√2)|(1/√2){|uu> ⊗ (√2-1)/2√2|ud>}

    = <ud|(1/√2)(1/√2)(√2-1)/2√2|ud>

    = (1/2)(√2-1)/2√2

    = 0.075

Likewise, compute the probability P(↗,→).

<ψ|(I/2 + (σ_x + σ_z)/2√2)(I/2 + τ_x/2)|ψ> 

P(↗,→) = 0.075

Finally, for P(↑,→).

<ψ|(I/2 + σ_z/2)(I/2 + τ_x/2)|ψ> 

<(<ud| - <du|)(1/√2)|ℙ_↑ℙ_↗|(1/√2)(|ud> - |du>)

P(↑,→) = 0.25

P(↑,↗) + P(↗,→) ≱ P(↑,→) 

There is another way to look at this.
 
     A      B
    ↑↗→    ↑↗→
    ===    ===

N₁  +++    ---   + = aligned, - = counter aligned

N₂  ++-    --+

N₃  +-+    -+-

N₄  +--    -++

N₅  -++    +--

N₆  -+-    +-+

N₇  --+    ++-

N₈  ---    +++

Experimentally, we can compute the probabilities 
as follows:

P(↑,↗) = (N₃ + N₄)/N_T

P(↗,→) = (N₂ + N₆)/N_T

P(↑,→) = (N₂ + N₄)/N_T

P(↑,↗) + P(↗,→) ≥ P(↑,→) 

∴ (N₃ + N₄)/N_T + (N₂ + N₆)/N_T ≥ (N₂ + N₄)/N_T

∴ (N₃ + N₄) + (N₂ + N₆) ≥ (N₂ + N₄)

∴ (N₂ + N₄) + (N₃ + N₆) ≥ (N₂ + N₄)

Components of Spin
------------------

The eigenvectors of the Pauli matrices represents 
an observable describing the spin of a spin 1/2 
particle in the three spatial directions.

      -    -      - -    - -
σ_z:  | 1  0 | => | 1 |, | 0 |
    | 0 -1 |    | 0 |  | 1 |
      -    -      - -    - -

      -   -            - -          -  -
σ_x:  | 0 1 | => (1/√2)| 1 |, (1/√2)|  1 |
    | 1 0 |          | 1 |        | -1 |
      -   -            - -          -  -

 - -                          - -          -  -
| 1 | = (1/√2)(χ_+x + χ_-x) = (1/2)| 1 | + (1/2)|  1 |
| 0 |                        | 1 |        | -1 |
 - -                          - -          -  -

      -    -            - -          -  -
σ_y:  | 0 -i | => (1/√2)| 1 |, (1/√2)|  1 |
    | i  0 |          | i |        | -i |
      -    -            - -          -  -

 - -                          - -          -  -
| 1 | = (1/√2)(χ_+y + χ_-y) = (1/2)| 1 | + (1/2)|  1 |
| 0 |                        | i |        | -i |
 - -                          - -          -  -

Therefore, P(|χ_+x>) = P(|χ_+y>) = 1/2

Alternatively, we can look at this as the overlap 
of χ_+x with χ_+z.  Then:
                                                                  
P(χ_+x) = <u_+z|u_+x><u_+z|u_+x>^* 

       = |<u_+z|u_+x>|²

         -   -  -    -
<u_+z|u_+x> = | 1 0 || 1/√2 | = 1/√2
         -   - | 1/√2 |
                -    -

So, P(χ_+x) = (1/√2)(1/√2) = 1/2
                                                    
Therefore, if you line up spin in any direction 
and measure an orthogonal component, it will have 
an equal probability of being up or down.  We can 
generalize this to find the components of spin in
any direction when the electron is line up along 
the +z axis.

Let n be a unit vector with components n_x, n_y, n_z 
that represents the measurements axis.  Therefore, 
σ.n represents the component of σ along n, i.e.

σ.n = σ_xn_x + σ_yn_y + σ_zn_z

       -    -     -      -     -    -    
    = | 0 n_x | + | 0 -in_y | + | n_z 0  |
      | n_x 0 |   | in_y 0  |   | 0 -n_z |
        -    -     -      -     -    -    
       -         -
    = |n_z    n_x-in_y|
      |n_x+in_y  -n_z |
       -         -
       -     -
    = |n_z    n_-|   where n_- = n_x - in_y and n₊ = n_x + in_y
      |n₊   -n_z|
       -     -

This is Hermitian with eigenvalues of +/-1.

