# Bell's Theorem: The Quantum Venn Diagram Paradox

Henry: \hIf you have polarized sunglasses, you have a quantum measurement device.
Grant: Each of these pieces of glass is what's called a "polarizing filter", which means
when a photon of light reaches the glass, it either passes through, or it doesn’t.
And whether or not it passes through is effectively a measurement of whether that photon is polarized
in a given direction.
Henry: \hTry this: Find yourself several sets of polarized sunglasses.
Look through one set of sunglasses at some light source, like a lamp, then hold a second
polarizing filter, between you and the light.
As you rotate that second filter, the lamp will look lighter and darker.
It should look darkest when the second filter is oriented 90 degrees off from the first.
What you're observing is that the photons with polarization that allows them to pass
through a filter along one axis have a much lower probability of passing through a second
filter along a perpendicular axis – in principle 0%.
Grant: Here's where things get quantum-ly bizarre.
All these filters do is remove light – they “filter” it out.
But if you take a third filter, orient it 45 degrees off from the first filter, and
put it between the two, the lamp will actually look brighter.
This is not the middle filter generating more light – somehow introducing another filter
actually lets more light through.
With perfect filters, if you keep adding more and more in between at in-between angles,
this trend continues –\hmore light!
Henry: \hThis feels super weird.
But it’s not just weird that more light comes through; when you dig in quantitatively
to exactly how much more comes through, the numbers don’t just seem too high, they seem
impossibly high.
And when we tug at this thread, it leads to an experiment a little more sophisticated
than this sunglasses demo that forces us to question some very basic assumptions we have
about the way the universe works – like, that the results of experiments describe properties
of the thing you’re experimenting on, and that cause and effect don’t travel faster
than the speed of light.
Grant: \hWhere we're headed is Bell’s theorem: one of the most thought-provoking discoveries
in modern physics.
To appreciate it, it’s worth understanding a little of the math used to represent quantum
states, like the polarization of a photon.
We actually made a second video showing more of the details for how this works, which
you can find on 3blue1brown, but for now let’s just hit the main points.
First, photons are waves in a thing called the electromagnetic field, and polarization
just means the direction in which that wave is wiggling.
Grant: Polarizing filters absorb this wiggling energy in one direction, so the wave coming
out the other side is wiggling purely in the direction perpendicular to the one where energy
absorption is happening.
But unlike a water or sound wave, photons are quantum objects, and as such they either
pass through a polarizer completely, or not at all, and this is apparently probabilistic,
like how we don’t know whether or not Schrodinger’s Cat will be alive or dead until we look in
the box.
Henry: For anyone uncomfortable with the nondeterminism of quantum mechanics, it’s tempting to imagine
that a probabilistic event like this might have some deeper cause that we just don’t
know yet.
That there is some “hidden variable” describing the photon’s state that would
tell us with certainty whether it should pass through a given filter or not, and maybe that
variable is just too subtle for us to probe without deeper theories and better measuring
devices.
Or maybe it’s somehow fundamentally unknowable, but still there.
Henry: \hThe possibility of such a hidden variable seems beyond the scope of experiment.
I mean, what measurements could possibly probe at a deeper explanation that might or
might not exist?
And yet, we can do just that.
Grant:...With sunglasses and polarization of light.
Grant: Let’s lay down some numbers here.
When light passes through a polarizing filter oriented vertically, then comes to another
polarizing filter oriented the same way, experiments show that it’s essentially guaranteed to
make it through the second filter.
If that second filter is tilted 90 degrees from the first, then each photon has a 0%
chance of passing through.
And at 45 degrees, there’s a 50/50 chance.
Henry: What’s more, these probabilities seem to only depend on the angle between the
two filters in question, and nothing else that happened to the photon before, including
potentially having passed through a different filter.
Grant: But the real numerical weirdness happens with filters oriented less than 45° apart.
For example, at 22.5 degrees, any photon which passes through the first filter has an 85%
chance of passing through the second filter.
To see where all these numbers come from, by the way, check out the second video.
Henry: What’s strange about that last number is that you might expect it to be more like
halfway between 50% and 100% since 22.5° is halfway between 0° and 45° – but it’s
significantly higher.
