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Entangled States
----------------
Consider 2 electrons 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 1 we can write:
σ_{x}|u> = |d>
σ_{x}|d> = |u>
σ_{y}|u> = i|d>
σ_{y}|d> = -i|u>
σ_{z}|u> = |u>
σ_{z}|d> = -|d>
- - - -
Where, |u> = | 1 | and |d> = | 0 |
| 0 | | 1 |
- - - -
Similarly, for electron 2 we can write:
τ_{x}|u> = |d>
τ_{x}|d> = |u>
τ_{y}|u> = i|d>
τ_{y}|d> = -i|u>
τ_{z}|u> = |u>
τ_{z}|d> = -|d>
Examples:
σ_{x}|ud> = |dd>
σ_{x}|dd> = |ud>
σ_{z}|ud> = |ud>
σ_{z}|dd> = -|dd>
Likewise
τ_{y}|ud> = -i|uu>
We can define a general state for 2 electrons as a linear
superposition as follows:
α|uu> + β|ud> + γ|du> + δ|dd>
such that:
αα* + ββ* + γγ* + δδ* = 1
At first sight it might seem that only 4 are required to describe
such a state. However, this is not true. Consider:
α, β, γ, δ are complex => 8 variables.
α^{2} + β^{2} + γ^{2} + δ^{2} = 1 => 7 variables
Phase doesn't matter => 6 variables
So in fact 6 variables are needed to describe the state. This is
our first insight into the concept of ENTANGLED STATES. To explore
this further consider 2 electrons that are physically located at
different points in space. Because they are at different places,
the Pauli exclusion principle does not come into play. For each
electron we can write:
state 1 = α_{1}|u> + β_{1}|d>
state 2 = α_{2}|u> + β_{2}|d>
The joint state of the 2 electrons can be expressed as the product of
the respective individual electron states. Thus,
(α_{1}|u> + β_{1}|d>)(α_{2}|u> + β_{2}|d>) = α_{1}α_{2}|uu> + α_{1}β_{2}|ud> + β_{1}α_{2}|du> + β_{1}β_{2}|dd>
We only need four real numbers (α_{1}α_{2}, α_{1}β_{2} etc) to describe a product
state, as opposed to six for the general state, hence this product
state is not entangled. This is because the two single-electron
states are both phase-independent and unit length, hence we can
reduce the total of real numbers needed by two for each state.
Product states correspond to the situation in which each electron
is prepared independently, and can be measured independently. This
means that, for a product state, there is always a direction along
which you will 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.
This is equivalent to saying that each electron can be measured
independently.
Now construct the following 2 electron states:
1/√2{|ud> + |du>} ... TRIPLET STATE
1/√2{|ud> - |du>} ... SINGLET STATE
By definition, these states are systems where the second electron
is in the opposite configuration to the first. A pair of electrons in
this state are said to be ENTANGLED. It turns out to be energetically
favorable for the electrons to end up in the singlet state. The singlet
state results by bringing 2 electrons close enough together so that
spins interact. Again, as long as the electrons are not on top of each
other, the PE principle is not violated. Each electron is like a little
magnet and after a while they will orient themselves to form the
lowest energy state of the 2 electron system. A photon may or may
not be emitted in the process depending on the initial orientation of
the electrons.
For any direction we can show that the averages (expectation values)
of the associated spins are zero, meaning that measurements along
this direction are equally likely to be +1 or -1. Recall, the
expectation value is given by <a|O|a>.
{<ud| ± <du|}σ_{x}{|ud> ± |du>} => {<ud| ± <du|}{|dd> ± |uu>} = 0
{<ud| ± <du|}σ_{y}{|ud> ± |du>} => {<ud| ± <du|}{i|dd> ±' i|uu>} = 0
{<ud| ± <du|}σ_{z}{|ud> ± |du>} => {<ud| ± <du|}{|ud> ±' |du>} = 0
±' ≡ -/+
Likewise, if we did the samething for τ we would get the same
result.
Clearly, this is a different situation from the single electron
system where there is always an axis where <σ.n> = 1. In contrast,
in the entangled case, any axis you measure has a 50/50 probability
of being up or down (in other words, the expectation value is 0).
This is quite different from the product state described above.
There is a significant difference between the two states:
1/√2{|ud> + |du>}
and
1/√2{|ud> - |du>}
One way to distinguish them is to compute the result of the sum of the
two spin operators, σ and τ, for each direction on each state (+ and -)
This is analagous to the vector sum of a pair of directions, n and m.
