Redshift Academy


Projection Operator Representation
----------------------------------

A projection operator projects a vector onto a sub 
space.  Note:  The projection operator produces 
another vector and is not the same as the dot 
product, u.v = |u||v|cosθ, which produces a scalar.

Example:

Consider the following:



The vector, u, is defined in 3D space.  We want to 
find the linear operator that returns the projection 
of this vector onto the subspace with the orthonormal 
basis of {x,y}.

     - -         - -
    | 1 |       | 0 |
x = | 0 |   y = | 1 |
    | 0 |       | 0 |
     - -         - -

The projection operator that projects u onto the 
x-y plane is:

ℙ_xy := Σ|n><n| where n are the subspace orthogonal 
       ⁿ
basis vectors.  So,

ℙ = |x><x| + |y><y|

     - -  -     -     - -  -     -
    | 1 || 1 0 0 | + | 0 || 0 1 0 |
  = | 0 | -     -    | 1 | -     -
    | 0 |            | 0 |
     - -              - -
     -     -     -     -
    | 1 0 0 | + | 0 0 0 |
  = | 0 0 0 |   | 0 1 0 |
    | 0 0 0 |   | 0 0 0 |
     -     -     -     -
     -     -
    | 1 0 0 |
  = | 0 1 0 |
    | 0 0 0 |
     -     -
                              - -
So clearly, if have a vector | x | then the   
                             | y |
                             | z |
                              - -
projection onto the x-y plane is:

 -     -   - -     - -
| 1 0 0 | | x |   | x |
| 0 1 0 | | y | = | y | as expected
| 0 0 0 | | z |   | 0 |
 -     -   - -     - - 

If the space and subspace are the same, the  
projection operator is the unitary matrix, I, 
and the vector returned is just the original 
vector.

Properties of ℙ
---------------

Idempotent:

The projection operator satisfies the operator 
equation ℙℙ = ℙ.  This means that acting twice 
with a projection operator on a vector gives the 
same result as acting once.  

ℙ² = (|ψ><ψ|)(|ψ><ψ|)

   = |ψ><ψ|ψ><ψ|

   = |ψ><ψ| if <ψ|ψ> = 1 (normalized)

ℙ|ψ> = λ|ψ>

ℙ²|ψ> = λ²|ψ>

This result implies that the eigenvalues of ℙ 
can only be 0 or 1.

Hermitian:

ℙ^† = (|ψ><ψ|)^†

   = (|e_i><e_i|)(|e_i><e_i|) 

   = |e_i> (<e_i|e_i>) <e_i|

   = |e_i> 1 <e_i|

   = |e_i><e_i|

   = ℙ

Density Matrix Representation
-----------------------------

A density matrix is an alternative way to describe 
the quantum state of system.  Instead of using state 
vectors or wavefunctions, the states are represented 
by matrices.  Wavefunctions can only represent pure 
states while density matrices can be used to represent 
both pure and mixed states. 

Pure States
-----------

The density matrix, ρ, for a pure state is simply 
the projection operator, i.e.

ρ = |ψ><ψ| = Σα_iα_j*|e_i><e_j|  
             ^ij 

[ρ_ij = <e_i|ρ|e_i> = <e_i|ψ><ψ|e_i> = a_ia_j^*]

Which is just the projection operator.

A density operator represents a pure state if and 
only if: 

1.  ρ² = ρ 

2.  Tr(ρ) = 1

3.  Tr(ρ²) = 1 (Purity)

Consider the pure state:

|ψ> = 1/√2(|u> + |d>)

ρ = |ψ><ψ|

  = (1/2)(|u><u| + |u><d| + |d><u| + |d><d|)

     -       -
  = | 1/2 1/2 |
    | 1/2 1/2 |
     -       -

Note:  The cross terms remain because >< is the 
tensor product, not the dot product, i.e.

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

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

The non-zero off-diagonal elements represen the 
interference (coherence) terms associated with 
quantum superposition.  

