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Units, Constants and Useful Formulas

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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:

ℙxy := 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|)†

= (|ei><ei|)(|ei><ei|)

= |ei> (<ei|ei>) <ei|

= |ei> 1 <ei|

= |ei><ei|

= 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| = Σαiαj*|ei><ej|
ij

[ρij = <ei|ρ|ei> = <ei|ψ><ψ|ei> = aiaj*]

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 ρAB?  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 = ΣPkρk| where ΣPk = 1 and ρk 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, <σx> = Tr(ρσx) 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.

ρA = TrB(|ψi><ψj| ⊗ |φi><φj|)

= |ψi><ψj|Tr(|φi><φj|)

= |ψi><ψj|<φi|φj>)

For the singlet:

ρAB = |ψ><ψ|

= (1/2)((|uA><uA|)|dB><dB| - (|uA><dA|)|dB><uB|

- (|dA><uA|)|uB><dB| + (|dA><dA|)|uB><uB|)

ρB = TrA(ρAB)

ρB = (1/2)(Tr(|uA><uA|)|dB><dB| - Tr(|uA><dA|)|dB><uB|

- Tr(|dA><uA|)|uB><dB| + Tr(|dA><dA|)|uB><uB|)

= (1/2)((<uA|uA>)|dB><dB| - (<uA|dA>)|dB><uB|

- (<dA|uA>|)|uB><dB| + (<dA|dA>)|uB><uB|)

= (1/2)(|dB><dB| + |uB><uB|)

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

Likewise,

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

In this specific example, ρA and ρB are equal,
but this is not always the case.

This says that the subsystems A and B are both
in mixed states because Tr(ρA) ≠ Tr(ρA2) and
Tr(ρB) ≠ Tr(ρB2).  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)| ρAB = ΣPkρA ⊗ ρB |  ψAB = ψA ⊗ ψB
|       k          |
---------------+------------------+---------------------
Reduced(5)     |   ρA = TrB(ρAB )  |
---------------+------------------+---------------------
TDSE(6)        |   (1/ih)[H,ρ]    | d/dt|ψ> = (1/ih)H|ψ>
---------------+------------------+---------------------

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

= Σ<ei|ρ|ei><ei|O|ei>
i

= ΣρiiOii
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 Pk = 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.  ρA = TrB(ρAB)

= TrB(ρA ⊗ ρB)

= ρATrB(ρB)

= ρA

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)

= -Σλilogλi
i
Where λi are the eigenvalues of ρ

In classical mechanics this is equivalent to:

S = -ΣiPilogPi

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>}.

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

and for the entropy we get:

SA = -Σλilogλi
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)(|uAuB> + |dAuB>)

= (1/√2)(|uA> + |dA> ) ⊗ |uB>

ρAB = |ψ><ψ|

= (1/2)(|uAuB> + |dAuB>)(<uAuB| + <dAuB|)

= (1/2)(|uAuB><uAuB| + |uAuB><dAuB|

+ |dAuB><uAuB| + |dAuB><dAuB|)

= (1/2)(|uA><uA|(|uB><uB|) + |uA><dA|(|uB><uB|)

+ |dA><uA|(|uB><uB|) + |dA><dA|(|uB><uB|))

ρA = TrB(ρAB)

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

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

SA = -Σλilogλi
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.