Extreme Finitism presents
Entanglement and Bell Test Experiments: A Local Hidden Variable Solution
Einstein believed any apparent spooky action at a distance could be explained using local hidden variables
Many people thought this was his biggest ever mistake
During the 1920s and 30s
Niels Bohr and Albert Einstein held a series of public debates about quantum mechanics in
particular these debates questioned an element of quantum theory called nonlocality
Bohr's position was that there was no way to measure the state of a quantum particle
without disturbing it and Einstein wanted to disprove this so in
1935 Einstein proposed a thought experiment in which two particles with corresponding properties are produced from a single quantum event
these are called entangled particles
He claimed a measurement on one particle would give us information about the other particle without disturbing it
In 1964 John Stuart Bell used a principle from set theory in an attempt to resolve the issue
He produced a mathematical expression called Bells inequality that appeared to show that certain correlations between entangled particles
cannot be reproduced if all objects have just local properties or what Einstein called local hidden variables
A theorem was formed and named after Bell. The theorem states that local hidden variables cannot reproduce the predictions of quantum mechanics
But there's a problem with using Bells inequality in relation to quantum mechanics. If the detection of a quantum particle
itself is a quantum event then the very nature of particle detection should be probabilistic
Bells inequality holds if a hundred percent of particles are detected, but if detection is probabilistic
then we could never get 100% detection rate and so Bells inequality simply does not apply
If we visualise photons as small coins, then it's easy to imagine how its probability of being detected could be altered
Coins approached the detector with a full face showing and more likely to be detected than those just a thin edge showing
Now let's cover some of the basics
One way to create entangled photons is to fire high-energy ultraviolet light into a nonlinear crystal
most of the high-energy photons pass straight through
But occasionally one will hit the crystal lattice and split into two photons each with half the energy of the original
Now let's just suppose a photon has a property that we'll refer to as a polarisation angle. For example, this one is five degrees
It's entangled partner will have a related property and let's say the polarisation angles of entangled particles differ by 180 degrees
Even though we might assume that this property exists we can't be sure because we've no way to measure it. All we can do is
force or resolve it into one of two states, and then try to detect that state
So assuming the polarisation angle property exists its exact value is inaccessible to us as we've no way to measure it
Its value is entirely hidden from us
Such properties are local to photons or other entities and their
values can vary, as such they're called local hidden variables
By way of introduction to Bell test correlations consider some photons passing through two polarising filters
Half the photons pass through the first filter and when both filters are in alignment
all the photons that pass through the first filter also pass through the second
Now if we turn the second filter, so it's at an angle of 45 degrees from vertical
we find that only half the photons that reach it to get through
At 90 degrees none of the photons get through
But things get a lot less linear when we consider an angle like 22 and 1/2 degrees at this angle
85% of the photons that reach the second filter pass through
It appears that the probability of a photon passing through the second filter is
given by the cosine squared of the angle between the two filter alignments
Now what if we were to separate the two filters and send one of an entangled pair of photons through each of them
What quantum mechanics says about this is that entangled photons
do not have a single polarisation state they exist in all possible states at the same time. This is called superposition
The photons approach their respective filters, and one will reach its filter first
At this point the photon either takes on the same alignment as the filter
or it takes on an alignment perpendicular to the filter alignment either way the state of the superposition ends
Only the photons that assume the same alignment as the filter will pass through it
As soon as the first particle hits its filter and is forced into one of two states
Then instantaneously its entangle partner will end its superposition, and it will somehow acquire equal and opposite properties
It Is this mysterious communication between two different physical locations that Einstein called spooky action at a distance?