Now,
        
(σ.n)² = (σ_xn_x + σ_yn_y + σ_zn_z)(σ_xn_x + σ_yn_y + σ_zn_z) 

      = n_x² + n_y² + n_z² + n_xn_y(σ_xσ_y + σ_yσ_x) 

        + n_xn_z(σ_xσ_z + σ_zσ_x) + n_yn_z(σ_yσ_z + σ_zσ_y)

      = n_x² + n_y² + n_z²

      = 1 (since σ_iσ_j = -σ_jσ_i)

Check:  If the electron spin is lined up along 
the +z axis and measured in the x direction we
get n_z = 1 and n_x = n_y = 0 which leaves us with the 
σ_z matrix and a positive eigenvector equal to:

 - -
| 1 | 
| 0 |
 - -

Then the probability, P_x = <u_+z|u_+x><u_+z|u_+x>^* 

                         = |<u_+z|u_+x>|²

         -   -  -    -
<u_+z|u_+x> = | 1 0 || 1/√2 | = 1/√2
         -   - | 1/√2 |
                -    -

So, P_+x = (1/√2)(1/√2) = 1/2 as before.

The General Case
----------------

Now, what about the situation where we prepare the 
spin in direction n and measure it in direction m 
where the angle between m and n is θ.

                 m         n
                  \       /
                   \     /
                    \ θ /
                     \ /
                      o

Let |σ.n=1> be an eigenvector of σ.n with eigenvalue 
+1 that satisfies: 

σ.n|σ.n=1> = +1|σ.n=1>  

This is interpreted as preparing the electron spin
in the up state along n and then measuring the 
component of the spin along the same axis.  Likewise, 
for direction, m:  
 
σ.m|σ.m=1> = 1|σ.m=1>

Note: For the spin down case we would write:

|σ.n=-1> and σ.n|σ.n=-1> = -1|σ.n=-1>

The eigenvectors of σ.n=1 can be found from:

 -      - -  -      -  -
|n_z    n_-|| α | = +1| α |
|n₊   -n_z|| β |     | β |
 -      - -  -      -  -

From which we get 

n_zα + n_-β = α 

n₊α - n_zβ = β

These equations are equivalent.  Solving for β 
gives:

n₊α/(1 + n_z) = α(1 - n_z)/n_-

∴ n₊/(1 + n_z) = (1 - n_z)/n_-

∴ n₊n_- = 1 - n_z²

∴ n_x² + n_y² + n_z² = 1

Choose β = α(1 - n_z)/n_-

Normalize:

 -            -  -           -
| α α(1 - n_z/n₊)||      α     |
 -           - | α(1 - n_z)/n_- |
                 -           -
   = α² + α²(1 - n_z)(1 - n_z)/n₊n_-

   = α²((n₊n_- + 1 - 2n_z + n_z²)/n₊n_-)

   = α²((1 - n_z² + 1 - 2n_z + n_z²)/n₊n_-)

   = α²(2 - 2n_z)/n₊n_-)

   = α²2(1 - n_z)/(1 - n_z)²

   = α²2(1 - n_z)/(1 - n_z)(1 + n_z)

   = 2α²/(1 + n_z)

So, to get the normalization factor, N, we need 
to find the inverse √ of this.  Thus, we get:

N = √{(1 + n_z)/2α²}

  = (1/α)√{(1 + n_z)2}

So the eigenvector of σ.n=1 is:

                             -            -
|σ.n=1> = (1/α)√{(1 + n_z)/2}|     α      |
                           | α(1 - n_z)/n_- |
                             -            -

           -                          -
        = |        √{(1 + n_z)/2}     |
          | √{(1 + n_z)/2}{(1 - n_z)/n_-} |
           -                          -

Now we can apply the same reasoning to get the 
eigenvector of σ.m=1:

           -                          -
|σ.m=1> = |        √{(1 + m_z)/2}     |
          | √{(1 + m_z)/2}{(1 - m_z)/m_-} |
           -                          -

The probability of measuring the state σ.m=1 
given that the electron has been prepared in the 
state σ.n=1 is:

P(σ.n=1 σ.m=1) = |<σ.n=1|σ.m=1>|²

Which, after a lot of tedious math leads to,

P(σ.n=1 σ.m=1) = (1 + n_xm_x + n_ym_y + n_zm_z)/2

               = (1 + n.m)/2

               = (1 + cosθ_mn)/2 since |n| = |m| = 1

               = cos²(θ_mn/2) (using trig identity)

Alternative derivation using polar coordinates:

n_x = cosφsinθ

n_y = sinφsinθ

n_z = cosθ

       -                         -
σ.n = |   cosθ       exp(-iφ)sinθ |   λ = ±1
      | exp(iφ)sinθ     -cosθ     |
       -                         -

det(σ.n - λI)v = 0.  For λ = +1 we get:

 -                          -  - -
|   cosθ - 1    exp(-iφ)sinθ || α | = 0
| exp(iφ)sinθ     -cosθ - 1  || β |
 -                          -  - -

