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CHAPTER 7
Spooky Barking at a Distance:
Quantum Entanglement
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Emmy is napping in the living room, but wakes up as I pass through. She stretches hugely, then follows me into the kitchen looking pleased with herself. “I’m going to measure a bunny,” she announces.
“Beg your pardon?” She’s always making these weird announcements.
“I’ve figured out how to measure both the position and the momentum of a bunny.”
“You have, have you? How are you going to do that?”
“I’m going to put a big grid of lines in the back garden, and then when the bunny is right on top of a grid mark, all I have to do is measure how fast it’s going.” She wags her tail proudly.
“Right … And how are you going to measure when the bunny is right on a grid mark?”
“What do you mean? I’m just going to look.”
“Okay, which means you’ll see the bunny, and the bunny will see you, and then it will change its velocity to run away.”
“Oh.” Her tail droops. “I didn’t think of that.”
“Look, we’ve been through this. There’s no way around the uncertainty principle. Really clever humans have tried to find a way around it, and it can’t be done. Einstein spent years arguing about it with Niels Bohr.”
“Did he come up with anything?”
“He tried lots of different arguments, but none of them actually worked. He even had a really clever argument that quantum mechanics was incomplete, involving two entangled particles, prepared so that their states are correlated.”
“Correlated how?”
“Well, let’s say I have two treats in my hand—stop drooling, it’s a thought experiment—and one of them is steak, and the other is chicken.”
“I like steak. I like chicken.” She’s drooling all over the floor.
“Yes, I know. Thought experiment, remember?” I grab some paper towels to mop up the floor. “Now, imagine I throw these two treats in opposite directions, one to you, and one to some other dog.”
“Don’t do that. Other dogs don’t deserve treats.”
“It’s a hypothetical, try to keep up. Now, if you got the steak treat, you would know immediately that the other dog got the chicken treat. And—why are you looking all sad?”
“I like hypothetical chicken treats.”
“You got a hypothetical steak treat.”
“Oooh! I like hypothetical steak.”
“The point is, by measuring what sort of treat you got, you know what the other treat is, without ever measuring it.”
“Yes, so? What’s weird about that?”
“Well, in the quantum version, the state of the particles is indeterminate until one of them is measured. When I throw the treats, until you get one and find out whether it’s steak or chicken, it’s not either. In some sense, it’s both.”
“Chickensteak! Steakchicken! Sticken!”
“You’re ridiculous. Anyway, Einstein thought this was a problem, and that the fact that you could predict the state of one particle by measuring the other particle meant that both of them had to have definite states the whole time.”
“That makes sense.”
“In a classical world, yes. Einstein’s argument fails, though, because he’s assumed what’s called ‘locality’—that measuring one particle does not affect the other. In fact, measuring the state of one determines the state of the other, absolutely and instantaneously.”
She looks really bothered by this. “I don’t like that idea. Wouldn’t that require a message to travel from one treat to the other?”
“That’s what bothered Einstein, and he called it spukhafte Fernwirkung.”
“ ‘Spooky action at a distance’?” she translates.
“Since when do you know German?”
“Just look at me.” She turns sideways for a second, showing off her black and tan coloring and pointed nose. “German shepherd, remember?”
“Of course, how silly of me. Anyway, yes, this bothered Einstein because information cannot pass between separated objects faster than the speed of light. But quantum mechanics is nonlocal, and the entangled particles act like a single object. A man named John Bell showed that it’s possible to put limits on what you can measure in theories where the particles have definite states, and showed that those limits are different than the limits for entangled quantum particles. People have done the experiments and found that the quantum theory is right. The state of the particles really is indeterminate until they’re measured.”
“So Einstein was wrong?”
“About this, yes. And generally, about the basis of quantum theory.”
“But he was really clever, wasn’t he?”
“Yes. Einstein was arguably cleverer than Bohr. Bohr won all their debates, though, because he had the advantage of being right.” I bend over to scratch behind her ears. “You’re pretty clever, but you’re no Einstein.”
“I’m, like, the canine Einstein, though, right?”
“All right then. As far as I know, you’re the Einstein of the dog world.”
“Can I have some steak, then? Or chicken?”
“Maybe.” I grab a treat out of the jar on the counter. “You’ll find out when you measure it.” I throw the treat out of the back door, and she goes bounding after it.
“Great! Indeterminate treats!”
