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  • 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 axisin principle 0%.

  • Grant: Here's where things get quantum-ly bizarre.

  • All these filters do is remove lighttheyfilterit 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 lightsomehow 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 workslike, 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 somehidden variabledescribing 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 betweenand 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 questionsWould it pass through

  • A”, “Would it pass through B” andWould 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 questionsWould it pass through A”, “Would it

  • pass through B” andWould 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 calledrealism”.

  • And the assumption that faster than light influence is not possible is calledlocality”.

  • 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 certainloopholes

  • 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

Henry:  If you have polarized sunglasses, you have a quantum measurement device.

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Bell's Theorem: The Quantum Venn Diagram Paradox

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    Summer posted on 2021/08/18
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