Quantum Mechanics, Superposition, Entanglement, EPR Paradox, Bell's Inequality, Malus Formula, all in 20-page summary

1. 2022 Nobel Physics Prize Topic
Entangled states – from theory to technology
Quantum Superposition, Entanglement, Violation of Bell Inequality, EPR Paradox
The pioneers of Quantum Information Science
Warning: The following presentation
contains graphical contents intended for
curious audiences. Viewer discretion is
advised.
How was Einstein proven wrong

2. What is Superposition ?
History: In 1900, Max Planck discovered that energy is quantized,
can only be emitted or absorbed in multiples of a small unit. A
letter/symbol, 'h', is used to denote Planck's constant, and it has a
value of 6.626 070 15 x 10-34 J Hz-1.
Now let's first talk about an experiment known as
the Stern-Gerlach experiment. This experiment
allows us to study what is known as "Superposition".
Hot Oven
with
Silver (Ag)
1.When Silver (Ag) gets heated up
in an oven, it will boil/evaporate
and emit atoms
The Stern Gerlach experiment
separates a beam of silver atoms into
two because of the direction of the
magnetic field of the unpaired electron
in each atom.
Magnetic Field,
Vertical
2.We then apply a
vertical Magnetic field
to these Silver atoms
3. Silver(Ag) atoms are
Non-magnetic. They are
not supposed to be
affected by our magnet.
But, in the experiment,
we ended up with 2
beams of silver atoms,
one on top of the other.
???? Why ????

3. Now we know: Silver atoms have an
unpaired electron that makes the atom spin.
Half of the atoms upwards, the other half
spin downwards.
Prior the entering the magnetic field, we DO
NOT know which direction each atom is
spinning, so we say that they are in a
"Superposition" state, with a 50-50 chance of
up and down spin respectively.
Next, we split the word "bracket" into two
words, "bra" and "ket". This is known
as Dirac's Bra-Ket notation, and is used to
denote the followings:
[U]p-spin state is denoted as: |u>
[D]own-spin state is denoted as: |d>
A silver atom can either spin upwards or downwards, there is no other
alternative. Therefore, we can use a math equation (Born's Function) to
describe the superposition state of a silver atom:
|Ψ> = |u> + |d>
For now this is all we need to know. Do NOT get intimidated by this
awkward looking equation, it just states that the atom can either spin
up or down, and that there is no third option.
Oven
with
Silver (Ag)
Magnetic Field,
at an angle θ
We now repeat the same
experiment, with the Magnet
twisted by an angle θ, then not
surprisingly, the Silver
atoms' trajectory would be bent by
the same angle.
This means that how the Silver atoms spin is DEPENDENT on how we decide to measure
them. If we choose a magnetic field that is not vertical, the atoms will be found to spin at
an angle from vertical.

4. Another example: Do you know that light is Polarized?
When we buy sunglasses, we tend to buy the 'Polarized'
variety. Light (photon) is either polarized [H]orizontally
or [V]ertically, ie it is in a Superposition state.
The intensity of light is denoted with the letter "I".
The original intensity of an incoming light beam is often written
as I0
Assume now we have only 1 photon on hand. When we send it
towards a polarized lens, without knowing whether it is polarized
Horizontally or Vertically, we can only say that it has a 50%
chance of passing through that lens.
Our Photon, therefore, by definition, is in
a "Superposition" state. It can be
polarized either Horizontally or Vertically,
and no mortal on earth can tell for sure
which way it is polarized.
Vertically Polarized
lens
Photon
|Ψ> = |H> + |V>
H=horizontal
V=vertical
Incoming Intensity=I0
Output Intensity= 0.5I0

5. Sidenote: Polarization of Light
The direction of the electric field vector in
electromagnetic wave is called the polarization
direction.
Most naturally produced light (e.g. thermal or
atomic radiation) has polarization vectors
oriented randomly in all directions
(“unpolarized”).
A polaroid filter absorbs light polarized along one
axis and transmits light polarized along the other.

