The strangeness of Entanglement
In the last century physicists have developed quantum
mechanics, a theory required to explain observations made at the
atomic level. Quantum mechanics is a very successful theory:
It can precisely predict the colours emitted by atoms and give an
explanation for the properties of the elements. It also
describes a property discovered in quantum system called
"entanglement", but in doing so it forces us to reject locality or
Our common experience of the world is
local and real. It is taught to us by experience and
expressed with statements such as: "Touch the cup of tea to feel
if it's warm" or "The gold is in the safe."
Locality expresses a known interaction observed
between two objects when they touch each other. This
two objects to influence each other in a predictable way.
Your hand must touch the cup of tea to feel the warmth.
Reality expresses a relationship between our
knowledge and the world. When something is real our
knowledge of it can be demonstrated. Open the safe and you
will find the gold.
Entanglement is a statistical property of quantum
systems. Because entanglement is not observed in familiar
objects, it is a concept which might be difficult to grasp.
Its description involves statistical analysis, quantum mechanics
and Bell's inequalities. In this paper, however, I present
entanglement from a simpler point of view. The statistical
analysis is simplified to a level comparable in difficulty with
the analysis of a coin toss or a roll of the die. Moreover,
most of the difficulties related to quantum mechanics are avoided
by using a much more familiar classical scenario.
The scenario of a classical experiment is presented
first. Based on locality and reality, only one model
explains the experimental results. Then, the model is used
to make a complementary prediction which is confirmed by the
results. In the second part of this paper, a quantum
experiment is described where entanglement is
observed. The description is phenomenological and does not
require a knowledge of quantum mechanics beyond what is expected
from the reader. However, since this quantum experiment was
carried out in a laboratory, results are available for this
discussion. Based on locality and reality, the experimental results are explained by only one model which turns out to be the same model
found for the classical scenario.
Because of the strange properties of entanglement, however,
the complementary predictions made by the model disagree with the
results of the quantum experiment. The only explanation for
the discrepancy is that locality or reality must be rejected.
This is inspired by the papers from Mermin and Bell.
A classical experiment
1) Classical experiment and correlations
Imagine this scenario which describes a classical
Three people are involved in this experiment: you, myself
and a common friend who writes to both of us. Each
week our friend sends a pair of postcards, one for you
and the other one for me. Both postcards are sent
at the same time via Canada Post. In this scenario, the
mail service is the fastest way to communicate between people:
it takes at least two days to deliver the mail, but never more
Upon receiving several postcards, I observe the
following. The postcards always have three flaps labeled
1, 2 and 3 and a date written on the postmark. When a flap
is lifted a single word is exposed which can be either "heads"
or "tails". However, the postcards have a special
mechanism: Opening a flap makes ink flow behind the other two
and covers the other words. So although I am free to
choose which flap to lift, I can only read one of the three
Having figured out so much, I decide to start recording
my data in a systematic way. Upon receiving the postcard,
I pick a random number from 1 to 3 and lift the flap labeled
with that number. On the first week I choose the number 1
and lift the corresponding flap; the word "heads" appears.
The next week, I choose 3 and the word "heads" appears.
The following postcard has "tails" underneath flap 1.
After two months my list of flap-numbers and words is: 1H, 3H, 1T, 1H, 2H, 2T, 3H, 3H.
I am intrigued by the behaviour of our friend and wonder
what he writes on your postcards. So I send you my
list (also via Canada post) with the date of the first
entry. Since great minds think alike, you also have been
recording data obtained by lifting flaps at random. You
send me your data, starting on the same date as mine: 2H, 3H, 3H, 1H, 1T, 2T, 1H, 2T.
Being an investigative person, you combine our lists to
make word-pairs with my data on the left and yours on the right:
1H2H, 3H3H, 1T3H, etc. You discover
something interesting: Word-pairs with the same flap-numbers
always have the same words! The combined
observations give the following word-pair list:
3H3H <- flap 3 lifted on both
postcards, same words exposed: "heads" and "heads"
1H1H <- flap 1 lifted on both
postcards, same words exposed: "heads" and "heads"
2T2T <- flap 2 lifted on both
postcards, same words exposed: "tails" and "tails"
This continues for several years until we each have
received over 400 such postcards. We combine our lists to
get the word-pairs shown below in Table 1. Although the
words "heads" and "tails" seem to be randomly distributed,
word-pairs with the same flap-numbers always have the same
They are highlighted in
red to make them stand out.
TABLE 1. Partial list of 432 word-pairs
from the postcards received every week. The green
rectangle highlights my
first entries. The red entries are those for which we
randomly chose the same flap-numbers.