Density Matrix for the Singlet
------------------------------

Matrix Vectorization:

Vectorization of a matrix is a linear transformation 
which converts the matrix into a column vector. 
Specifically, the vectorization of a m × n matrix A, 
denoted vec(A), is the mn × 1 column vector obtained 
by stacking the columns of the matrix A on top of 
one another.  Example:

     - -        - -
u = | 1 |  d = | 0 |
    | 0 |      | 1 |
     - -        - -
               - -
     -   -    | a |
vec | a b | = | c |
    | c d |   | b |
     -   -    | d |
               - -
⊗                                         - - 
⊗                 -   -        -   -     | 1 |
|uu> ≡ |u ⊗ u> = | 1 0 |  vec | 1 0 | => | 0 |        
⊗                | 0 0 |      | 0 0 |    | 0 |
⊗                 -   -        -   -     | 0 |
⊗                                         - -

We can apply vectorization of a matrix to the
Singlet state.  Therefore,

      ⊗                -    - 
       - -⊗    - -    |  0   |
(1/√2)| 1 | ⊗ | 0 | = | 1/√2 | ... (1)
      | 0 |⊗  | 1 |   |  0   |
       - -⊗    - -    |  0   |
      ⊗                -    - 

      ⊗                -    - 
       - -⊗    - -    |  0   |
(1/√2)| 0 | ⊗ | 1 | = |  0   | ... (2)
      | 1 |⊗  | 0 |   | 1/√2 |
       - -⊗    - -    |  0   |
      ⊗                -    - 

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

              -      -                       
             |   0    | -              -    
ρ = |ψ><ψ| = |  1/√2  || 0 1/√2 -1/√2 0 | 
             | -1/√2  | -              -    
             |   0    |                     
              -      -                       

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

ρ² = ρ therefore this is a pure entangled state.  
We know this is an entangled state because it 
cannot be factored but can we determine this 
by just looking at ρ_AB?  The answer is yes but 
we first have to construct the REDUCED DENSITY
MATRIX.  This is discussed in the next section. 

Density Operator for Mixed States
---------------------------------

A mixed state is a statistical ensemble of 
normalized pure states.  It is not a superposition 
and because it is statistical in nature, there is 
limited or no knowledge about the state of the 
system the state represents.  Unlike a pure state, 
which can be unequivocally defined by the single 
state vector, |ψ>, which has complete information 
about the system, a mixed state cannot be described 
by a wave function.  Instead, it requires a density 
matrix, ρ, for its description.  The density 
operator for a mixed state is:

ρ = ΣP_kρ_k| where ΣP_k = 1 and ρ_k is for a pure state.
    ^k
It satisfies:

1.  ρ² ≠ ρ 

2.  Tr(ρ) = 1

3.  Tr(ρ²) ≠ 1 

With a coherent superposition there is a well 
defined phase relationship between the pieces in 
the superposition, which means that there can be 
interference between these pieces in subsequent 
operations.  Consider the state:

|ψ> = 1/√2(exp(iθ)|d> + exp(iφ)|u>)

If the phase, (θ - φ), is constant, the density 
matrix for this state looks like:

     -                             -
ρ = |       1/2       exp(i(θ-φ))/2 |
    | exp(-i(θ-φ))/2       1/2      |  
     -                             -

Since ρ² = ρ, this state is pure.

Now, suppose that (θ - φ) is random (i.e. the 
phase difference between |u> and |d> is random).  
In this case the expectation value of exp(i(θ-φ)) 
is 0.  The density matrix becomes:

      -       -
ρ' = | 1/2  0  | 
     |  0  1/2 |    
      -       -

ρ no longer represents a pure state.  This is 
simply a classical statistical probability law. 
The off-diagonal elements of ρ have disappeared
meaning that all the coherence has been lost 
resulting in a completely mixed state.

It is also possible to have a partially mixed 
state where the off-diagonal elements of the 
density matrix are not equal to 0.  This indicates 
that the state is a combination consisting of a 
statistical ensemble of basis elements and a 
quantum superposition of the same basis elements.  
In other words, the off-diagonal entries represent 
the coherences in the mixed state.  