6. Sidenote: Polaroid Filters and Unpolarized Light
When unpolarized light passes through a polaroid
filter, two things happen:
• Half the intensity is absorbed by the filter
• The transmitted light becomes linearly polarized
in the same direction as the filter’s polarizing axis
The polaroid filter has long molecules (polymers)
stretched perpendicular to the polarizing axis that
strongly absorb electric fields in that direction
(only)

7. Sidenote: Polaroid Filters and Linearly Polarized Light
When linearly polarized light passes
through a polaroid filter:
• The intensity is reduced depending
on the angle 𝜙 between the
polarization direction and the
polarizing axis (Malus’ Law):
𝐼 = 𝐼0 cos2
𝜙
• The transmitted light becomes
linearly polarized in the same
direction as the filter’s polarizing axis
Malus’ Law only applies to light which is already linearly polarized!

8. Sidenote: Polarization of light, Conclusions
• Light is always polarized in the direction of the last
polaroid filter it passed through
• If light (polarized or unpolarized) is passed through
two ideal polaroid filters with perpendicular
polarizaration axes, it will be completely absorbed
• If light (polarized or unpolarized) is passed through a
series of three polaroid filters, the first one being
vertical, the second angled at 45o from the first, and
the third angled at 45o from the second (ie the first
and third polarizer being perpendicular to each other,
with a second polarizer in between), then the light will
NOT be completely absorbed.
Light Intensity values, I0, I1, I2, and I3 can be deduced using Malus formula:
I1 = I0*cos2(45) = 0.5I0
I2= I1*cos2(45) = 0.5I1 = 0.25I0
I3= I2*cos2(45) = 0.5I2 = 0.125I0
Please note that I3 is NO LONGER zero, due to the presence of the middle polarizer, without
which, I3 would be zero (no light passes through). When the middle Polarizer is removed:
I1=I0*cos2(45) = 0.5I0
I2 = I1 (no middle Polarizer) = 0.5I0
I3 = I2cos2(90) = 0

9. What, then, is Entanglement?
We know that electrons are negatively charged. There
are also positively charged electrons, known as
positrons.
(Side note: PET scans in hospitals are good examples of
Positrons in action, the "P" in "PET" stands for
Positron.)
When a Positron and an Electron meet, they will
annihilate each other, resulting in 2 photons,
shooting in opposite directions and with
opposite polarity.
We call these 2 Photons 'entangled' at birth,
their polarizations are opposite to each
other. When one's state is known, the other one
can be immediately determined.
Furthermore, each Photon has a superposition
of polarization (either horizontal, or vertical).
e
p
Electron
Positron
Photon1 Photon2
The Bohr vs Einstein debate
Bohr proposed that entities (such as electrons, photons) had
only probabilities if they weren't observed, their states are
non-deterministic.
Einstein said Bohr is wrong, “God does not play dice”, 'Do
you really believe the moon is not there when you are not
looking at it?'
Bohr argued that we could have only probabilistic
knowledge of a system: in Schrödinger’s thought
experiment, a cat in a box is both dead and alive until it is
seen. And the superposition state collapses to a single
eigen-state only when being observed/measured.
Einstein called this “spooky action at a distance”, that there
is a "Local Hidden Variable" that we have yet to discover.
Please note that the verb "collapse" is used to describe the
result of an observation/measurement.

