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Part 4: Investigating Entanglement
Timing: This full lesson requires 60 minutes and much of it is very challenging. A simpler lesson
involving just the concrete models and the video would only require 30 minutes and could be used in a
physics or computer science class.
Key Concepts: Entanglement is a key concept of quantum physics and very counter-intuitive.
Assumptions of Student Background Knowledge: This lesson assumes that the students have already
gone through Part 3: Investigating Quantum Polarization.
Materials: The teacher needs a pair of gloves, coins and a secretly prepared assistant. The teacher also
needs a data projector and internet access. Students need the worksheets and an ABCD booklet.
The answers and extra information for teachers is in red and need to be removed before photocopying and
printing the student worksheet.
Quantum physics is very different from classical physics. Some of the key concepts of quantum physics
are intrinsic randomness, wave-particle duality, measurement-disturbance, uncertainty and
superpositon. A fifth and perhaps the most important key concept is entanglement.
Certain crystals can take one photon and produce two photons with half the energy. These photon pairs
are said to be entangled because they have identical - but still random - polarization. If you measure the
polarization of one of them, you immediately know what the polarization of its partner is.
This splitting of photons is called spontaneous parametric down conversion - SPDC. Usually it takes an
ultraviolet photon and forms two infrared photons. Sometimes the photons have identical polarizations
and sometimes they are orthogonal. We will consider ones that have identical polarizations.
This process is a very rare occurrence and only one out of each thousand billion (1012) photons will split.
This means it is also a good way to produce single photons. A typical laser pointer produces a few mW of
light. A red photon has an energy of about 10-19 J. That means that laser pointer produces 1016 photons
every second. If that is reduced by SPDC there will be 10,000 photons per second, or one every 100 s.
This may seem fast, but single photon detectors can measure to a thousandth of a s.
1) Modelling Entanglement with Concrete Objects:
a) Explain how entangled gloves work.
You know that a pair of glove has a right hand glove and a left hand glove. Therefore, once you know
what glove you have, you also know what glove the other one has. No surprise.
How to do the demonstration: Hide a pair of gloves in a box. Have a volunteer take one glove without
letting you see it and walk far away. You look in the box to see what remains and ‘amaze’ people by
being able to say what kind of glove the volunteer has – right hand or left hand. The class won’t be
amazed, because the gloves have a handedness that is revealed when you ‘measure’ what remains in the
box. Einstein was convinced that entangled particles must work the same way. This model is called
‘hidden’ variables because it says that the photon pairs have identical but hidden properties which
determine their behaviour. The polarizers select for, but do not change these properties
b) Explain how entangled coins work.
This demonstration only works if you cheat.
How to do the demonstration: You need a ‘volunteer’ who has been secretly briefed on what to do.
Show the class two coins and give your volunteer one and have them stand far away. Each of you flips a
coin and one calls out what they have. The other one says “That’s exactly what I got!!” whether it is the
true or not. The class won’t be fooled for long. They will demand that there be a witness for each flip. If
you are lucky, you will manage to succeed a couple of times without cheating. However, the trick will be
exposed after a few tries. Let them know that quantum physics says entangled photons do this trick without cheating. Their polarization is intrinsically random and as soon as one photon is measured - the
polarization of both is set.
2) Exploring Entanglement with Probabilities:
Consider three types of polarizers separated by 120o. Let’s call them Vertical, Plus and Minus.
Entangled photons are sent in opposite directions towards one of these three types of filters.
i) Which of the following is true? If the filters are identical (both V or both P or both M) then
A) if a photon passes then its twin also passes.
B) if a photon fails to pass then its twin also fails to pass.
C) both A and B
D) Neither A nor B.
C: If the filters for the photon pair are identical, then 100% of the results will be the same, i.e.
either they both pass or they both fail. This could be caused either by hidden variables or by the
intrinsic randomness of quantum physics, so it isn’t able to tell us which model is correct.
ii) Now we will look at what will happen if the two filters are different. Suppose one of the photons
passes through a Vertical filter. What is the chance that its twin will have the same result and also pass
through the Plus or Minus filter? Hint: The twin must be a vertical photon and the second filter is at an
angle of 120o to vertical.
The chance of the same result - passing - is cos2 (120o) % = 25%.
iii) Suppose that the first photon failed to pass. What is the chance that its twin will have the same result
and fail to pass?
If the photon did not pass through a vertical filter, then it and its twin are horizontal photons. The chance
of the horizontal twin passing through one of the other filters is cos2 (120o - 90o) % = 75%. Therefore the
chance of the same result - failing to pass - is 25%.
If the filters for the photon pair are different, then 25% of the results will be the same.
Quantum physics says that these results are intrinsically probabilistic - like the demonstration with the
entangled coins. However, that means that as soon as one photon is measured, information must be
transferred instantaneously to its partner. This seems highly unrealistic. An explanation that makes much
more sense is that the entangled photons have hidden variables - like the handedness of the entangled
gloves - that determine whether they will pass through a given filter.
iv) The demonstration with the entangled gloves only had two types of gloves - left and right. To explain
the results with the photons we will need more. How many different types do we need? Explain. Hint:
They either pass or fail to pass each of three types of filter.
A) 3
B) 4
C) 8
D) 9
C: There are eight photon types, summarized in the first three columns of the table below. To simulate
this we could use entangled eight-sided dice.