Note that the expectation value of an operator 
corresponding to an observable acting on a state 
defined by ρ is given by <O> = Tr(ρO).  Since the 
diagonal elements of ρ are the same in each case,
there is no way to physically distinguish a coherent 
superposition from a mixed state.  The only way to
tell is if an operator exists that satisfies the 
definition of a coherent superposition as mentioned 
before.  For example, <σ_x> = Tr(ρσ_x) gives:

   -       -  -   -        -       -
Tr| 1/2 1/2 || 0 1 |  = Tr| 1/2 1/2 | = 1
  | 1/2 1/2 || 1 0 |      | 1/2 1/2 |
   -       -  -   -        -       -

   -       -  -   -        -       -
Tr| 1/2  0  || 0 1 |  = Tr|  0  1/2 | = 0
  | 0   1/2 || 1 0 |      | 1/2  0  |
   -       -  -   -        -       -

Mixed states are important in the fields of quantum 
computing and communication where there maybe noise 
in the system that could cause a state to flip with 
some probability, p.  For example, Alice could send 
Bob the state |u> and Bob could end up with the state 
|d> with probability, p, or the desired state with 
the probability, 1 - p.  So Bob's state is either |u> 
or |d> but not a quantum superposition of the two.  
A more practical example is the qubit.  Suppose we 
want to prepare a system in the following coherent 
state:

|ψ₁> = (1/√2)|u> + (1/√2)|d> 

However, because of noise in the system, there is 
only an 80% chance of achieving this. 10% of the 
time we get:

|ψ₂> = (1/2)|u> + (√3/2)|d>

and 10% we get: 

|ψ₃> = (√3/2)|u> + (1/2)|d> 

Since we do not know the outcome of our qubit 
every time we prepare it, we can represent it as 
a mixed state of the form:

ρ = (4/5)|ψ₁><ψ₁| + (1/5)|ψ₂><ψ₂| + (1/5)|ψ₃><ψ₃|

Where 4/5, 1/5 and 1/5 are the classical probabilities 
of obtaining the states |ψ₁>, |ψ₂> and |ψ₃>.

     -                        -
ρ = |      1/2     √3/20 + 2/5 |
    | √3/20 + 2/5       1/2    |
     -                        -

The matrix tells us that the state is not only a 
mixed state (ρ² ≠ ρ and Tr(ρ²) < 1) but also has 
some level of coherent superposition.  The ratio
coherence/mixed is dependent on the amount of coherence 
'leaked' off due to interactions with the local
environment.

Reduced Density Matrix
----------------------

The reduced density operator describes completely 
all the properties/outcomes of measurements of the 
subsystem B, given that system A is left unobserved 
(”tracing out”) subsystem A.  

ρ_A = Tr_B(|ψ_i><ψ_j| ⊗ |φ_i><φ_j|)

   = |ψ_i><ψ_j|Tr(|φ_i><φ_j|)

   = |ψ_i><ψ_j|<φ_i|φ_j>)

For the singlet:

ρ_AB = |ψ><ψ|

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

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

      - |d_Au_B><u_Ad_B| + |d_Au_B><d_Au_B|)

    = (1/2)((|u_A><u_A|)|d_B><d_B| - (|u_A><d_A|)|d_B><u_B| 

      - (|d_A><u_A|)|u_B><d_B| + (|d_A><d_A|)|u_B><u_B|)

ρ_B = Tr_A(ρ_AB)

ρ_B = (1/2)(Tr(|u_A><u_A|)|d_B><d_B| - Tr(|u_A><d_A|)|d_B><u_B| 

     - Tr(|d_A><u_A|)|u_B><d_B| + Tr(|d_A><d_A|)|u_B><u_B|)

   = (1/2)((<u_A|u_A>)|d_B><d_B| - (<u_A|d_A>)|d_B><u_B| 

     - (<d_A|u_A>|)|u_B><d_B| + (<d_A|d_A>)|u_B><u_B|)

   = (1/2)(|d_B><d_B| + |u_B><u_B|)

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

Likewise,

      -       -
ρ_A = | 1/2  0  |
     |  0  1/2 |
      -       -

In this specific example, ρ_A and ρ_B are equal, 
but this is not always the case.