This is an example of a correlation: it is a
predictable relationship between two different data. The
correlation is perfect for word-pairs with the same flap-number
because we can predict with 100% certainty that the outcome will
be a word-pair with the same words.
2) Statistical analysis
A statistical analysis is necessary to better understand
random data. The analysis gives the probability of obtaining
a certain outcome in a large data set. A probability is the
ratio of two numbers: the number of desired outcomes and the
number of all possible outcomes.
For example, the theoretical probability of winning a coin
toss if you bet "heads" is calculated as follows. The number
of desired outcomes is 1 since there is only one outcome that can
make you win. The total number of possible outcomes is 2
since the coin has a total of two sides, "heads" or "tails".
The probability of winning is 1÷2 = 50%.
Another example: what is the probability of throwing
"4" or "5" with a single die? The number of desired outcomes
is 2 since there is one "4" and one "5" on the die. The
total number of possible outcomes is 6 since a die has six
sides. The probability of throwing a "4" or a "5" is 2÷6 =
In an actual experiment, the number of occurrence of a
desired outcome must be counted on a large number of measurements
to obtain a good estimate of the probability. For
example, you might toss a coin 1000 times and obtain 489
"heads". The estimated probability of getting "heads" is
489÷1000 = 48.9%, which is consistent with the 50% probability
calculated from a fair coin. The estimate is less accurate
if fewer measurements are available.
After eight years of receiving postcards from our
(boring) friend, we make a statistical analysis of the entries
in Table 1.
A) For this first analysis we only look at individual
Although this analysis is overwhelming, the important
thing to retain is that the estimated probabilities are all
consistent with a probability of 50%.
Counting the entries in my list where I lifted flap 1: 1H is listed 78 times, 1T is listed 67 times, for a total of
145 words. Therefore the estimated probability of
occurrence for "heads" is 78÷145 = 54% and for "tails" is 67÷145
The same procedure gives for my other entries: 2H = 50%, 2T
= 50%, 3H = 43%, 3T = 57%, and independently of the flap
I chose H = 49%, and T =51%. For your entries: 1H = 49%, 1T
= 51%, 2H = 56%, 2T = 44%, 3H
= 40%, 3T = 60%, H = 48%, and T
If all entries are taken together, the estimated
probability of occurrence for "heads" is 48.6% and for "tails"
B) For this second analysis we look at the combined word-pair
entries, but only those with the same flap-numbers.
Word-pairs having the same words are listed 139 times,
word-pairs having different words are not found in the list,
for a total of 139 word-pairs with the same flap-numbers (shown
in red in Table 1 as 1T1T,
1H1H, 2T2T, 2H2H, 3T3T, and 3H3H).
The estimated probability of occurrence of the same
words when the same flap-numbers are chosen is 139÷139 = 100%.
3) A model for the observed statistics and correlations
We explain these results by
following this formal line of reasoning.
A) The conditions of the classical
i- A postcard has three flaps, each one hides a single word:
"heads" or "tails".
ii- Lifting a flap exposes a word and releases ink that covers
the other two, making them unreadable.
iii- Our friend sends pairs of postcards simultaneously via
Canada Post, one for you the other for me. A postmark with
the sent date is printed on the postcards.
iv- It takes at least two days to deliver the mail, never more
than three. This is the fastest way to communicate between
v- A random number is chosen just before lifting the flap
labeled with that number. This is done in order to sample
the words without any bias.
vi- For every postcard
I take note of the date, the chosen flap-number
and the exposed word. You follow the same procedure.
vii- There is no other common agreement between any of
you, me and our friend.
viii- There is no communication
between any of us except for:
- the postcards sent to us by our friend,
- the lists sent to each other.
ix- Our lists are combined into word-pairs for each pair of
postcards sent on the same date.
B) The individual observations are:
i- One of only two possibilities, "heads" or "tails", is
exposed when a flap is lifted. We never see an ink
ii- The occurrence of the words is random, with no discernible
iii- The words have a near equal chance of being used,
independently of which flap is lifted. This is the
result of the statistical analysis (2A).
C) The combined observations are:
i- Word-pairs with the same
flap-number always have the same words. This is the
result of the statistical analysis (2B).
D) We make
a few (reasonable but debatable) assumptions:
i- Our friend flips a coin to generate the random words on
the postcards. The statistical analysis (2A) is
consistent with the 50% probability expected from a coin
ii- The word exposed when I lift a flap tells me that a the
same word is written in your postcard under the
same-numbered flap, even if you never lift that flap.
called "element of reality": the word is determined even
if nobody looks. "The moon is there when nobody
iii- Since I am free to choose which flap to lift, all three
words on your postcard must already be determined when the
postcard is sent.