10. Einstein–Podolsky–Rosen (EPR) paradox
(David Bohm's version/explanation, 1951)
When sub-atomic particle "Pion" (π) decays, it disappears into an
Electron (negatively charged) and a Positron (positively
charged). The Electron and the Positron carry opposite
charges, and they also spin in opposite directions (either up or
down), ie they are entangled at birth.
We can use a vertical magnetic field at either end to
measure the spin directions of the Electron as well as that
of the Positron.
Pion
π
decays
Electron Positron
Magnetic Field Magnetic Field
The Electron (left) and the Positron (right) are entangled at birth. In the diagram above, the electron's spin direction Up. Therefore it
gets deflected upwards by our Magnetic field.
If the Electron's spin is measured to be Up, as in this example, then the Positron's spin will have to be Down. And vice versa.
They always spin in opposite directions. Measuring one will allow us to learn about the other.
The Positron and the Electron are said to be 'entangled' and they both carry a spin superposition.
Einstein, Podolsky, and Rosen co-wrote a
paper best known as the EPR Paradox.
Original Title: 《Can Quantum Mechanical Description of
Physical Reality be Considered Complete》

11. Pion
π
Electron Positron
Magnetic Field A
(vertical)
Observer: Alice
Magnetic Field B
(angled at θ)
Observer: Bob
Next, we repeat the previous experiment, with a slight twist:
The Magnetic field on the right side is now at an angle θ from vertical
+1
-1
Further, we define Up-spin as +1 (as illustrated on the left) and
Down-spin as –1 (as shown on the right side of the diagram).
The Electron and the Positron are entangled at birth. If the electron
(left side of diagram) is measured and found to be spinning UP, the
Positron on the right must be spinning DOWN. And vice versa.
With the Magnetic Field B on the right side placed at an angle θ (ie no longer
vertical), this experiment seems to demonstrate that Einstein has a point. The
Positron on the right, as measured by an observer Bob on the right, is
spinning DOWN at an angle θ (not straight Down), while the observer on the left,
Alice, upon noticing that the Electron is spinning straight Up, concludes that Bob's
Positron must be spinning straight Down, in obvious violation of Bob's observation.
Then John Stewart Bell
came along, in 1964.
JS Bell: the Belfast scientist
who proved Einstein wrong
θ
Sidenote: By convention,
the observer on the left is
called Alice, and on the
right is named Bob.
Einstein then concludes that Bohr's theory is "incomplete" (his exact word).

12. Bell's Inequality, 1964
Remember Venn Diagram?
Real life example:
Let's say we are performing a clinical trial of a miracle drug
that can conquer every ill. We have 100 patients, and we
classify them into 3 strata:
A. Male or Female
B. Smoker or non-Smoker
C. Over-40 or Under-40
In other words:
"A" denotes male patient
"not A" denotes female patient
"B" denotes smoker
"not B"denotes non-smoker
"C" denotes patient over 40
"not C" denotes patient under 40
Furthermore:
"A but not B" denotes Male non-Smoker patients
"B but not C" denotes Smoker under-40
"A but not C" denotes Male under-40
Bell's inequality states that
"A but not B" + "B but not C" have to be greater than or equal to "A
but not C" under all circumstances, no exceptions.
Conversely, "A but not C" must be smaller than or equal to the sum
of "A but not B" and "B but not C"
In plain English, "A but not C" (Male under 40) has to be less
than the sum of "A but not B"(Male non-Smoker) and "B but
not C"(Smoker under 40).
This makes sense. "A but not C" (Male under 40) did not
include female smokers who are either over 40 or under.
Bell's Inequality assumes that all our patients have been
assigned 3 pre-determined properties: Male/Female,
Smoker/non-Smoker, over-40/under-40.
If Einstein is right, that all matters are deterministic, then all our
experiments in the future will comply with Bell's Inequality.