Everything we have talked about so far has been a one-particle phenomenon. Most of the experiments need to be repeated many times to see the effects, using different individual particles prepared the same way, but at a fundamental level, all the interference, diffraction, and measurement effects we’ve talked about work with one particle at a time. Each particle in an interference experiment can be thought of as interfering with itself, and measurement phenomena like the quantum Zeno effect involve the state of a single particle.*
Of course, the world we live in involves a great many particles, so we need to look at what happens when we apply quantum physics to systems involving more than one particle. When we do, it’s no surprise that we find some weird things going on, starting with the idea of “entangled states.”
In this chapter, we’ll look at the idea of “entangled” particles, whose states are correlated so that measuring one particle determines the exact state of the other. Entangled particles are the basis for the most serious challenge Einstein mounted against quantum theory, known as the Einstein, Podolsky, and Rosen (EPR) paradox. We’ll talk about John Bell’s famous theorem resolving the EPR paradox, and its disturbing implications for the commonsense view of reality. Finally, we’ll talk about the experiments that prove Bell’s theorem, and show the lengths that physicists go to in challenging new ideas.
SLEEPING DOGS LET EACH OTHER LIE: ENTANGLEMENT AND CORRELATIONS
Entanglement is fundamentally about correlations between the states of two objects. To illustrate the idea, let’s think about two dogs—we’ll use my parents’ Labrador retriever, RD, and my in-laws’ Boston terrier, Truman—who can each be in one of two states: “awake” or “asleep.” If the dogs are completely separate from each other, there are four states we could find our two-dog system in: we can find both dogs awake, both dogs asleep, Truman awake while RD is asleep, or Truman asleep while RD is awake.
If we bring the two dogs together and allow them to interact, though, a correlation develops between the state of the two dogs. If Truman is asleep while RD is awake, RD will wake Truman up to play, and vice versa. You will either find both dogs awake or both dogs asleep, but never one awake and the other asleep. We go from four possible states to only two.
Moreover, this correlation allows us to know the state of one of the dogs without measuring it. If Truman is awake, we know that RD must be awake, and if Truman is asleep, we know that RD must be asleep. We can look at RD if we want, but we’ll just confirm what we already know. Measuring the state of one of the two dogs immediately and absolutely tells us the state of the other dog.
IS QUANTUM MECHANICS INCOMPLETE? THE EPR ARGUMENT
What does this have to do with Einstein? Einstein was a strong believer in a deterministic universe, in which we can always trace a clear path from cause to effect. He had major philosophical problems with quantum mechanics. In particular, he was bothered by the idea that properties of quantum particles are undefined until they are measured, and then take on random values.
From the late 1920s through the mid-1930s, Einstein had a series of arguments with Niels Bohr, who was also philosophically inclined* but was a champion of the quantum theory. Einstein first attacked the idea of uncertainty with a number of different ingenious thought experiments that would perform measurements forbidden by the uncertainty principle—measuring both the position and momentum of an electron, for example. Every time he did, Bohr found a semiclassical counterargument showing that Einstein’s proposed experiment had some hidden flaw.†
In the early 1930s, Einstein reconciled himself to uncertainty, but he remained troubled by quantum theory, and found a new problem to attack. He argued that the existing quantum theory did not contain all the information needed to describe a particle’s properties. In a 1935 paper titled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Einstein and colleagues Boris Podolsky and Nathan Rosen presented an ingenious argument for this claim, using the idea of an entangled state. They proposed an experiment to demonstrate this supposed incompleteness by entangling the states of two particles, and then separating them so that they no longer interact (but their states do not change). You can then measure the two in separate experiments that have no possible influence on each other, and see what happens.
In the EPR scheme, measuring the position of one of the two particles (Particle A) allows you to predict the position of the other (Particle B) with absolute certainty. At the same time, if you measured the momentum of Particle B, you would know with certainty the momentum of Particle A. According to Einstein, Podolsky, and Rosen, since there’s no way for measurements of Particle A to affect the outcome of measurements of Particle B, or vice versa, both the position and the momentum of each particle must have definite values the whole time. This suggests that quantum mechanics is incomplete: the information needed to describe the precise state of the particles exists, but is not captured by quantum theory.
“That’s just what I was saying!”
“What was?”
“A bunny does have a definite position and momentum. All that uncertainty business was just you being all confusing and stuff.”
“It sounds like a convincing argument, but if you remember, I also said it was wrong. It’s brilliantly wrong, but there’s still a flaw in one of their assumptions, namely the idea that it’s impossible for a measurement of one particle to affect the outcome of the state of the other particle.”