This says that the subsystems A and B are both 
in mixed states because Tr(ρ_A) ≠ Tr(ρ_A²) and 
Tr(ρ_B) ≠ Tr(ρ_B²).  This result might seem rather 
strange since we started out with a pure entangled 
state.  This reflects the fact that an observer 
who doesn't know about the entanglement (cut off 
from B) will describe A as a mixed state, while 
an observer who knows about the entanglement with 
B will describe them as entangled.  Therefore, in 
ignoring part of the information about the state
we are forced to deal with a mixed state.  In fact	
the state is said to be entangled only if the 
reduced density matrix describes a mixed state.
Otherwise, the state is separable.  The reduced
density matrix is used to quantify the amount of
entanglement in a system in terms of its entropy.


Summary:

               |   Density Op     |  State Vector
---------------+------------------+---------------------
   Description |      ρ           |     |ψ>
---------------+------------------+---------------------
   Expection⁽¹⁾   |   <O> = Tr(ρO)   |  <O> = <ψ|O|ψ>
---------------+------------------+---------------------
   Probability⁽²⁾ |   P(λ) = Tr(ρℙ)  |  P(λ) = <ψ|ℙ|ψ>
---------------+------------------+---------------------
   Eigenvalue  |                  |  O|ψ> = λ|ψ>
---------------+------------------+---------------------
   Separability⁽³⁾| ρ_AB = ΣP_kρ_A ⊗ ρ_B |  ψ_AB = ψ_A ⊗ ψ_B
               |       ^k          |            
---------------+------------------+---------------------
   Reduced⁽⁵⁾     |   ρ_A = Tr_B(ρ_AB)|  
---------------+------------------+---------------------
   TDSE⁽⁶⁾        |   (1/ih)[H,ρ]    | d/dt|ψ> = (1/ih)H|ψ>
---------------+------------------+---------------------

1.  Tr(ρO) = Σ<e_i|ρO|e_i> by definition
             ⁱ

           = Σ<e_i|ρ|e_i><e_i|O|e_i>
             _i

           = Σρ_iiO_ii
             ⁱ

           = Tr(ρO)

2.  Density Operator:

        -       -  -   -         -     -   
    Tr(| 1/2 1/2 || 1 0 |) = Tr(| 1/2 0 |) = 1/2
       | 1/2 1/2 || 0 0 |       | 1/2 0 |
        -       -  -   -         -     -

    State Vector:

               -      -  -    -  -    -     
    <u|ℙ|u> = | 1/√2 0 || 1  0 || 1/√2 | 
               -      - | 0  0 ||  0   |        
                         -    -  -    -               

               -      -  -    -
            = | 1/√2 0 || 1/√2 | = 1/2
               -      - |  0   |   
                         -    -

    In general:

               -   -  -   -  - -  
              | α β || 1 0 || α | = αα*
               -   - | 0 0 || β |
                      -   -  - -

3.  Pure state:

    For pure state P_k = 1.

    |ψ> = (1/√2)(|u> + |d>) ⊗ (1/√2)(|u> + |d>):

     -               -
    | 1/4 1/4 1/4 1/4 |    -       -
    | 1/4 1/4 1/4 1/4 | = | 1/2 1/2 | ⊗ | 1/2 1/2 |
    | 1/4 1/4 1/4 1/4 |   | 1/2 1/2 | r | 1/2 1/2 |
    | 1/4 1/4 1/4 1/4 |    -       -
     -               -

    Mixed state: 

    ρ = (1/4)((1/√2)|u> + (√1/2)|d>) 

            ⊗ (3/4)((1/√2)|u> + (1/√2)|d>):

     -               -                       
    | 1/4 1/4 1/4 1/4 |         -       -      -      -                
    | 1/4 1/4 1/4 1/4 | = (1/4)| 1/2 1/2 | ⊗ | 1/2 1/2 |
    | 1/4 1/4 1/4 1/4 |        | 1/2 1/2 | ⊗ | 1/2 1/2 |        
    | 1/4 1/4 1/4 1/4 |         -       -      -      -
     -               -            -      -      -       -
                          + (3/4)| 1/2 1/2 | ⊗ | 1/2 1/2 |
                                 | 1/2 1/2 | ⊗ | 1/2 1/2 |
                                  -      -      -       -
             