∴ β = exp(iφ)(1 - cosθ)α/sinθ

    = αexp(iφ)sin(θ/2)/cos(θ/2)

and,

β* = αexp(-iφ)sin(θ/2)/cos(θ/2)

Normalization:

α² + β² = α² + β*β = α²(1 + sin²(θ/2)/cos²(θ/2)) = 1

∴ α² = 1/(1 + sin²(θ/2)/cos²(θ/2))

Using the identity sinθ = 2sin(θ/2)cos(θ/2) this 
becomes:

α² = cos²(θ/2)

Which gives:

α = cos(θ/2) and β = sin(θ/2)

|+> = cos(θ/2)|u> + exp(iφ)sin(θ/2)|d>

This represents the qubit in terms of the Bloch 
sphere.  Likewise:

|-> = sin(θ/2)|u> - exp(iφ)cos(θ/2)|d>

Density Operator Formulation:
  
ℙ_n = (I + σ.n)/2

           -   -          -          -
   = (1/2)| 1 0 | + (1/2)| cosθ  sinθ |
          | 0 1 |        | sinθ -cosθ |
           -   -          -          -
      
           -   -          -         -          -          -
   = (1/2)| 1 0 | + (1/2)|  0   sinθ | + (1/2)| cosθ   0   |
          | 0 1 |        | sinθ  0   |        |  0   -cosθ |
           -   -          -         -          -          -

      -                     -
   = |  cos²(θ/2) (1/2)sinθ  |
     | (1/2)sinθ   sin²(θ/2) |
      -                     -

This is the probability to measure |σ.n=1> if the 
initial state is |σ.m=1> if the state is prepared
in a direction θ ...

|ψ> = (1/√2)|u> + (1/√)|d>

                   -     -
ρ = (1/2)|u><u| = | 1/2 0 | 
                  |  0  0 |
                   -     -

∴ P(ℙ_n) = Tr(ρℙ_n) = (1/2)cos²(θ/2)

                    -      -
ρ' = (1/2)|d><d| = | 0   0  |
                   | 0  1/2 |
                    -      -

∴ P(ℙ_n) = Tr(ρ'ℙ_n) = (1/2)sin²(θ'/2)

 σ.n=1
    |   /
    |θ /
    | /
    |/
    o
    |\
    | \
    |θ'\
    |   \
 σ.n=-1


If an electron is prepared in a certain direction 
and measured in another, the result of the measurement 
will be either spin up or spin down.  The interesting 
thing is that if the exact same experiment is repeated 
a second time, the result may or may not be the same.  
If the experiment is repeated multiple times, a 
probability distribution of the measured spins will 
be obtained.  The above equation shows that the smaller 
the angle between the prepared state and the measured 
state, the greater the probability of spin up occurrences.  
For example if the prepared and measured state are in 
the same direction the probability will be 1 as expected.  
If the prepared state and the measured state are in 
opposite directions, then the probability will be 0 
(θ = 180).  If the prepared state and the measured 
state are orthogonal then the probability will be 1/2 
(θ = 90) as previously found.

The above is the result for a single electron.  In the 
case of the singlet we need to modify this to account 
for the normalization factors of (1/√2).  We get:

P(σ.n=1 σ.m=1) = (1/2)|<σ.m=1|σ.n=1>|² 

           = (1/2)(1 + cosθ)/2

           ≡ (1/2)cos²(θ/2) (trig identity)

We can now find the probabilities as follows:

P(↑↗) case:

       ↑|   /↗
        |  /       θ = π/4
        |θ/  
        |/     
        o   
       /    
      /      
     /~↗

Counterclockwise angle between ↑ and ~↗ = 5π/4

P(↑↗) = (1/2)cos²((5π/4)/2) = 0.075

P(↑→) case:

       ↑|   
        |  
        |          θ = π/2
        |θ   
  ------o------   
  ~→

Counterclockwise angle between ↑ and ~→ = 3π/2

P(↑→) = (1/2)cos²((3π/2))/2 = 0.25

P(↗→) case:

       ↑|   /↗
        |  /       θ = π/4
        |θ/ 
        |/     
  ------o   
  ~→

Counterclockwise angle between  ~→ and ↗ = 5π/4

P(↗→) = (1/2)cos²((5π/4))/2 = 0.075
 
Clearly, the last inequality cannot be true and Bell's 
theorem is violated.  The implication of this is that
the hidden variables that says each particle carries 
with it all of the required information at the time of 
separation, and nothing needs to be transmitted from 
one particle to the other at the time of measurement, 
cannot be true.  Understanding 'spooky' action at a
distance is one of the major unsolved mysteries of
Physics. The study of String Theory and the idea of
special extra spatial dimensions may offer some hope
in this regard.