Bell's Theorem: The Quantum Venn Diagram Paradox

Summary

This script explores the quantum phenomenon of polarization through the lens of sunglasses, delving into the concept of quantum measurement and the probabilistic nature of photon behavior. It introduces Bell's theorem, a cornerstone of modern physics, which challenges our understanding of realism and locality. The script uses the analogy of polarizing filters and entangled photons to explain complex quantum mechanics, suggesting that the universe may not operate under the deterministic rules we expect. The discussion highlights the impossibility of hidden variables in quantum states and the implications of faster-than-light communication, ultimately presenting a fascinating insight into the quantum world.

Takeaways

• 🕶️ Polarized sunglasses act as quantum measurement devices, using filters to determine if light photons are polarized in a certain direction.
• 🌟 The polarization of light is a quantum phenomenon where the direction of the light wave's oscillation is measured by whether it passes through a polarizing filter.
• 🔄 Rotating a second polarizing filter relative to the first can cause the light source to appear brighter or dimmer, with the darkest point occurring at a 90-degree angle difference.
• 📈 Adding a third filter at a 45-degree angle between two filters can paradoxically allow more light to pass through, contrary to classical expectations.
• 🤔 The increase in light transmission through multiple filters challenges the idea of a deterministic universe and suggests the presence of 'hidden variables' in quantum mechanics.
• 🧩 Bell's theorem is introduced as a significant concept in modern physics that questions the fundamental assumptions about the nature of reality and the speed of causal influence.
• 📊 The script explains the mathematics behind quantum states, such as photon polarization, and how it relates to the behavior of light through filters.
• 🚫 The probabilities of photons passing through filters at various angles suggest that there cannot be a hidden variable that determines each photon's state with respect to every filter.
• 🔬 Experiments with entangled photons show that the behavior of one photon can instantaneously affect its entangled partner, regardless of the distance between them.
• 🔗 The concept of 'entanglement' is crucial in demonstrating that quantum mechanics cannot be explained by local realism alone.
• 📚 The script highlights the simplicity of the counting argument at the heart of Bell's Theorem and contrasts it with the profound implications it has for our understanding of the universe.

Q & A

What is a polarizing filter and how does it relate to quantum measurement?

A polarizing filter is a type of glass that allows light to pass through it when the light's polarization aligns with the filter's orientation. It effectively measures whether a photon is polarized in a certain direction, either allowing it to pass or blocking it.

How does the orientation of polarizing filters affect the amount of light passing through them?

When two polarizing filters are aligned at the same angle, light passes through both. If they are perpendicular (90 degrees apart), no light passes through. At 45 degrees, there's a 50/50 chance for each photon to pass through the second filter after the first.

What is the phenomenon where adding a third filter between two others makes the light source brighter?

This occurs when a third filter is placed at 45 degrees to the first and causes more light to pass through the system. It's counterintuitive because it seems like the middle filter is generating more light, but it's actually altering the probabilities of light passing through the system.

Why do the numbers seem 'impossibly high' when analyzing the light passing through filters?

The numbers appear impossibly high because the probabilities of light passing through filters at certain angles do not follow the expected linear relationship. For example, at 22.5 degrees, the probability is 85%, which is significantly higher than the expected halfway point between 50% and 100%.

What is Bell's theorem and why is it significant in modern physics?

Bell's theorem is a principle that challenges the concept of local realism in quantum mechanics. It suggests that either the universe does not operate on hidden variables (realism) or that information can be exchanged faster than the speed of light (locality), or some combination of both.

How do the concepts of 'realism' and 'locality' relate to the discussion of hidden variables in quantum mechanics?

Realism is the assumption that particles have definite properties, even when not being measured, while locality is the principle that information cannot travel faster than light. The experiment with entangled photons suggests that one or both of these assumptions must be incorrect.

What is the significance of the experiment with entangled photons in the context of Bell's theorem?

The entangled photon experiment is significant because it shows that the outcomes of measurements on entangled particles cannot be explained by local hidden variables. This supports the violation of Bell inequalities and challenges the principles of realism and locality.