“Oh, really? Prove it.”
“I’ll get there. Just give me a minute …”
“DON’T KNOW” VS. “CAN’T KNOW”: LOCAL HIDDEN VARIABLES
Bohr’s initial response to the EPR argument was rushed and nearly incomprehensible.* He refined this later, but he was never able to come up with a convincing semiclassical counterargument, in the way that he had in all his other debates with Einstein. The reason is simple: there is no such argument. Quantum mechanics is a “nonlocal” theory, meaning that measurements separated by a large distance can affect one another in ways that wouldn’t be allowed by classical physics.
The sort of theory preferred by Einstein, Podolsky, and Rosen is called a local hidden variable (LHV) theory, after the underlying assumptions that make up the model. “Hidden variable” means that all quantities that might be measured have definite values, but those values are not known to the people doing the experiment. “Local” means that measurements and interactions at one point in space can only instantaneously affect things in the immediate neighborhood of that point. Long-distance interactions are possible, but those interactions must take some time to be communicated from one place to another, at a speed less than or equal to the speed of light.*
Locality is so central to classical physics that it may seem too obvious to challenge. Locality says that some time must pass between causes and effects. When a human calls to a dog out in the garden, the dog won’t come running until enough time has passed for the sound of the call to travel from the human to the dog.† Nothing the human does can have any influence on the dog’s actions before that time.
Locality is what makes the EPR argument a paradox. Nothing in the proposed experiment limits the time between the two measurements. You can keep Particle A at Princeton, and send Particle B to Copenhagen, and agree to measure the position of A and the momentum of B at, say, one nanosecond past noon, Eastern Standard Time. There is no possible way for any message to travel from Princeton to Copenhagen in time to influence the outcome of the second measurement. Hence, assuming locality is true, the two measurements are completely independent of each other, and each must reflect some underlying reality.
As obvious as the assumption of locality seems, this is exactly the point where the argument fails. Quantum mechanics is a nonlocal theory, and a measurement made on one of two entangled objects will affect measurements made on the other instantaneously, no matter how far apart the two are. A measurement in Princeton can determine the result of a measurement in Copenhagen, provided the objects being measured are entangled.
Because quantum mechanics is nonlocal, the state of two entangled particles remains indeterminate until one of the two is measured. Not only do you not know the state of the particles, you can’t know it. In terms of our dog example (page 143), until somebody measures the state of one of the two dogs, both dogs are simultaneously asleep and awake—the wavefunction for the system has a part corresponding to “Truman asleep and RD asleep” and a part corresponding to “Truman awake and RD awake,” but neither dog is definitely asleep or awake. The dogs exist in a superposition, like a friendlier version of Schrödinger’s cat.
The state of a given dog takes on a definite value only when it is measured, and when that happens, the state of the other dog is simultaneously determined. The instant that you measure one, you determine the state of both, no matter where they are. If Truman is awake, so is RD, and if Truman is asleep, so is RD. If you take them into different rooms before measuring their states, you’ll still find them correlated, despite the fact that measuring Truman’s condition does not directly affect RD, and no information passes between them. The two separated dogs are a single quantum system, and a measurement of any part of that system affects the whole.
Nonlocality prevents the EPR experiment from being able to circumvent the uncertainty principle. A measurement of Particle A does perturb the state of Particle B, exactly as if the measurement had been made on Particle B. This holds true no matter how carefully the two are separated before the measurement—the entangled particles are a single, nonlocal quantum system.
Nonlocality presents a philosophical challenge to the basis of classical science as profound and disturbing as the issues of probability and measurement discussed in chapters 3 and 4. The instantaneous projection of the entangled objects onto definite states* is a conclusion that we’re forced to by quantum theory, and there’s nothing like it in classical mechanics.
With the EPR paper, quantum physics reached a philosophical impasse. Supporters of Bohr’s orthodox quantum theory were unconvinced by the EPR argument, but could not present a compelling counterargument. Meanwhile, people like Einstein who were bothered by the implications of quantum theory pointed to the EPR argument as suggesting some deeper theory that would make sense of this weird and unpleasant quantum business. More people took Bohr’s side than Einstein’s, because quantum theory provided such accurate predictions of atomic properties, but neither side could think of a definitive experiment.