This is because it is potentially
possible for me to predict which word will be written under
any of your flaps.
iv- You can deduce the same thing about the words written in
v- Transmission of information is
impossible at a speed faster
than Canada Post can send a message.
assumed: a message cannot influence the other at another
location faster than the maximum speed of transmission
E) Given that we accept what is written above, we
conclude the following:
i- In order to explain observation (3C-i), our friend must
send us identical postcards chosen from one of these eight
templates. There is no other possibility.
Template My postcard Your
- 2H - 3H 1H
- 2H - 3H
2 1H - 2H - 3T 1H - 2H - 3T
3 1H - 2T - 3H 1H - 2T - 3H
4 1H - 2T - 3T 1H - 2T - 3T
5 1T - 2H - 3H 1T - 2H - 3H
6 1T - 2H - 3T 1T - 2H - 3T
7 1T - 2T - 3H 1T - 2T - 3H
8 1T - 2T - 3T 1T - 2T - 3T
ii- Each template has the same probability of being
chosen. This is necessary in order to explain (2A).
iii- The postcard cannot be changed during transit (e.g. by a
criminal tampering with the mail). If the criminal
opened a flap, we would sometimes expose an ink covered word
(which does not happen according to 3B-i). Even if a new
postcard was forwarded by the criminal, they could only read
one word from off the original and not know which other two
words to fill in (because of 3B-ii they cannot predict the
words written on the postcards). Forwarding a new
postcard would destroy the perfect correlation (2B).
Even if an eavesdropper saw which flap I opened, there would
not be enough time for them (because of 3A-iv) to mail that
information to an accomplice who could intercept your postcard
and modify it.
4) A complementary prediction
make a prediction:
We now want to calculate the probability of obtaining
the same words in a word-pair.
This is obtained by taking the template table and
reducing it to the case where I lift flap 1. We don't
need to know what is behind the other flaps:
Template Mine Yours
1 1H 2H - 3H
1H 2H - 3T
1H 2T - 3H
4 1H 2T - 3T
5 1T 2H - 3H
6 1T 2H - 3T
7 1T 2T - 3H
8 1T 2T - 3T
From (E-ii) we know that each template has the same probability of
being chosen. If you lift flap 2, we get the
same words for templates 1, 2, 7 and 8 out of eight templates:
that is 4÷8 = 50% of the time. If you lift flap 3, we
get the same words for templates 1, 3, 6 and 8 out of eight
templates: that is also 50% of the time. The same
probability of 50% is also obtained if the table is reduced to
the case where I lift flap 2 or 3. (Left as an
we chose to lift a different flap on
our respective postcards, the probability of obtaining the
same words in a word-pair must be 50%.
B) Comparison with the complementary analysis:
Given that we accept 3A), 3B), 3C) and 3D), we predict
that if we lift different flaps the probability of getting the
same words in a word-pair is 50%.
The following table lists the results obtained from a
statistical analysis of word-pairs with different flap-numbers
taken from Table 1.
|100% are the
(1T1T or 1H1H)
|55% are the same
(1T2T or 1H2H)
|61% are the same
(1T3T or 1H3H)
|54% are the same
(2T1T or 2H1H)
|100% are the
(2T2T or 2H2H)
|45% are the same
(2T3T or 2H3H)
|55% are the same
(3T1T or 3H1H)
|49% are the same
(3T2T or 3H2H)
|100% are the
(3T3T or 3H3H)
TABLE 3: Statistical properties of 432
word-pairs obtained with a classical experiment. Green
of getting the same words for the 293 word-pairs with
The analysis is therefore compatible with the predicted
probability of 50%.
A quantum experiment
5) Quantum experiment
The experimental setup is described here for reference
only. More details can be found in original papers such as
but are very specialized. It is best to think of this
description in terms of black boxes and compare it with the
classical scenario described above. As with the classical
scenario, the quantum experiment has two receivers (called A and
B), a choice between three possible measurements (called 1, 2, or
3) and only two possible outcomes (called H or T).
A real quantum experiment can be made using
a system with a single Source and two Detectors. The
Source emits a pair of simultaneous light-pulses in opposite
directions: one light-pulse toward Detector A and the other
toward Detector B (see Figure A).
The detectors measure the polarization of the light-pulses using
an analyzer and two photodetectors. When a
photodetector detects a light-pulse,
it amplifies the signal to produce an electrical output
large enough to be easily measured. The analyzer
redirects the light-pulses
according to their polarization: If a light-pulse
through the analyzer, the photodetector labeled "heads"
generates an output; if a light-pulse is reflected by the analyzer, the
photodetector labeled "tails" generates an output. It is possible to produce
light-pulses which will only trigger one photodetector
at a time. Either "heads" or "tails" will
produce an output at "H" or "T" respectively, never
both at the same time. Hence the name "photon"
is used because the indivisible quantized light-pulse
can't be split: it will either go straight into
photodetector "heads" or be reflected into
Simplified schematics of an experimental
setup. Two Detectors measure the polarization
of the received light.