How do the probabilities of light passing through filters differ when considering entangled photons?

With entangled photons, the probabilities of both photons passing through or being blocked by filters remain consistent with the non-entangled case, but the correlations between the outcomes are stronger, regardless of the distance between the photons, suggesting non-local interactions.

What is the significance of the first 'loophole-free' test of Bell's theorem in 2015?

The first loophole-free test of Bell's theorem in 2015 was significant because it provided strong evidence against local realism by closing potential experimental flaws that could have otherwise explained the observed violations of Bell inequalities.

What is the role of the polarization of a photon in the context of the quantum measurement with sunglasses?

The polarization of a photon is the direction in which its electromagnetic wave oscillates. In the context of the quantum measurement with sunglasses, the polarization determines whether a photon will pass through a polarizing filter, which is a key aspect of demonstrating quantum behavior.

How does the script relate the simple act of using polarized sunglasses to the complex principles of quantum mechanics?

The script uses the act of using polarized sunglasses as a simple, relatable demonstration to explain the complex principles of quantum mechanics, such as quantum superposition and entanglement, making these abstract concepts more accessible to a general audience.

Outlines

00:00

🕶️ Quantum Measurement with Polarized Sunglasses

The script introduces the concept of quantum measurement using polarized sunglasses as an analogy. It explains how polarizing filters act as quantum measurement devices for photons, determining their polarization state. The phenomenon of light intensity changing with the orientation of the filters is discussed, highlighting the quantum behavior where additional filters can paradoxically allow more light to pass through. This leads to a discussion on Bell's theorem and the quantum states of photons, suggesting that the traditional notions of cause and effect and experimental results may need reevaluation.

05:03

🔬 Probing Hidden Variables with Polarization

This paragraph delves into the concept of hidden variables in quantum mechanics, proposing that there might be underlying properties determining the behavior of particles like photons. It challenges the idea through the use of polarized filters and a thought experiment involving three filters, which shows that the probabilities of photons passing through do not align with the presence of a hidden variable. The script uses a Venn diagram to illustrate the contradiction, suggesting that the act of measurement may affect the outcome in a way that cannot be explained by local hidden variables.

10:03

🌌 Entanglement and Nonlocality in Quantum Mechanics

The final paragraph explores the implications of quantum entanglement on the principles of realism and locality. It describes an experiment involving entangled photons and polarizing filters at different locations, which, despite the distance, show correlated behaviors. The discussion centers on Bell inequalities and how they are violated by quantum mechanics, indicating that particles cannot have predetermined properties independent of measurement. The script concludes with the acknowledgment of the difficulty in conducting loophole-free tests of Bell's theorem and the profound implications of these findings for our understanding of the universe.

Keywords

💡Polarized sunglasses

Polarized sunglasses are a type of eyewear that contain polarizing filters to reduce glare and enhance visual clarity. In the context of the video, they serve as an accessible tool to demonstrate the principles of quantum mechanics, specifically quantum measurement and the polarization of light. The script uses polarized sunglasses to introduce the concept of light polarization and how it can be measured, leading to a deeper discussion about quantum states and measurements.

💡Polarizing filter

A polarizing filter is a device that allows light waves vibrating in certain directions to pass through while blocking others. It is integral to the video's demonstration of quantum measurement, where the filter's orientation determines whether a photon's polarization aligns with the filter's pass-through axis. The script explains how rotating a second polarizing filter in relation to the first affects the amount of light passing through, illustrating quantum behavior.

💡Photon

A photon is a quantum of light and other electromagnetic radiation, exhibiting both particle-like and wave-like properties. The video script uses photons to discuss the polarization phenomenon and to introduce the concept of quantum measurement. The behavior of photons passing through polarizing filters is central to the explanation of quantum mechanics and the video's exploration of Bell's theorem.

💡Quantum measurement

Quantum measurement refers to the process of observing a quantum system, which can affect the system's state. In the video, the act of a photon passing through a polarizing filter is described as a quantum measurement, determining the photon's polarization state. The script explains how this measurement can have probabilistic outcomes, challenging classical deterministic views of physics.