SETTLING THE DEBATE: BELL’S THEOREM
This impasse lasted for almost thirty years, until the Irish physicist John Bell came up with a way to distinguish between the predictions of quantum theory and those of the local hidden variable models preferred by Einstein. Bell realized that LHV theories have definite particle states and only local interactions, and are thus limited in ways that quantum theory is not. He proved a mathematical theorem stating that entangled quantum particles have their states correlated in ways that no possible local hidden variable theory can match. These correlations can be measured experimentally; a measurement showing correlations beyond the LHV limits would conclusively prove that Bohr was right, and Einstein was wrong.
Bell’s theorem is critical to the modern understanding of quantum mechanics, so it’s worth exploring in some detail. It can’t be demonstrated with dogs, but it’s not too hard to do using the polarized photons we talked about in chapter 3 (page 65). To be concrete, let’s think about two photons whose polarizations are the same—if one is measured to be horizontal, the other is also horizontal; if one is measured at a 45º angle, the other is at the same 45º angle. Then we look at three different possible measurements.
The traditional arrangement calls for two experimenters—they’re usually called “Alice” and “Bob,” but we’ll stick with “Truman” and “RD,” because they’re good dogs—to each receive one of the two photons. Truman and RD are each given a polarizer and a photon detector, which combine to make detectors that register either a “1” or a “0,” depending on whether the photon makes it through the polarizer or not. For example, if the polarizer is set to vertical, a vertically polarized photon will be transmitted and give a “1,” while a horizontally polarized photon will be blocked, and give a “0.” If the polarizer is set at 45° to the vertical, a vertically polarized photon has a 50% chance of making it through and being recorded as a “1,” otherwise it will be blocked and recorded as a “0.”
The experiment is simple: each dog sets his polarizer at one of three angles, a, b, or c. He then records the detector reading (“0” or “1”) for one photon. Then they change the detector settings, and do it again. After repeating this over and over, they will have tried all the possible combinations of detector settings many times, and then they compare their results.
Schematic of a measurement to test Bell’s theorem. Truman and RD each take one photon from an entangled photon source, and measure its polarization along one of three angles using a polarizing filter and a photon detector. They can distinguish between quantum mechanics and a local hidden variable theory by measuring how often they detect the same thing when their polarizers are at different angles.
When they compare results, they’ll notice two things. When their polarizers are set to the same angle, they’ll see that both get the same answer (either “1” or “0”), every time. They’ll also see that no matter what angle they choose, they get equal numbers of “0” and “1” results—if they repeat the experiment 1,000 times at a given angle, they will get 500 “0”s and 500 “1”s. These two observations are true whether they’re dealing with a quantum entangled state, or a state governed by an LHV theory.
“Wait, shouldn’t it depend on the angles?”
“What angles?”
“Your a, b, and c angles. Why do they get equal numbers of ‘0’s and ‘1’s? Shouldn’t the measurement results depend on which angle they choose? Like, if they have their polarizers set vertically, they always detect a ‘1’?”
“No, the states we’re dealing with are states of indeterminate polarization. In the quantum picture, the polarization is undefined, while in the LHV picture, it’s equally likely to be either horizontal or vertical.”
“Doesn’t that mean they’re at 45º? Then shouldn’t they get ‘1’ every time when they put the polarizers at 45º?”
“No, they get the same result at 45º. The photons are equally likely to be 45º counterclockwise from vertical, or 45º clockwise, or any other angle. It really doesn’t matter what angles they choose for a, b, and c—even ‘vertical’ and ‘horizontal’ are kind of arbitrary.”
“No they’re not.”
“Yes they are. When I say something weird, and you look at me sideways—like you’re doing right now—that changes what ‘vertical’ looks like, right?”
“I guess. Everything looks different from an angle, and sometimes weird human stuff makes more sense.”
“It’s the same thing here. The angles they set for the polarizers determine what ‘0’ and ‘1’ will mean, in the same way that tilting your head changes your perception of ‘horizontal’ and ‘vertical.’ They still have an equal chance of getting either result. What you see depends on what you’re looking for. To go back to the treat analogy from page 140, it’s like a treat where if you’re looking for ‘meat,’ you get either steak or chicken, but if you’re looking for ‘not meat,’ you get either peanut butter or cheese.”
“Those treats sound good. You should buy me some of those.”
“I don’t think they have them at the pet shop, but I’ll look.”
To test Bell’s theorem, we ask how often they get the same answer with their detectors at different settings. That is, how many times did Truman record a “0” with the detector in position “a,” while RD got a “0” in position “c,” or Truman a “1” in position “b” and RD a “1” in position “a,” and so on. The probability of both dogs getting the same result with different detector settings is very different for LHV theories and quantum mechanics.
