How to Teach Physics to Your Dog (20 page)

“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 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.

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
non
local 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.

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.