💡Hidden variable

The concept of a hidden variable suggests that there may be underlying properties of a quantum system that are not directly observable but determine the outcomes of quantum measurements. The script discusses this idea in the context of trying to explain the probabilistic nature of quantum mechanics, suggesting that there might be unknown variables that dictate whether a photon passes through a filter.

💡Bell's theorem

Bell's theorem is a result in quantum mechanics that shows certain correlations between distant particles cannot be explained by hidden variables that adhere to classical notions of locality and realism. The video script delves into Bell's theorem to highlight the apparent paradoxes in quantum mechanics, particularly the contradiction between observed quantum behavior and the idea that particles have definite properties prior to measurement.

💡Entanglement

Entanglement is a quantum phenomenon where pairs or groups of particles interact in such a way that the state of each particle cannot be described independently of the others, even when separated by large distances. The script uses entanglement to illustrate how measurements on one particle can instantaneously affect the state of another, leading to discussions about the violation of local realism.

💡Locality

Locality is the principle that physical processes occurring at one location do not depend on the properties of objects at other locations. The video script challenges this principle through the discussion of entanglement and Bell's theorem, suggesting that the behavior of entangled particles implies non-local interactions that could potentially occur faster than light.

💡Realism

In the context of the video, realism refers to the belief that objects have definite properties, independent of whether they are measured or observed. The script explores how quantum mechanics, particularly through Bell's theorem, challenges this notion by showing that particles do not seem to have pre-determined properties prior to measurement.

💡Bell inequality

A Bell inequality is a mathematical inequality that must be satisfied by certain correlations in a system that obeys local realism. The video script explains how quantum mechanics, and specifically entangled states, are predicted to violate these inequalities, which has been confirmed by experiments, thus providing evidence against local realism.

💡Loophole

In the context of Bell's theorem experiments, a loophole refers to potential issues or shortcomings in the experimental setup that might allow for alternative explanations to the observed phenomena, thus not fully closing the case against local realism. The script mentions that the first loophole-free Bell test occurred in 2015, indicating a significant advancement in the confirmation of quantum mechanics' predictions.

Highlights

Polarized sunglasses act as quantum measurement devices, demonstrating the polarization of light.

Polarizing filters work by either allowing light to pass through or blocking it, effectively measuring photon polarization.

Experiment with multiple sets of polarized sunglasses to observe light intensity changes, indicating quantum behavior.

Adding a third polarizing filter at 45 degrees can paradoxically increase the amount of light passing through.

The increase in light through additional filters challenges our understanding of cause and effect and the speed of light.

Bell’s theorem is introduced as a key concept challenging basic assumptions about the universe's operation.

Quantum states, such as photon polarization, are represented using mathematical models.

Polarizing filters' effect on light waves is probabilistic, akin to Schrodinger’s Cat thought experiment.

The concept of 'hidden variables' is proposed to explain the probabilistic outcomes in quantum mechanics.

Experiments with polarized light and sunglasses can test the existence of hidden variables.

Probabilities of photon passage through filters at various angles reveal unexpected results.

Inserting a filter at 22.5 degrees between two others results in a paradoxical increase in light transmission.

The numerical outcomes of these experiments suggest the impossibility of hidden variables with definite answers.

Venn diagrams are used to illustrate the contradiction between experimental results and hidden variable theories.

Entanglement is key to a stronger test of hidden variables, requiring non-local interactions between particles.

Entangled photons exhibit correlated behavior regardless of the distance between them, challenging locality.

Bell inequalities provide a mathematical framework to test the validity of local realism in quantum mechanics.

Experiments have shown violations of Bell inequalities, suggesting the non-viability of local realism.

The simplicity of the Bell test using inexpensive materials underscores the profound implications of quantum mechanics.

Transcripts

Henry:  If 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:  Try 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 – more light!

Henry:  This 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:  Where 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:  The 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:  A, 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.  That 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.  And 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

are made.

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

their answers to these questions.

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:  Well...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:  We 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:  So 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:  But 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.  Likewise,

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.  And 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.  Likewise, 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.  Why?

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:  Again, 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