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Quantum Randomness

If there’s no predeterminism in quantum mechanics, can it output numbers that truly have no pattern?

Scott Aaronson

Enter the Quantum

At this point, the impatient reader might shout: If you want guaranteed randomness, why not just use quantum mechanics? After all, quantum mechanics famously says that you can’t predict with certainty whether, say, a radioactive atom will decay within a specified time period, even given complete knowledge of the laws of physics as well as the atom’s initial conditions. The best you can do is to calculate a probability.

But not so fast! There are two problems. Quantum-mechanical random number generators do exist and are sold commercially. But hardware calibration problems really can make the numbers predictable if they’re not fixed. Furthermore, even if you couldn’t predict the numbers better than chance, how could you be sure there wasn’t some subtle regularity that would let someone else predict them?

The second problem is philosophical, and goes back to the beginning of quantum mechanics. It’s well known that Einstein, in his later years, rejected quantum indeterminism, holding that “God does not play dice.” It’s not that Einstein thought quantum mechanics was wrong, he merely thought it was incomplete, needing to be supplemented by “hidden variables” to restore Newtonian determinism. Today, it’s often said that Einstein lost this battle, that quantum indeterminism triumphed. But wait! Logically, how could Einstein’s belief in a hidden determinism ever be disproven? Can you ever rule out the existence of a pattern, just because you’d failed to find one?

The answer to this question takes us to the heart of quantum mechanics, to the part that popular explanations usually mangle. Quantum mechanics wasn’t the first theory to introduce randomness and probabilities into physics. Ironically, the real novelty of quantum mechanics was that it replaced probabilities—which are defined as nonnegative real numbers—by less intuitive quantities called amplitudes, which can be positive, negative, or even complex. To find the probability of some event happening (say, an atom decaying, or a photon hitting a screen), quantum mechanics says that you need to add the amplitudes for all the possible ways that it could happen, and then take the squared absolute value of the result. If an event has positive and negative amplitudes, they can cancel each other out, so the event never happens at all.

The key point is that the behavior of amplitudes seems to force probabilities to play a different role in quantum mechanics than they do in other physical theories. As long as a theory only involves probabilities, we can imagine that the probabilities merely reflect our ignorance, and that a “God’s-eye view” of the precise coordinates of every subatomic particle would restore determinism. But quantum mechanics’ amplitudes only turn into probabilities on being measured—and the specific way the transformation happens depends on which measurement an observer chooses to perform. That is, nature “cooks probabilities to order” for us in response to the measurement choice. That being so, how can we regard the probabilities as reflecting ignorance of a preexisting truth?

Well, maybe it’s possible, and maybe not. The answer’s not obvious, and wasn’t until the 1960s—after Einstein had passed away—that the situation was finally clarified, by physicist John Bell. What Bell showed is that, yes, it’s possible to say that the apparent randomness in quantum mechanics is due to some hidden determinism behind the scenes, such as “God’s unknowable encyclopedia” listing everything that will ever happen. That bare possibility has no experimental consequences and can never be ruled out. On the other hand, if you also want the hidden deterministic variables to be local—that is, to obey the inherent impossibility of faster-than-light communication—then there’s necessarily a conflict with the predictions of quantum mechanics for certain experiments. In the 1970s and 1980s, the requisite experiments were actually done—most convincingly by physicist Alain Aspect—and they vindicated quantum mechanics, while ruling out local hidden variable theories in the minds of most physicists.

To clarify, these experiments didn’t rule out the possibility of any determinism underneath quantum mechanics; that’s something that no experiment can do in principle. There’s even a theory, Bohmian mechanics (named after physicist David Bohm), that reproduces all the predictions of quantum mechanics, by imagining that besides the amplitudes, there are also actual particles that move around in a deterministic way, guided by the amplitudes. Bohmian mechanics is extremely strange, because it involves instantaneous communication between faraway particles, though not of a kind that could actually be used to send messages faster than light. In essence, what Bell’s theorem shows is that if you want a deterministic theory underpinning quantum mechanics, then it has to be strange in exactly the same way Bohmian mechanics is strange: It has to be “nonlocal,” resorting to instantaneous communication to account for the results of certain measurements. In fact, understanding this aspect of Bohmian mechanics is what motivated Bell to prove his theorem in the first place.