THE EPR OPTION: LOCAL HIDDEN VARIABLE PREDICTION
The key to Bell’s theorem is that all the predictions of a local hidden variable theory can be written down in advance, so let’s do that. Each photon has a well-defined state, and we can represent that state by a set of three numbers, each giving the definite outcome of a measurement in polarizer position a, b, or c. The two-photon system offers a total of eight possible states, which we can represent in a table:
To test Bell’s theorem, we need the probability of both dogs getting the same answer with different detector settings. Looking at the table, we see that no matter what angles we pick, four of the eight possible states give the same answer. For example, if Truman sets his detector to position a, and RD sets his to b, states 1 and 2 will give them each a “1” and states 7 and 8 will give them each a “0.” If Truman chooses c and RD chooses a, the four states giving the same answer are 1, 3, 6, and 8, and so on.
We’re not stuck with exactly 50% probability of getting the same answer for different settings, though. We’re free to adjust the probability of the photons being in a particular state—say, making state 1 more likely, and state 6 less likely—though any change we make has to end up with equal probabilities of finding “0” or “1” for each detector setting.
If we play around with the probabilities of the individual states, we find that we can cover a limited range of possible probabilities. We can make the maximum probability of both dogs getting the same result 100%, but the minimum probability is 33%, not 0%. No matter what we do, we can never make the probability lower than 33%.*
Notice that we haven’t said anything about what causes those states, or how they are chosen. We don’t need to the mere fact that we can write down the limited number of possible results places restrictions on the experiment. No model in which the two photons have well-defined states when they leave the source can give a probability of less than 33% for the two measurements to give the same outcome. The probability must be less than or equal to 100%, and greater than or equal to 33%.† Similar limits hold true for any LHV theory you can dream up.
THE BOHR OPTION: QUANTUM MECHANICAL PREDICTION
To prove quantum mechanics correct, then, we need to find some detector angles for which the probability of both dogs getting the same answer with different settings is less than 33%. Bell showed that this can be done, thanks to entanglement: measuring the polarization of one of the two photons instantaneously determines the polarization of the other.
In the quantum picture, the state of the two photons is indeterminate until the instant when one of the two is measured, when it has a 50% chance of ending up as a 0 or 1. At that instant, the polarization of the second photon is set to the same angle as the first, whatever that is. If the first photon passed through a vertical polarizer, recording a “1,” the second photon is now vertically polarized. If the first photon was blocked by the vertical polarizer, recording a “0,” the second photon is now horizontally polarized. The possible outcomes of the second measurement are then determined by the first polarizer angle.
To prove Bell’s theorem, let’s imagine Truman sets his detector to vertical polarization (which we’ll call “a”). RD sets his detector to either 60° clockwise from vertical (“b”), or 60° counterclockwise from vertical (“c”). What are the possible ways to get the same answer for both dogs when they have different polarizer settings?
Well, half of the time, Truman will detect a “1” with his detector, which means that we want the probability of RD also getting a “1.”* Since Truman’s polarizer is vertical, the entangled photon hitting RD’s detector is also vertically polarized. If his detector is set to position “b,” then the angle between the vertical photon and RD’s polarizer is 60°, and the probability of the photon passing through the polarizer is 25%. The same holds for position “c,” which is 60° from “a” in the other direction.
The other half of the time, Truman measures a “0,” and both entangled photons are horizontal. RD’s photon again has a 25% chance of being blocked and giving a “0,”† for either angle.
No matter what value Truman measures, then, quantum theory tells us that there is only a 25% chance that RD will get the same value with his detector at a different polarizer setting. This directly contradicts the prediction of the local hidden variable theory, which gave a minimum chance of 33%. Only one in four of RD’s measurements is the same as Truman’s, where LHV says that at least one in three should be the same.
You might think that the two theories should give the same results, because they’re describing the same system, in the same way that the different interpretations of quantum mechanics all give the same predictions. That’s what most physicists thought, until Bell showed otherwise. The core assumptions of the local hidden variable theories mean that they are subject to strict limits—you can write down a table like the one above showing all possible results. Quantum theories do not have the same limitations, so a clever experiment can distinguish between them.*
The results are different because quantum mechanics is nonlocal—the polarization of RD’s photon is not set in advance, but is determined by the outcome of Truman’s measurement. The probability of getting the same result with different settings is lower because the two measurements affect each other, no matter how far apart they are, or when they’re made. Einstein called this “spooky,” and it’s hard to argue with him.