The probability that B passes through its filter is therefore influenced by photon A's earlier interaction
This concept forms the basis of Bell test experiments
But in order to do the Bell test properly we need a way of counting the photons that don't pass through the filter
Fortunately there's device called a polarising beam splitter that can do the job
It's not a hundred percent efficient, but it's just about good enough for our purposes
So with a polarizing beam splitter we can identify photons that pass straight through as
detected at one and
reflected photons as detected zero
Now if we include a source of entangled particles we now have the configuration of a Belt test experiment
Here's a table that shows a set of Bell test correlations
Note that the important thing is whether or not the detections are the same at both sides or different
This table shows what quantum mechanics predicts you're not supposed to be able to reproduce these figures using local hidden variables
But this table of quantum counts prediction says nothing about how many cases of one-sided detections we should expect
And in real experiments these counts are the biggest most significant counts of all
Bell test experiments are prone to a lot of experimental errors
But it's only by dismissing all the one-sided detections as experimental errors that case can be made in favour of spooky action
So how might we explain this huge number of one-sided detections?
Well if on average there was say a 50% chance of detection
then the expected breakdown of
detected or not on either side could easily be predicted, but as there's no way to recall cases of not detected at both sides
We're left with elevated perceived probabilities for the three remaining scenarios
Then given that there's bound to be some experimental errors
we end up with the prediction that only about 30% or less of the photon pairs will get detected at both sides and
this prediction matches actual experimental results
The figures used in Bells inequality relate to the long-term expected correlations between 'same' outcomes
where there's one at both sides or zero at both sides and
'different' outcomes where there's one at one side and a zero at the other
From the experimental results we convert the 'same' and 'different' counts into probabilities
And then we subtract one from the other to get the correlation values
We find that real experiment correlations bear some rough similarity to these highlighted values that are predicted by quantum mechanics
Note that a correlation of 1 means that the outcomes are a hundred percent
correlated ie they are all the same, and a correlation value of minus 1 means that the outcomes are 100%?
anti-correlated
meaning they're all different
When we talk about Bells inequality we usually mean this variation of it
The original bells inequality was valid for a property like polarisation, but not valid for some other properties like spin
But this generalised version works for any property that we're measuring
Its proper name is a CHSH inequality where the letters stand for the names the people involved with its creation
So how can we do an experiment to test for this inequality?
First we need to choose two angles to use at side a and two angles to use at side B
Some choices of angles won't violate the inequality
But angles like the ones selected here should lead to the biggest violation according to quantum mechanics
Then we run the experiment for each combination of filters and in theory according to quantum mechanics
We should get correlation values somewhere close to these 0.7 values shown in the table
We then plug the results into the formula and see what we get
If we get a value greater than 2, then we've violated Bells inequality. It's important to note that in real experiments
we don't get anywhere close to the correlations predicted by quantum mechanics. This is all put down to experimental inefficiencies
Now I think it's important to just clarify what this video is all about
It's not anyway an attempt to say what's really happening at the quantum level
It's the put forward an argument that the results of Bell test experiments can be explained using local hidden variables
Now we can unveil our hidden valuable solution
Each photon has three hidden variables a polarization angle a detection probability and an entangled flag
When the photon reaches the filter its detection probability is assigned a value a different process is used if the photon is entangled
The logic that determines if the photon goes straight through to the one detector
Or is reflected down to the zero detector is also different if the photon is entangled
Finally a new value is assigned to the polarisation angle and the entangled flag is reset
First let's consider the case of normal non-entangled photons going through normal polarised filters
The detection probability will always be set to 0.77
If it's reflected or not will be determined by a pseudo-random number tested against a cosine squared operation
If it passes through, then we set the polarisation angle to be equal to the filter's alignment
With no filter
the default detection probability is no 0.77
So we can detect quite a lot of photons
If we introduce a polarising filter
Then we detect about half the number of photons
Each individual photon has the probability of passing through a polarised filter as
cosine squared of it's polarisation angle
and all of the 80 photons that pass through will now have a polarisation angle of 0 degrees
So if we introduce a second filter with an alignment of 22 and 1/2 degrees we can easily calculate that on average