Henry: To see concretely how strange this is, let’s look at a particular arrangement
of our three filters: \hA, oriented vertically, B, oriented 22.5 degrees from vertical, and
C, oriented 45 degrees from vertical.
We’re going to compare just how many photons get blocked when B isn’t there with how
many get blocked when B is there.
When B is not there, half of those passing through A get blocked at C. \hThat is, filter
C makes the lamp look half as bright as it would with just filter A.
Henry: But once you insert B, like we said, 85% of those passing through A pass through
B, which means 15% are blocked at B. \hAnd 15% of those that pass through B are blocked
at C. But how on earth does blocking 15% twice add up to the 50% blocked if B isn’t there?
Well, it doesn’t, which is why the lamp looks brighter when you insert filter B, but
it really makes you wonder how the universe is deciding which photons to let through and
which ones to block.
Grant: In fact, these numbers suggest that it’s impossible for there to be some hidden
variable determining each photon’s state with respect to each filter.
That is, if each one has some definite answers to the three questions “Would it pass through
A”, “Would it pass through B” and “Would it pass through C”, even before those measurements
Grant: We’ll do a proof by contradiction, where we imagine 100 photons who do have some
hidden variable which, through whatever crazy underlying mechanism you might imagine, determines
And let’s say all of these will definitely pass through A, which I’ll show by putting
all 100 inside this circle representing photons that pass through A.
Grant: To produce the results we see in experiments, about 85 of these photons would have to have
a hidden variable determining that they pass through B, so let’s put 85 of these guys
in the intersection of A and B, leaving 15 in this crescent moon section representing
photons that pass A but not B. Similarly, among those 85 that would pass through B,
about 15% would get blocked by C, which is represented in this little section inside
the A and B circles, but outside the C circle.
So the actual number whose hidden variable has them passing through both A and B but
not C is certainly no more than 15.
Grant: But think of what Henry was just saying, what was weird was that when you remove filter
B, never asking the photons what they think about 22.5 degree angles, the number that
get blocked at C seems much too high.
So look back at our Venn diagram, what does it mean if a photon has some hidden variable
determining that it passes A but is blocked at C?
It means it’s somewhere in this crescent moon region inside circle A and outside circle
C.
Grant: Now, experiments show that a full 50 of these 100 photons that pass through A should
get blocked at C, but if we take into account how these photons would behave with B there,
that seems impossible.
Either those photons would have passed through B, meaning they’re somewhere in this region
we talked about of passing both A and B but getting blocked at C, which includes fewer
than 15 photons.
Or they would have been blocked by B, which puts them in a subset of this other crescent
moon region representing those passing A and getting blocked at B, which has 15 photons.
So the number passing A and getting blocked at C should be strictly smaller than 15 +
15...but at the same time it’s supposed to be 50?
How does that work?
Grant: Remember, that number 50 is coming from the case where the photon is never measured
at B, and all we’re doing is asking what would have happened if it was measured at
B, assuming that it has some definite state even when we don’t make the measurement,
and that gives this numerical contradiction.
Grant: For comparison, think of any other, non-quantum questions you might ask.
Like, take a hundred people, and ask them if they like minutephysics, if they have a
beard, and if they wear glasses.
Well, obviously everyone likes minutephysics.
Then among those, take the number that don’t have beards, plus the number who do have a
beard but not glasses.
That should greater than or equal to the number who don’t have glasses.
I mean, one is a superset of the other.
But as absurdly reasonable as that is, some questions about quantum states seem to violate
this inequality, which contradicts the premise that these questions could have definite answers,
right?
Henry: \hWell...Unfortunately, there’s a hole in that argument.
Drawing those Venn diagrams assumes that the answer to each question is static and
unchanging.
But what if the act of passing through one filter changes how the photon will later interact
with other filters?
Then you could easily explain the results of the experiment, so we haven’t proved
hidden variable theories are impossible; just that any hidden variable theory would have
to have the interaction of the particle with one filter affect the interaction of the particle
with other filters.
Henry: \hWe can, however, rig up an experiment where the interactions cannot affect each
other without faster than light communication, but where the same impossible numerical weirdness
persists.
The key is to make photons pass not through filters at different points in time, but at
different points in space at the same time.
