by CHAD ORZEL
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 45° angle, the other is at the same 45° 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.
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.
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.
“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 45°? Then shouldn’t they get ‘1’ every time when they put the polarizers at 45°?”
“No, they get the same result at 45°. The photons are equally likely to be 45° counterclockwise from vertical, or 45° 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.”
“Oooh! Those treats sound good. You should buy me some of those.”
“I don’t think they have them at the pet store, 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 c
hange 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 interpretation, like the Copenhagen interpretation or many-worlds pictures that we talked about earlier. It adds a little more math 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 bugs 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.
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.
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.†
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 abso lutely 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.