68 out of every 80 will get through
and if we introduce a third
filter with an alignment of 90 degrees then a local valuable
calculation tells us that on average 10 out of 68 will get through and
about 77 out of every hundred that reach the detector will be detected
Now consider the case of entangled photons
we'll use a different technique to determine the detection probability and
a really simple angle test to determine if it's reflected or not
Then we'll assign a new polarisation angle just like we did for the non-entangled case
Let's suppose that the closer the
polarisation angle is to either the horizontal or the vertical axis of the filter the higher the probability of detection
so polarisation angle on a filter axis would have a high probability of being detected possibly certainty and
a polarisation angle right in between the axes would have a low probability being detected possibly zero
We can get this probability distribution by using the absolute value of the cosine of two times the polarisation angle
But this doesn't work if we want to match the quantum mechanics predictions at all the Bell test angles
To match those predictions, we can use this expression for our distribution
This means that our lowest probability of detection is now 0.37 instead of zero
Going back to the coin analogy this makes some sense
because the thin edge of a coin will still have some area that can be detected and
when we calculate the average probability of being detected we get 0.77
Now let's see how all this can be applied to a Bell test where the filter on side A is set to twenty two and
a half degrees and the filter on side B is set to zero degrees the
dotted lines mark out twenty two and a half degree sectors, and if the photons angle maps onto a dark sector
then it will be reflected down to be detected as zero as opposed to going straight through and being detected as one
The entangled photons travel to their respective sides
In this case side A should detect a one whereas site B. Should detect a zero assuming both sides get detected
These highlighted sectors are the only ones that will give us 'different' readings all the other sectors will result in 'same' readings
In terms of area alone this would result in 75% same and
25% different
But if we include the probability of detection these figures will change
We can get a result that almost exactly matches the quantum mechanics predictions
We have created the illusion of the spooky correlation by only counting the outcomes where we've got detection on both sides
If we measure a property called spin instead of polarisation, then we get two different results
These tables show how the quantum mechanics predictions differ in the two cases
We can extend the hidden variable solution to cover the measurement of spin
For example consider the case where there's 45 degrees between the filter alignments
We use a different arrangement of the light and dark areas when working with spin
We simply have light and dark semi circles
These highlighted sectors are the ones that lead to 'different' results and again they form one quarter of the whole area
We will use two new hidden variables: spin angle and spin detection
And we simply use theta instead of two theta in the probability distribution expression
Perhaps we shouldn't abandon local reality just yet
detection probability distribution might help explain other quantum phenomena and
spooky quantum weirdness might not really exist after all
This video was brought to you by Extreme Finitism dot com
to 75% different
then if our sample of outcomes isn't big enough we could easily just be unlucky and get 20% same and
80% different so our sample will have been basically unfair
In 2013 a belt test was run over several days
where they got coincidence counts which is detection of both sides of over 1 million for each of the four runs
This may have closed the fair sampling loophole, but with an efficiency of around 25%
it comes nowhere near to eliminating the detection probability issue
This begs the question
Can Bell's inequality be of any use? If we assume 100%
efficiency then hidden valuable theory cannot violate the inequality
But with anything less than 100% efficiency it can
so by necessity the inequality requires 100% efficiency and nothing less
If Bell test experiments could ever run at 100% efficiency then local hidden variables would have a problem
It would also mean the some quantum events can be a hundred percent deterministic
So I look forward to that day if it ever happens because the results would certainly be very interesting
Do Bell test experiments support spooky quantum weirdness and is it likely any time soon that we will perform a Bell test with
100% efficiency?
The answer to both these questions is a big no
Despite what you may have heard elsewhere
Bell test experiments provide no evidence whatsoever to support the notion of spooky action at a distance
Three final thoughts; perhaps we shouldn't abandon local reality just yet
detection probability distribution might help explain other quantum phenomena and
spooky quantum weirdness might not really exist after all
This investigation of Bells inequality was made by me, Mark Payne. I'm based in the UK and my background is in computing
Currently I can only produce content like this in my spare time
I'd like to investigate the fundamentals of maths and science on a full-time basis, but in order to do this
I need funding if you can help in any way towards this goal then please contact me via regainer.com, many thanks
This video was brought to you by Extreme Finitism dot com