by CHAD ORZEL
“What do you mean?”
“Well, the Sun shines because of fusion reactions in the core, right?”
“Everybody knows that. Even the beagle down the street knows that, and that dog is really dumb.”
“Yes. Well. Anyway, fusion works by sticking protons together to make helium from hydrogen. Because protons are positively charged, they repel one another, and that repulsion sets up a barrier. And as hot as the Sun is, the protons in the Sun still don’t have enough energy to get over that barrier directly.”
“So they tunnel through?”
“Exactly. The probability of any given proton tunneling through the barrier is pretty low, but there are lots and lots of protons in the Sun, and enough of them do tunnel through to keep the reaction going. So it’s really tunneling that lets the Sun shine.”
“Hmm. I guess that is pretty cool.”
“I’m so glad you approve . . .”
“Can we go play fetch, now?”
* Temperature measures the energy due to the motion of the individual atoms making up an object, and “absolute zero” is the imaginary temperature at which that motion would cease. No real object can be cooled all the way to absolute zero, though, and even if one could it would still have zero-point energy, as discussed in chapter 2 (page 49).
* Potential energy is generally much easier to calculate than kinetic energy. Potential energy usually depends only on the positions of the interacting objects, while the kinetic energy depends on the velocity, which depends on what has happened in the recent past. The easiest way to tackle an energy problem is usually to calculate the potential energy using the position, and find the kinetic energy by process of elimination. For example, when a roller coaster pauses at the top of a big hill, we know that all of its energy is potential energy. Later on, we can easily calculate the potential energy from the height of the track, and that lets us find the kinetic energy (and thus the speed) without needing to know what happened in between.
* The total energy of an object can be increased, by adding energy from some other source, in the same way that a dog’s treat jar can be refilled by a friendly human. The extra energy does not come for free, though—the energy of the outside object has to decrease, in the same way that a human’s bank balance will decrease in order to supply the treats. The total energy of the entire universe—balls, dogs, treats, and humans—is a constant, and has not increased or decreased in the fourteen billion years since the Big Bang.
* Strictly speaking, the probability is never exactly zero—the mathematical function describing the probability is an exponential, and while it gets closer to zero as the electron moves into the barrier, it never gets all the way there. Quantum physics predicts a tiny probability that a ball thrown in the air will tunnel through the forbidden region that starts at its classical maximum height, and end up on the Moon. That’s not a good bet, though—the probability is so small that it’s indistinguishable from zero, for all practical purposes.
* Notice that the uncertainty in the position is very large—the electron could be just about anywhere to the left of the forbidden region.
* A potential energy barrier doesn’t have to be a solid physical object. An air gap will do just fine, which is why you can’t make a lightbulb light up by just holding it close to the socket.
CHAPTER 7
Spooky Barking at a Distance: Quantum Entanglement
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 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 backyard, 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. “Uncertainty, unschmertainty.”
“Uh-huh. 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.”
“Sure, 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 smart 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.”
“Yeah, 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, sure. 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?”
“Dude, 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 guy 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.”
“Yes. Einstein was arguably smarter 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 smart, but you’re no Einstein.”
“I’m, like, the canine Einstein, though, right?”
“Sure. 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 the back door, and she goes bounding after it.
“Oooh! 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 so 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, yeah? 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 yard, 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 o
f 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.