Thus,
(σ_{n} + τ_{n}){|ud> ± |du>} neglecting 1/√2 coefficient for now
which is equivalent to,
σ_{n}{|ud> ± |du>} + τ_{n}{|ud> ± |du>}
Consider {|ud> + |du>}, we get
σ_{z}{|ud> + |du>} = { |ud> - |du>}
τ_{z}{|ud> + |du>} = {-|ud> + |du>}
∴ σ_{z}{|ud> + |du>} + τ_{z}{|ud> + |du>} = 0
σ_{x}{|ud> + |du>} = {|dd> + |uu>}
τ_{x}{|ud> + |du>} = {|uu> + |dd>}
∴ σ_{x}{|ud> + |du>} + τ_{x}{|ud> + |du>} = 2{|dd> + |uu>}
σ_{y}{|ud> + |du>} = { i|dd> - i|uu>}
τ_{y}{|ud> + |du>} = {-i|uu> + i|dd>}
∴ σ_{y}{|ud> + |du>} + τ_{y}{|ud> + |du>} = 2i{|dd> - |uu>}
Now consider {|ud> - |du>}, we get
σ_{z}{|ud> - |du>} = { |ud> + |du>}
τ_{z}{|ud> - |du>} = {-|ud> - |du>}
∴ σ_{z}{|ud> - |du>} + τ_{z}{|ud> - |du>} = 0
σ_{x}{|ud> - |du>} = {|dd> - |uu>}
τ_{x}{|ud> - |du>} = {|uu> - |dd>}
∴ σ_{x}{|ud> - |du>} + τ_{x}{|ud> - |du>} = 0
σ_{y}{|ud> - |du>} = { i|dd> + i|uu>}
τ_{y}{|ud> - |du>} = {-i|uu> - i|dd>}
∴ σ_{y}{|ud> - |du>} + τ_{y}{|ud> - |du>} = 0
Thus, the expectation value of (σ_{n} + τ_{n}) is 0*.
* From <ud| - <du|}(σ_{n} + τ_{n}){|ud> - |du>}
Now, we can prove that fact is true for any arbitrary direction, n
(n is a unit vector with components n_{x}, n_{y}, n_{z}) by taking the dot
product as follows:
(σ + τ).n = (σ_{x} + τ_{x})n_{x} + (σ_{y} + τ_{y})n_{y} + (σ_{z} + τ_{z})n_{z}
Therefore, since we have previously proved that all components of
(σ + τ) are 0, (σ + τ) in the n direction must also equal 0.
Einstein - Podolsky - Rosen Correlation
---------------------------------------
As mentioned previously, a pair of electrons will ultimately produce
a singlet (lowest energy) state if they are brought together so that
their magnetic fields interact. Since the sum of the spins is always
0, it has the special property that if you measure a component of
spin for one of the electrons, you instantly know the same component
of spin for the other electron.
This is known as the EINSTEIN - PODOLSKY - ROSEN CORRELATION.
The curious fact is that the 2 electrons will remain entangled
regardless of the separation between them unless either is
'disturbed' by a measurement. A measurement results in the
collapse of the state of the of the original entangled system.
This is equivalently described as the entanglement of the
electron with the measurement apparatus.
If you measure electron 1's spin, electron 2's gets set by the
measurement ... but somehow electron 2 also instantly "knows"
what spin it is supposed to take on. This seemingly involves
communication between the two particles at speeds greater
than the speed of light, which is in conflict with Einstein's theory
of relativity (violation of locality). Einstein himself supported
an alternative approach called 'hidden variables theory' which
suggested that quantum mechanics was incomplete. However,
this theory has the problems that violates Bell's theorem. This
is discussed here.
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.
The Density Matrix
------------------
A density matrix is a matrix that describes a quantum system in
a mixed state, a statistical ensemble of several quantum states.
This should be contrasted with a single state vector that describes
a quantum system in a pure state. The density matrix is written
for a 4 state system as:
- -
| ρ_{1} 0 0 0 |
ρ = | 0 ρ_{2} 0 0 |
| 0 0 ρ_{3} 0 |
| 0 0 0 ρ_{4} |
- -
where ρ_{i} is the probability for being in the ith state. As
all probabilities must sum to 1 the trace of ρ, Tr(ρ), must
therefore = 1.
The density matrix is an Hermitian operator.