Photon Type
V
P
Pass
Pass
Pass
Pass
Fail
Pass
Fail
Pass
Fail
Pass
Fail
Pass
Fail
Fail
Fail
Fail
M
Pass
Fail
Pass
Fail
Pass
Fail
Pass
Fail
Combination of Filters
VP
VM
same
same
different
same
different
same
different
different
different
different
different
same
different
same
same
same
PM
same
different
different
same
same
different no
different
same
% same result
100
33
33
33
33
33
33
100
v) How often will we get the same results for an entangled pair? Hint: Consider what happens with each
photon type if the filter combinations are Vertical/Plus, Vertical/Minus and Plus/Minus.
A) 33%
B) 50%
C) 62.5 %
D) 100%
It depends: The results are summarized in the last four columns of the table above. The first and last
types give 100% and the six others give 33%. If each type is equally probable - which is reasonable
because of symmetry – the average is 62.5 %. No matter what type shows up, the results for the photon
pair will be the same at least 33% of the time.
If the filters for the photon pair are different, then the results will be the same 33% of the time or
more. This contradicts the quantum physics prediction of 25%. Now we can tell the difference
between the two models.
The original argument about entanglement appeared in an article by Einstein, Podolsky and Rosen in
1935 and it is called the EPR paradox. At that time there was no way to determine whether hidden
variables or quantum physics was correct because both models predicted the same results. In 1964 John
Bell considered testing the entangled twins differently - similar to what we have done here - and
developed what is known as Bell’s inequality. In 1983 Alain Aspect tested the predictions. This
experiment and every other one since has confirmed the predictions of quantum physics and disproved
hidden variables.
You can see Alain Aspect give a public lecture about this sponsored by PI. http://pirsa.org/06070050/
3) Understanding Entanglement with the Double-Slit Experiment:
i) One of a pair of entangled photons is sent toward double slits that normally produce an interference
pattern. These slits are covered with polarizing filters, one vertical and the horizontal. The partner photon
is sent off in a completely different direction - it never goes near the slits. This is repeated many times and
the number of photons (vertical) vs. detector location is graphed. Which of the following graphs shows
what you will get if you graph the number of photons (vertical) vs. detector location?
A)
B)
C)
Explain:
C: This is review of the concepts in part 3) Quantum Polarization. However, it uses histograms of the
results (like in Part 1) rather than showing the interference patterns that are seen on a screen. The lack of
an interference pattern can be explained by noting that vertical photons
cannot interfere with horizontal photons. It can also be viewed as a sum of
the in-phase and an out-of-phase interference patterns.
ii) The experiment is repeated, but now the photon that does not go near the slits is sent thorough a third
polarizing filter which is at 45o to vertical. Which of the above graphs shows what you will get? Explain:
C: Measuring the partner, sets the polarization of each photon, but it does not change the total pattern that
we see. It looks like the measurement had no effect.
iii) The experiment is repeated, but we only keep the data for photons whose partners made it through the
filter. Which of the above graphs shows what you will get? Explain:
A or B: Measuring the photon that does not go through the slits does affect its entangled partner. It
determines whether the photon will be found in the in-phase or out-of phase pattern. This effect is
instantaneous. This experiment and others with single photons is discussed in more detail in Fig. 4 of
http://arxiv.org/abs/1304.4943
4) Applications of Entanglement: Quantum Computation
Watch the first 15 minutes of the video Manipulating Light
http://www.learner.org/courses/physics/unit/unit_vid.html?unit=7&vidNum=0
i) How are bits different from qubits?
A single bit is a 0 or a 1. A qubit is a superposition of those states. If the information is help by
polarization, a classical bit would be photon which is either vertical or horizontal. A qubit would be 45o
photon which is in a superposition of horizontal and vertical. If you have three times the number of bits,
you can do three times the computations. However, tripling the number of qubits gives you 23 or eight
times as many computations. It grows exponentially.
Note: Polarization is just one way of forming qubits. There are many other physical systems.
ii) How do the crystals change the photons? They
A) split all the photons, increase their energy
B) split all the photons, decrease their energy
C) split some of the photons, increase their energy D) split some of the photons, decrease their energy
D) Only one photon in a billion is split. The sum of the energy of the entangled pair is equal to the
original photons energy so, each photon will have a decreased energy, roughly half. That is why blue light
change to red light.
iii) What aspects of the photons are entangled?
A) momentum, direction, polarization
C) energy, momentum, polarization
B)
B) energy, direction, polarization
D) energy, momentum, direction
iii) What weird features of quantum physics are used in a quantum computer?
A) superposition, conservation of energy
B) entanglement, superposition
C) entanglement, conservation of energy
D) all three
B) All three concepts are used, but conservation is not a weird quantum feature.
This video features the work of Paul Kwiat who wrote the Scientific American article that was the source
of the quantum editor demonstration in the Quantum Polarization lesson. This video is very well-done,
with lots of analogies to help understand quantum information, superposition and entanglement.
The second 15 minutes deals with Bose-Einstein condensates and slowing light and is not nearly as
successful. This video is part of Physics for the 21st C which is a good place to send a very strong student
who wants to learn more about modern physics. There are nine units and each unit has a 30 minute video
that looks at the work of two physicists. There are also activities. The best of these are from PI!