“Can’t you just make a better theory?”
“What kind of better theory?”
“A better hidden variable theory. That matches the predictions better.”
“That’s the whole point. Bell didn’t look at a particular theory—what he showed is that there’s no possible local hidden variable theory that can reproduce all the predictions of quantum mechanics. If the two measurements are independent of each other, there’s no way to arrange things so that the measurements show the same correlation that you see with quantum mechanics.”
“So make the measurements depend on each other.”
“That works, but that isn’t a local hidden variable theory anymore. In fact, David Bohm worked out a version of quantum mechanics that uses nonlocal hidden variables, and reproduces all the predictions of quantum theory using particles with definite positions and velocities.”
“That sounds nice. Why don’t people use that?”
“Well, Bohm’s theory introduces an extra ‘quantum potential,’ a function that extends through the entire universe and changes instantaneously when you change some property of the experiment. It’s a really weird object, and it’s a headache when doing calculations. It’s also easier to extend regular quantum mechanics to be compatible with relativity, in what’s known as quantum field theory.”
“It’s not wrong, though?”
“No, it predicts the same things as regular quantum theory. You can look at it as an extreme version of a quantum inter pretation, like the Copenhagen interpretation or many-worlds pictures that we talked about earlier. It adds a little more maths to the theory, but doesn’t predict anything different in practical terms.”
“Hmmm.”
“The important thing for this discussion is that Bohm’s theory is nonlocal, which is what the EPR paradox and Bell’s theorem are really about. From those, we know that quantum theory can’t be a strictly local theory, where measurements in two different places have no effect on each other.”
“That still annoys me. How do we know that that’s really true?”
“I’m glad you asked that …”
This example is a specific demonstration of Bell’s theorem, but it captures the flavor of the general theorem. What Bell showed is that there are limits on what can be achieved with LHV theories in general, and that under certain conditions, quantum mechanics will exceed those limits. A clever experiment can determine once and for all whether quantum mechanics is right, or whether it could be replaced by a local hidden variable theory as Einstein hoped.
LABORATORY TESTS AND LOOPHOLES: THE ASPECT EXPERIMENTS
Bell published his famous theorem in 1964. In 1981 and 1982, the French physicist Alain Aspect and colleagues tested Bell’s prediction with a series of three experiments that are generally considered to conclusively rule out local hidden variable theories.* They needed all three experiments to close a series of “loopholes,” gaps in their results that some local hidden variable models might slip through.
We’ll describe all three experiments here, because they’re outstanding examples of the art of experimental physics. More than that, though, they demonstrate the lengths you need to go to if you want to convince physicists of something. You need to answer not only the obvious objections, but also objections that are improbable enough to seem a little ridiculous.
The first experiment, published in 1981, was essentially the same as our thought experiment with Truman and RD. Aspect’s group made calcium atoms emit two photons within a few nanoseconds of each other, heading in opposite directions. These photons are guaranteed to have the same polarization—it’s equally likely to be either horizontal or vertical (or any other pair of angles), but if one photon is horizontal, the other must also be horizontal. This is exactly the entangled state you need in order to test Bell’s theorem.
In the first experiment, they placed two detectors on opposite sides of their entangled photon source, with a polarizer in front of each detector. The polarizers were set to various different angles, and they measured the number of times they counted a photon at both detectors—that is, both detectors reading “1,” in terms of our example above.
Physicists like to deal with numbers, and for the specific configuration they used, a local hidden variable treatment predicts that their results should boil down to a number between -1 and 0. When they did the experiment, they measured a value of 0.126, with an uncertainty of plus or minus 0.014.* The difference between the maximum LHV value and their measurement is nine times larger than the uncertainty in the measurement, meaning that there’s a one in 1036 probability of this happening by chance.†
The first Aspect experiment. An excited calcium atom emits two photons with entangled polarizations. Each photon heads toward a single detector with a polarizing filter in front of it, set to an appropriate angle.
So, that’s the end of LHV theories, right? It looks just like our imaginary experiment above, and that’s an astonishingly small probability of this happening by accident. Why did they need to do a second experiment, let alone a third?
Unfortunately, there’s a loophole in their result that allows some LHV theories to survive. In our thought experiment, we imagined Truman and RD with photon detectors that were absolutely perfect, because they’re very good dogs. Aspect and his coworkers are only human, though, and so were stuck using detectors with limited efficiency. On rare occasions, a detector would fail to record a photon that was really there.