13. Bell's Inequality: what's the relevance?
How was Einstein proven wrong?
Coming back to our photon example, let us define the followings. We have 100
random (non-polarized) photons.
Let us define:-
A = Photons that can pass through a θ=00 (vertical) polarizer
B = Photons that can pass through a θ=22.5o polarizer
C = Photons that can pass through a 2θ=45o polarizer
We now summon Malus formula. Malus formula
states that the intensity of the plane polarized light
that passes through the analyzer is directly
proportional to the square of the cosine of the angle
between the plane of the polarizer and the
transmission axis of the analyzer.
When linearly polarized light passes
through a polaroid filter:
• The intensity is reduced
depending on the angle 𝜙
between the polarization direction
and the polarizing axis (Malus’
Law):
𝐼 = 𝐼0 cos2
𝜙
• The transmitted light becomes
linearly polarized in the same
direction as the filter’s polarizing
axis
We have 100 random(non-polarized) photons. If we send them onto a polarizer we only get 50 polarized
photons on the other side. Now, if those 50 newly-polarized photons are directed onto another polarizer
at an angle θ, then Malus' formula states that the resulting photons would be reduced by a factor of
cos2(θ).
In other words, applying Bell's inequality notations, we can determine how many photons can
pass through our Polarizer A, B or C.
"A" = 100 * 0.5 = 50 (We lose half the random photons during the first pass)
"A and B" = 50*cos2θ (Malus formula, we lose more photons after the second pass)
"A but not B" = 50*(1-cos2θ) = 50(1-cos222.5) = 7.32 (negate B from above)
"B but not C" = 50*(1-cos2θ) = 50(1-cos222.5) = 7.32 (same logic as A but not B)
"A but not C" = 50*(1-cos22θ) = 50(1-cos245) = 25 (same logic, but the angle is now 45
For Einstein to be proven right, Bell's Inequality must hold true.
"A but not B" + "B but not C" must be greater than or equal to "A but not
C". Our math calculations above, unfortunately, seem to indicate
otherwise. Bell's Inequality did not hold true.
In this example, we have just proven him wrong, on paper.

14. We now need to perform experiments to demonstrate Bell's inequality is violated.
• John Clauser demonstrated that one Bell's Inequality is violated in practice, not just on paper
• Einstein's assumption, that each photon has a pre-determined characteristic at birth, is incorrect.
• Unlike in our example with 100 Patients during a clinical trial, where each patient has a pre-
determined characteristic (age, sex, and smoker/non-smoker) , Photons/electrons do not. Their
characteristics are non-deterministic and can only be described using probabilities.
Clauser's experiments were criticized, however, for having the 2 filters too close to each other,
allowing the entangled photons a chance to communicate with each other.

15. • Alain Aspect extended the distance between filters to 13m
• He would switch different settings at a rate of 10 nono-seconds, faster than a photon can travel from one side to the
other, thereby eliminating any chances of one photon communicating with another in time to modify characteristics
• However, he was criticized for not developing a truly RANDOM settings
• Anton Zeilinger used light from distant galaxies
to generate random settings
• Performed "Big Bell Test" on Nov 30, 2016
• Recruited 100,000 participants worldwide to
contribute unpredictable sequences of zeros
and ones (bits) through an online video game.
• The results of the experiments confirm the
violation of Bell inequalities by a more rigorous
methodology than ever
• More importantly, Zeilinger demonstrated
Quantum Teleportation

16. Image Credit:
NASA/JPL-Caltech

17. Image Credit:
NASA/JPL-Caltech

18. Sidenote: If you are a diabetic, don't even think that by not measuring your HbA1C, your chance of being a
diabetic will be reduced to 50%.
Sidenote: If you have hypertension, don't even think that by not measuring your Blood Pressure, you can
reduce your chance of being Hypertensive to 50%.
Sidenote: Also, don't even think about avoiding looking into the mirror to reduce the chance of getting
old to 50%. Nor avoiding the doctor would reduce your chance of getting sick to 50%.
Sidenote: Quantum Mechanics, after all, is Physics/Math. It cannot replace
Biology/Chemistry. Superposition does NOT exist in the latter (as far as I can tell).
Unless, of course, if you are a cat, and you belong to a man called Erwin Schrödinger.
We have witnessed the birth of Quantum Mechanics, and with it Quantum Information Science.
Quantum Teleportation was also demonstrated in front of our eyes by Anton Zeilinger.
We have learnt that things are not necessarily binary. They are probabilities.