Entanglement Entropy
--------------------
How do we quantify the degree (amount) of entanglement? One
way to do this to compute the QUANTUM ENTROPY defined as:
S = -Tr(ρlogρ)
In classical mechanics this is equivalent to:
S = -Σ_{i}P_{i}logP_{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. There will be only one non-zero entry in the density matrix
and the system is in a PURE STATE. 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. The system
is in a MIXED STATE and there will be more than non-zero entry
in the density matrix. If we have absolutely no knowledge about
how the sustem was prepared, all the elements in the density
matrix will be equal.
Consider the observable, O. O is an Hermitian operator.
<O> = Tr(Oρ) which is similar to the classical, <O> = Σ_{i}O_{i}P_{i}
Tr(Oρ) = Σ_{i}<i|Oρ|i> by definition
= Σ_{ij}<i|O|j><j|ρ|i>
= Σ_{ij}<i|O|i>ρ_{ij} since <j|ρ|i> = ρ_{ij}
This is the expectation value of O for state i weighted by the
probability that the state is i
Consider a state consisting of 2 systems a and b. If system a
has N possible states and system b has n states, then the total
number of states - n x N. For example, in the case of 2 spins
we could have uu, dd, ud and du.
The general state of the combined system can be written as:
Σ_{ab}ψ(a,b)|ab> where ψ(a,b) are complex coefficients such that
Σ_{ab}ψ^{*}ψ = 1
Now consider a measurement only on system 1 only with no
change to system b. The observable is represented by the
operator, O. Now compute the expectation value of O.
<O> = Σ_{aa'bb'}ψ(a',b')^{*}<a'b'|O|ab>ψ(a,b)
Since b is completely unchanged, b' = b, and we can write:
<O> = Σ_{aa'b}ψ(a',b)<a',b)|O|ab>ψ^{*}(a',b)ψ(a,b)
= Σ_{aa'b}<a'b|O|ab>ψ^{*}(a',b)ψ(a,b)
= Σ_{aa'}<a'b|O|ab>Σ_{b}ψ^{*}(a',b)ψ(a,b)
Now b is completely passive in <a'b|O|ab>. Therefore:
= Σ_{aa'}<a'|O|a>Σ_{b}ψ^{*}(a',b)ψ(a,b)
= Σ_{aa'}O_{a'a}Σ_{b}ψ^{*}(a',b)ψ(a,b)
= Σ_{aa'}O_{a'a}ρ_{aa'}
where
ρ_{aa'} ≡ Σ_{b}ψ^{*}(a',b)ψ(a,b)
<O> = Tr(Oρ)
Proof:
Tr(Oρ) = <a'|Oρ|a'>
= Σ_{aa'}<a'|O|a><a|ρ|a'>
= Σ_{aa'}O_{a'a}ρ_{a'a}
What this means is that if you have a combined system in a
pure state, ψ(a,b), 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>}. This is an entangled
state.
ψ(u,u) = 0
ψ(u,d) = 1/√2
ψ(d,u) = -1/√2
ψ(d,d) = 0
Now ρ_{aa'} ≡ Σ_{b}ψ^{*}(a',b)ψ(a,b). This leads to:
ρ_{uu} = ψ_{uu}ψ^{*}_{uu} + ψ_{ud}ψ^{*}_{ud} = 1/2
ρ_{dd} = ψ_{du}ψ^{*}_{du} + ψ_{dd}ψ^{*}_{dd} = 1/2
ρ_{ud} = ψ_{uu}ψ^{*}_{du} + ψ_{ud}ψ^{*}_{dd} = 0
ρ_{du} = ψ_{du}ψ^{*}_{uu} + ψ_{dd}ψ^{*}_{ud} = 0
The desity matrix becomes:
- -
ρ = | 1/2 0 |
| 0 1/2 |
- -
and for the entropy we get:
S = -Tr(ρlogρ) = -{(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. This represents a density matrix where all
of the elements are equal.
Therefore, we have a combined system in a pure 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).
There a specific case where this is not true. Consider a
product state. This is a non-entangled state.
ψ(a,b) = ψ(a)χ(b)
ρ_{aa'} = Σ_{b}φ(a)χ(b)φ^{*}(a')χ^{*}(b)
_{ } = φ(a')φ^{*}(a)Σ_{b}χ(b)χ^{*}(b)
_{ } = φ(a)φ^{*}(a')
<O> = O_{aa'} = <φ|O|φ>
In this case, each sub-sub system is in pure state, the density
matrix is 0 and the entropy is 0. We know everthing about the
combined system and its sub-systems.