This is a problem, because their experiment recorded a “0” when they expected a photon and didn’t see one—they assumed that those photons were blocked by the polarizers. But because their detectors sometimes failed to detect photons, it’s conceivable that the first Aspect experiment just looked like it violated the LHV prediction. If some of their “0”s really should’ve been “1”s, that could confuse their results.
For LHV theories to slip through this loophole the universe would need to be somewhat perverse, but it’s possible, so they did a second experiment, published in 1982, using two detectors for each photon.
They closed the detector efficiency loophole by directly detecting both possible polarizations, and only counting experiments where they detected one photon on each side of the apparatus. They replaced the polarizers with polarizing beam splitters that directed each polarization to its own detector. If one of the detectors failed to record a photon, that run of the experiment was discarded.
Their measured value in the second experiment exceeded the LHV limit by an astonishing 40 times the uncertainty, and the odds of that happening by chance are so small it’s ridiculous. So, why did they do the third experiment? As impressive as the second experiment was, it still left a loophole, because something could have passed messages between their detectors and their source.
The second Aspect experiment. The entangled photons leave the source and head toward a pair of detectors with a polarizing beam splitter in front of them. These beam splitters direct the “0” polarization to one detector and the “1” polarization to another, ensuring that no photons are missed in the experiment.
To test Bell’s theorem, it needs to be impossible for the measurement at one detector to depend on what happens at the other detector without some faster-than-light interaction. If there’s a way to send messages between the detectors at speeds less than that of light, all bets are off. In the first two experiments, they chose the detector settings in advance, and left them set for much longer than it took light to pass between the source and the detector. Something might have communicated the polarizer settings from the detectors to the source, which then sent out photons with definite polarization values chosen to match the quantum predictions. When the experimenters changed the angles, the new values would be sent to the source, which would change the polarizations sent out. Their results seemed to prove quantum theory, but they might have been the victims of a cosmic conspiracy.
The third experiment found an ingenious way to close that loophole, as well. Aspect and his colleagues ruled out any possibility of some sort of universal conspiracy mimicking the quantum results by changing their detector settings faster than light could go from the source to the detector.
They replaced the beam splitters with fast optical switches that could direct the photons to one of two detectors, each set for a different polarization. The switches flipped between the detectors every 10 nanoseconds, while it took the photons 40 ns to reach the detector. In effect, which detector a given photon would hit was not decided until after the photon had already left the source.
The third experiment’s results exceeded the LHV limit by five times the uncertainty. The chances of such a result happening by accident were about one in a hundred billion—better than the chances for the other two experiments, but still low enough to be convincing.
The third Aspect experiment. The two entangled photons leave the source, and head toward fast optical switches that send each photon toward one of two different polarizers, with the choice not being made until after the photons have left the source.
Even the third experiment doesn’t close every loophole,* but Aspect stopped there, because the experiments were extraordinarily difficult. A number of people have repeated these experiments, using more modern sources of entangled photons,† and a 2008 experiment has even tested Bell’s theorem using entangled states of ions instead of photons, but no loophole-free test has been done. As a result, there are still a few people who argue that LHV theories have never been completely ruled out.
These few die-hard theorists aside, the vast majority of physicists agree that the Bell’s theorem experiments done by Aspect and company have conclusively shown that quantum mechanics is nonlocal. Our universe cannot be described by any theory in which particles have definite properties at all times, and in which measurements made in one place are not affected by measurements in other places.
Aspect’s experiments represent a resounding defeat for the view of the world favored by Einstein and presented in the Einstein, Podolsky, and Rosen paper in 1935. But while the EPR paper is wrong, it’s brilliantly wrong, forcing physicists to grapple with the philosophical implications of nonlocality. Exploring the ideas raised in the paper has deepened our understanding of the bizarre nature of our quantum universe. The idea of quantum entanglement exploited in the EPR paper also turns out to allow us to do some amazing things using the nonlocal nature of quantum reality.
“Physicists are really weird.”
“Yes, nonlocality is strange.”
“Not that, the loopholes. Do physicists really believe that there are messages being passed back and forth between different bits of their apparatus? What would carry the messages?”
“I’m not sure anybody ever suggested a plausible mechanism, but it really doesn’t matter. They could be carried by invisible quantum bunnies, for all the difference it makes.”
“Quantum bunnies?”
“Invisible quantum bunnies. Moving at the speed of light. Don’t get your hopes up.”