And for this, you need entanglement.
Henry: For this video, what we'll mean when we say two photons are "entangled" is that
if you were to pass each one of them through filters oriented the same way, either both
pass through, or both get blocked.
That is, they behave the same way when measured along the same axis.
And this correlated behavior persists no matter how far away the photons and filters are from
each other, even if there's no way for one photon to influence the other.
Unless, somehow, it did so faster than the speed of light.
But that would be crazy.
Grant: \hSo now here’s what you do for the entangled version of our photon-filter experiment.
Instead of sending one photon through multiple polarizing filters, you’ll send entangled
pairs of photons to two far away locations, and simultaneously at each location, randomly
choose one filter to put in the path of that photon.
Doing this many times, you’ll collect a lot of data about how often both photons in
an entangled pair pass through the different combinations of filters.
Henry: \hBut the thing is, you still see all the same numbers as before.
When you use filter A at one site and filter B at the other, among all those that pass
through filter A, about 15% have an entangled partner that gets blocked at B. \hLikewise,
if they’re set to B and C, about 15% of those that do pass through B have an entangled
partner that gets blocked by C. \hAnd with settings A and C, half of those that through
A get blocked at C.
Grant: Again, if you think carefully about these numbers, they seem to contradict the
idea that there can be some hidden variable determining the photon’s states.
Here, draw the same Venn Diagram as before, which assumes that each photon actually does
have some definite answers to the questions “Would it pass through A”, “Would it
pass through B” and “Would it pass through C”.
Grant: If, as Henry said, 15% of those that pass through A get blocked at B, we should
nudge these circles a bit so that only 15% of the area of circle A is outside circle
B. \hLikewise, based on the data from entangled pairs measured at B and C, only 15% of the
photons which pass through B would get blocked at C, so this region here inside B and outside
C needs to be sufficiently small.
Grant: But that really limits the number of photons that would pass through A and get
blocked by C. \hWhy?
Well the region representing photons passing A and blocked at C is entirely contained inside
the previous two.
And yet, what quantum mechanics predicts, and what these entanglement experiments verify,
is that a full 50% of those measured to pass through A should have an entangled partner
getting blocked at C.
Grant: If you assume that all these circles have the same size, which means any previously
unmeasured photon has no preference for one of these filters over the others, there is
literally no way to accurately represent all three of these proportions in a diagram like
this, so it’s not looking good for hidden variable theories.
Henry: \hAgain, for a hidden variable theory to survive, this can only be explained if
the photons are able to influence each other based on which filters they passed through.
But now we have a much stronger result, because in the case of entangled photons,
this influence would have to be faster than light.
Henry: The assumption that there is some deeper underlying state to a particle even if it’s
not being probed is called “realism”.
And the assumption that faster than light influence is not possible is called “locality”.
What this experiment shows is that either realism is not how the universe works, or
locality is not how the universe works, or some combination (whatever that means).
Henry: Specifically, it’s not that quantum entanglement appears to violate realism or
the speed of light while actually being locally real at some underlying level - it the contradictions
in this experiment show it CANNOT be locally real, period.
Grant: What we’ve described here is one example of what's called a Bell inequality.
It's a simple counting relationship that must be obeyed by a set of questions with
definite answers, but which quantum states seem to disobey.
Grant: In fact, the mathematics of quantum theory predicts that entangled quantum states
should violate Bell inequalities in exactly this way.
John Bell originally put out the inequalities and the observation that quantum mechanics
would violate them in 1964.
Henry: Since then, numerous experiments have put it into practice, but it turns out it’s
quite difficult to get all your entangled particles and detectors to behave just right,
which can mean observed violations of this inequality might end with certain “loopholes”
that might leave room for locality and realism to both be true.
The first loophole-free test happened only in 2015.
Grant: There have also been numerous theoretical developments in the intervening years, strengthening
Bell’s and other similar results (that is, strengthening the case against local realism).
Henry: In the end, here’s what I find crazy: Bell’s Theorem is an incredibly deep result
upending what we know about how our universe works that humanity has only just recently
come to know, and yet the math at its heart is a simple counting argument, and the underlying
physical principles can be seen in action with a cheap home demo!
It’s frankly surprising more people don’t know about it