I’d like to explain Bell’s idea as a simple mathematical game (called the CHSH game, after its inventors, Clauser, Horne, Shimony, and Holt). This game involves two players, Alice and Bob, who are cooperating rather than competing. Alice and Bob can agree on a strategy in advance: There can be unlimited “classical correlation” between them. Once the game starts, however, no further communication is allowed. (We can imagine, if we like, that Alice stays on Earth while Bob travels to Mars.)

After they separate, Alice and Bob each open a sealed envelope, and find either a red or a blue card inside. The cards were chosen randomly, so that there are four equally likely possibilities: blue/blue, blue/red, red/blue, and red/red. After examining her card, Alice can raise either one or two fingers, and after examining his card, Bob can do likewise. Alice and Bob win the game if either both cards were red and they raised different numbers of fingers, or at least one card was blue and they raised the same number of fingers.

The question that interests us is, what is the best strategy for Alice and Bob—the one that lets them win this strange game with the maximum probability? As an exercise, I invite you to check that the best strategy is a rather boring one: Alice and Bob both just ignore their cards and raise one finger. Using this strategy, our heroes will win 75 percent of the time, whenever one or both cards are blue. The Bell inequality is simply the statement that no strategy does better than this.

On the other hand, suppose that before Alice and Bob separate, they put two electrons into a so-called Einstein-Podolsky-Rosen pair, a configuration where there’s some amplitude for both electrons to be spinning left, and an equal amplitude for both electrons to be spinning right. This pair is the most famous example of an entangled state: Roughly speaking, a combined quantum state of several particles that can’t be factored into states of the individual particles. (For our purposes, it doesn’t matter what it means for an electron to be “spinning left or right,” it’s just some property of the electron.) Then, when they separate, Alice takes one electron and Bob takes the other.

Click to Enlarge ImageIn this case, Bell showed that there’s a clever strategy where Alice and Bob use measurements of their respective electrons to correlate their responses—in such a way that no matter which cards they get, they win the game 85.4 percent of the time. Because 85.4 percent is more than 75 percent, there can be no way to describe the state of Alice’s electron separately from the state of Bob’s electron—if there were, then Alice and Bob couldn’t have won more than 75 percent of the time. To explain their success in the game, you need to accept that no matter how far apart Alice and Bob are, their electrons remain quantum mechanically entangled. (See the figure on the first page and table above.)

Before going further, it’s worth clarifying two crucial points. First, entanglement is often described in popular books as “spooky action at a distance”: If you measure an electron on Earth and find that it’s spinning left, then somehow, the counterpart electron in the Andromeda galaxy “knows” to be spinning left as well! However, a theorem in quantum mechanics—appropriately called the No-Communication Theorem—says that there’s no way for Alice to use this effect to send a message to Bob faster than light. Intuitively, this is because Alice doesn’t get to choose whether her electron will be spinning left or right when she measures it. As an analogy, by picking up a copy of American Scientist in New York, you can “instantaneously learn” the contents of a different copy of American Scientist in San Francisco, but it would be strange to call that faster-than-light communication! (Although this might come as a letdown to some science fiction fans, it’s really a relief: If you could use entanglement to communicate faster than light, then quantum mechanics would flat-out contradict Einstein’s special theory of relativity.)

However, the analogy with classical correlation raises an obvious question. If entangled particles are really no “spookier” than a pair of identical magazines, then what’s the big deal about entanglement anyway? Why can’t we suppose that, just before the two electrons separated, they said to each other “hey, if anyone asks, let’s both be spinning left”? This, indeed, is essentially the question Einstein posed—and the Bell inequality provides the answer. Namely, if the two electrons had simply “agreed in advance” on how to spin, then Alice and Bob could not have used them to boost their success in the CHSH game. That Alice and Bob could do this shows that entanglement must be more than just correlation between two random variables.

To summarize, the Bell inequality paints a picture of our universe as weirdly intermediate between local and nonlocal. Using entangled particles, Alice and Bob can do something that would have required faster-than-light communication, had you tried to simulate what was going on using classical physics. But once you accept quantum mechanics, you can describe what’s going on without any recourse to faster-than-light influences.

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