“Awww …”
“Anyway, the third Aspect experiment pretty much rules out any means of carrying messages between parts of the apparatus, involving bunnies or anything else. The point is, prior to that, it was at least possible in principle for there to be another explanation. And in science, you have to rule out all possible explanations, even the ones that seem really unlikely, if you want to convince anybody of an extraordinary claim.”
“Even the ones involving bunnies?”
“Even the ones involving bunnies. And anyway, the idea that distant particles can be correlated in a nonlocal fashion isn’t all that much weirder than quantum bunnies would be.”
“Good point. So, what’s this good for?”
“What do you mean?”
“You dropped a really heavy hint in that last paragraph that this entanglement stuff is good for something. What’s it good for, sending messages faster than light?”
“No, you can’t use it for faster-than-light communication, because the detections are random. There are correlations between particles, but the polarization of each pair will be random. I can’t send a message to somebody else using EPR correlations—all I can send is a random string of numbers.”
“So what good is it?”
“Well, random strings of numbers can be useful for quantum cryptography, making unbreakable codes. And the idea of entanglement is central to quantum computing, which could solve problems no normal computer can tackle. And there’s quantum teleportation, using entanglement to move states from one place to another. There’s all sorts of stuff out there, if you look for it.”
“Teleportation sounds cool! Talk about that.”
“Well, that’s next …”
* The process of decoherence (described in chapter 4) involves the interaction of a single quantum particle with a much larger environment, but we care only about the state of the single particle.
* Werner Heisenberg, who developed the uncertainty principle while working with Bohr, once described Bohr as “primarily a philosopher, not a physicist.”
† In almost all of those cases, Bohr’s argument depended on the effect of measurement on the system. Something in the process by which Einstein proposed to measure the position would cause a change in the momentum (as in the case of the Heisenberg microscope thought experiment discussed in chapter 2 [page 38]), or vice versa. Measuring the system requires an interaction, and that interaction changes the state of the system in a way that introduces some uncertainty in the quantities being measured.
* Bohr was somewhat famous for the opacity of his writing, but he outdid himself in this case. The crucial paragraph of his paper refers to the quantum connection between distant objects as “an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system” (italics in original), and declares that the quantum view “may be characterized as a rational utilization of all possibilities of unambiguous interpretation of measurement, compatible with the finite and uncontrollable interaction between the objects and the measuring instruments of quantum theory.”
* This light-speed limit is one of the main consequences of Einstein’s theory of relativity, and thus very important to his conception of physics.
† Or even longer, depending on what the dog is doing when called.
* If you prefer the Copenhagen view, this projection involves a real collapse of the wavefunction into a single state. If you prefer many-worlds, the apparent projection onto a single state comes because we perceive only a single branch of the wavefunction. In either case, the resulting correlation is the same, and the effect is instantaneous.
* We get the maximum value of 100% if the system has a 50% chance of being in state 1 and a 50% chance of being in state 8. We get the minimum value of 33% by never letting the system be in state 1 or state 8, and making the other six states equally likely. If you look at states 2 through 7, you’ll see that no matter what two different angles you choose, there are always two states that give you the same answer for both detectors.
† As a result the predictions of Bell’s theorem are often called “Bell inequalities.”
* We’re assuming that Truman’s photon is measured first, for the sake of clarity. The result is the same if we assume RD’s photon is the first one measured.
† The horizontal photon has a 75% chance of passing through the polarizer to the detector in either position. A “0” to match Truman’s result happens only when RD’s photon is blocked, a 25% probability.
* This raises the question of whether a sufficiently clever experiment might distinguish between, say, the Copenhagen interpretation and the many-worlds interpretation. This is a much harder problem than distinguishing between quantum and LHV theories. Some future John Bell may yet come along and find the right test, but no one has managed yet.
* John Clauser and a couple of other people had done earlier tests, but the Aspect (pronounced “As-PAY”) experiments had better precision, and so are regarded as the definitive tests.
* This uncertainty is a technical limitation based on the details of their experiment, and not anything to do with the Heisenberg uncertainty principle.
† 1036 is a billion billion billion billion, a number so large that it might have made even Carl “Billions and Billions” Sagan blink.
* The third experiment actually reopens the detector efficiency loop-hole, because they used only one detector for each polarizer.
† One experiment by Paul Kwiat (who was part of the Innsbruck–Los Alamos team doing quantum interrogation experiments in chapter 5) and colleagues at Los Alamos saw an effect a mind-boggling 100 times larger than the uncertainty.
