How to Teach Physics to Your Dog (19 page)

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.

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
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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 . . .”

Bohr’s initial response to the EPR argument was rushed and nearly incomprehensible.
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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.
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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
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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.

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.”