The Source produces simultaneous photon-pairs.
After the photon-pair is
emitted, the Detectors are rotated
individually about the axis of the light beam.
The entire assembly analyzer-photodetectors is
Detectors are set randomly to any of three positions:
Setting 1: aligned with the vertical at 0°,
rotated by 60° with
respect to the vertical,
rotated by 120° with respect
to the vertical.
Figure B shows an example with Detector A rotated
and Detector B aligned
with the vertical at 0°.
FIGURE B: Configuration with
Detector A at Setting 2 and Detector B at Setting 1
6) Results and
photon-pairs are detected, the settings of the Detectors
are recorded with the observed output-pairs. A
list of output-pairs is produced which uses the same
notation used for the classical experiment described
above. For example, 2T1H means that
Dectector A was set to 2 and measured a photon on
"tails", and Detector B was set to 1 and measured a
photon on "heads".
Such a setup
which uses the above concepts has been built Ref. .
analysis of output-pairs gives the following results:
A) The probability for producing an output "heads"
is 50% and the probability of obtaining "tails" is
B) The probability of the same
outputs for an output-pair is near 100% with
the same detector setting.
reveals that the quantum experiment
produces probabilities compatible with
the classical experiment for individual
detector probabilities and for
output-pairs with the same detector
At this point,
we have the same situation as with the classical experiment
described above. The setting of the Detector corresponds
to the flap-number. The outcomes "heads" or "tails" have
the same statistical properties as those described in 2A and
2B for the classical experiment. We can therefore follow
the same logic we followed in (3) to find the same model.
7) Model and comparison
with experimental results
In this case, the model as described in (4) makes the
same prediction for the quantum experiment:
When we chose a different setting for
the Detectors, the probability of obtaining
the same outputs in an output-pair must be 50%.
Comparison with experiment:
When the output-pairs from the quantum experiment Ref.
 are analyzed, the following results
are obtained which can be summarized in the following table:
|~25% are the same
|~25% are the same
|~25% are the same
||~25% are the same
|~25% are the same
||~25 % are the same
TABLE 4: Statistical properties of measured
occurrence of output-pairs. The results
obtained with the quantum experiment for different settings
compatible with the 50% predicted by the classical theory.
When the Detectors have different settings, the
probability of getting the same outputs in an output-pair is
near 25%. This is incompatible with the prediction of
a probability of 50%. (Another experiment Ref.  uses 143 km between the emitter and
receiver. It also shows incompatibility with the classical
prediction, although the long distance introduces noise and
reduces the probabilities to 80% for the same setting and 28% for
It is as if one Detector knows that the other one does
not have the same setting and, by some spooky action at a
distance, is able to change the statistics of the
measurements. This is, of course, impossible because the
detector settings are chosen randomly a few microseconds before
detection, a time too short to propagate any signal back to the
other detector even at the speed of light.
Experimental results show that the classical model
incorrectly predicts the statistical properties of quantum
experiments. This implies that the assumptions that were
used are incorrect: locality or reality are incompatible with
quantum physics. This is what Bell showed formally in his
describes the statistical property resulting from only one
element of reality for two separated signals and detectors
at the same time, independent of the distance between the
detectors. The signals reaching the detectors and
the position of the detectors behave as a whole, even if the
setting of one detector can be changed randomly just before a
measurement made by the other detector. This has been
Entanglement, as described by quantum mechanics, is a
solution which predicts experimental results. However,
it rejects "reality" (Einstein would not reject elements of
reality!) and "locality" (Einstein called this spooky action
at a distance!) This tells us that the behaviour of the
quantum world is far from our common experience.
 David Mermin, "Is the moon there when
nobody looks", Physics Today, pp. 38-47, April 1985.
 John Bell, "On the Einstein Podolsky Rosen Paradox," Physics.
1 (3), 195–200 (1964).
 Alain Aspect et al., "Experimental Test of Bell's
Inequalities Using Time-Varying Analyzers", Physical Review
Letters 45, no. 25, p. 1804, 20 Dec 1982.
 Xiao-song Ma et al., "Quantum
teleportation over 143 kilometres using active feed-forward,"
Nature 489, 269–73 (2012), also https://arxiv.org/abs/1205.3909.
© Louis Marmet 2018