                           -               -          
                          | 1/4 1/4 1/4 1/4 |              
                        = | 1/4 1/4 1/4 1/4 | 
                          | 1/4 1/4 1/4 1/4 |               
                          | 1/4 1/4 1/4 1/4 |               
                           -               -         

5.  ρ_A = Tr_B(ρ_AB)

       = Tr_B(ρ_A ⊗ ρ_B)

       = ρ_ATr_B(ρ_B)

       = ρ_A

6.  ihd/dt|ψ> = H|ψ>

    ihd/dt|ρ> = ihd/dt|ψ><ψ|

              = (d/dt|ψ>)<ψ| + |ψ>(<d/dt|<ψ|)

              = (1/ih)H|ψ><ψ| - (1/ih)|ψ><ψ|H

              = (1/ih)Hρ - (1/ih)ρH

              =  (1/ih)[H,ρ]

Examples:

     |        Not entangled        |     Entangled
-----+-----------------------------+-------------------------
Pure | |ψ> = (1/√2)(|uu> + |du>)   | |ψ> = (1/√2)(|ud> - |du>)
-----+-----------------------------+-------------------------
Mixed| ρ = (1/4)|uu> + (3/4)|du>   | ρ = (1/4)|ud> - (3/4)|du>)
-----+-----------------------------+--------------------------

Entanglement Entropy
--------------------

How do we quantify the degree of entanglement between 
2 subsystems constituting a two-part composite quantum 
system?  One way to do this to compute the QUANTUM 
ENTROPY (von Neumann) defined as:

S = -Tr(ρlogρ)

  = -Σλ_ilogλ_i
     ⁱ
Where λ_i are the eigenvalues of ρ

In classical mechanics this is equivalent to:

S = -Σ_iP_ilogP_i

The entropy is measure of our degree of knowledge 
about the state of the system.  In the case where 
there is only one state we know exactly what state 
the system is in and the entropy is 0.  Conversely, 
if the number of states is large, then we have very 
little knowledge about the precise state of the 
system and the entropy is large.  

To compute the entropy of an entangled state it is
only necessary to look at the reduced density matrix.
What this means is that if you have a combined 
system in a pure entangled state and you measure 
either sub-system (in this case A) alone they will, 
in general, be described by a mixed state with an 
associated density matrix.

Consider the singlet 1/√2{|ud> - |du>}.  

     -       -
ρ_A = | 1/2  0  | with λ = 1/2 with multiplicity 2
    |  0  1/2 |
     -       -

and for the entropy we get:

S_A = -Σλ_ilogλ_i
      ⁱ

   = -{(1/2)log(1/2) + (1/2)log(1/2)} 

   = log(2)

It is fairly easy to show that 2 is the maximum 
entropy ever allowed.  Therefore, we have a 
combined system in a pure entangled state but 
we have complete ignorance about either of the 
sub-systems (we have focussed on A here but the 
same is true for B).

Now consider the product state:

|ψ> = (1/√2)(|u_Au_B> + |d_Au_B>)

  = (1/√2)(|u_A> + |d_A> ) ⊗ |u_B>

ρ_AB = |ψ><ψ|

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

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

      + |d_Au_B><u_Au_B| + |d_Au_B><d_Au_B|)

    = (1/2)(|u_A><u_A|(|u_B><u_B|) + |u_A><d_A|(|u_B><u_B|) 

      + |d_A><u_A|(|u_B><u_B|) + |d_A><d_A|(|u_B><u_B|))

ρ_A = Tr_B(ρ_AB)

      -       -
   = | 1/2 1/2 |
     | 1/2 1/2 |
      -       -

This has eigenvalues, λ = 0 and 1.  For the 
entropy we get:

S_A = -Σλ_ilogλ_i
      ⁱ

   = -{(1)log(1) + (0)log(0)} 

   = 0

Monogamy of Entanglement
------------------------

If 2 electrons, are maximally entangled with each 
other they cannot be entangled at all with a third 
electron.  In general, there will be a trade off 
between the amount of entanglement between 1 and 
2 and 1 and 3 or 2 and 3.  This is referred to as 
the MONOGAMY OF ENTANGLEMENT.  In general, the 
more parties, the less entanglement between them.