Computing with Quantum Cats (16 page)

One of the places where Benioff presented his ideas was MIT, where he put them to a meeting in 1981 where Feynman was the keynote speaker. Feynman opened the proceedings with a talk titled “Simulating Physics with Computers,” in which he credited Ed Fredkin with stimulating his interest in the subject, and tackled two questions: Is it possible to simulate physics (meaning quantum physics) with a quantum computer? And, is it possible to simulate (quantum) physics with a classical computer (by simulating “probability” in some way)?

Curiously, Feynman referred to the discussion of quantum simulators as “a side-remark,” which he dealt with briefly before moving on to the second question, which he regarded as more important. He gave an example of how a “universal quantum simulator” might work, and said, “I therefore believe it's true that with a suitable class of quantum machines you could imitate any quantum system, including the physical world,” but was unable to offer a definitive proof that this is the case. He was, though, able to offer a definitive proof that quantum systems can
not
be “probabilistically simulated by a classical computer.”
¹⁶

The proof depends on the properties of pairs of particles that interact with one another and then fly off in different directions. Like the two-cat version of Schrödinger's puzzle, the behavior of one particle depends on what happens to the
other particle even when they are far apart; this is known as “entanglement,” a term coined by Schrödinger himself. What the argument boils down to is the comparison of a simple pair of numbers, which can be (and have been) measured in experiments. If one number is bigger than the other, there is no way in which quantum mechanics can be simulated perfectly by a classical computer, and therefore no way in which the world can be simulated perfectly by a classical computer. Feynman was delighted by this argument. “I've entertained myself always,” he said, “by squeezing the difficulty of quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item. It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than another. But there you are.”

What Feynman, who was not always scrupulous about giving credit to others, neglected to tell his audience was that the entire argument had been taken, lock, stock and barrel, from the work of CERN physicist John Bell, and is usually known as the Bell Inequality. Although Bell himself had not applied his ideas to quantum computers—or quantum simulators, in Feynman's terminology—the idea is so important, both in terms of our understanding of reality and in terms of quantum computation, that it will be the subject of my
next chapter
. Feynman's conclusion from 1981 is, though, still apposite: “Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical.”

Feynman himself did think further about computation in general and about quantum computers. He gave a course on computation at Caltech in the mid-1980s, and a talk in
Anaheim in 1984 in which he described the basis of a quantum mechanical computer,
¹⁷
along the lines of Benioff's quantum Turing machine, using reversible gates.
¹⁸
In that talk he came up with another of his memorable comments: “It seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds dominant sway.” But he never seems to have put two (from his 1981 lecture) and two (from his 1984 lecture) together and realized that such a computer would be fundamentally different from a classical computer not just in terms of its physics but in terms of the kinds of problems it could solve. That leap of inspiration came from Oxford theorist David Deutsch, also in the mid-1980s, and will form the heart of
Part Three
of this book.

The tangled story of entanglement
¹
begins, inasmuch as it has a beginning, with the work of a French nobleman, Louis de Broglie, in the 1920s. De Broglie, who held the honorary title of “Duke,” was a latecomer to research in physics. Born in 1892, as the younger son of an aristocratic family he was expected to make a career in the diplomatic service; but under the influence of his elder brother Maurice,
²
who became a physicist in spite of the strenuous objections of their father, he too began to study physics, alongside his “proper” course in history, at the Sorbonne in 1909. He hoped to move on to research, but his career was interrupted by the First World War, during which he served in the radio communications branch of the army, including a spell based at the Eiffel Tower, which was used as a radio mast. So it was not until 1924, when he was already in his early thirties, that de Broglie was able to submit a thesis for his PhD; but what a thesis!

De Broglie was not a great mathematician but he did have very good physical insight. He was one of the first people to fully accept the idea of light quanta (what we now call photons), and picked up on a curious feature of the equations that Einstein had used to describe these entities. The equations showed a relationship between the wave properties of a photon (its wavelength or frequency) and its particle properties (such as momentum and energy). They showed that light “waves” could also be treated as particles, and that if you knew the wavelength of a photon you could calculate its momentum. De Broglie pointed out that the same equations worked in reverse—that “particles” (specifically, electrons) should also behave as waves, if the equations were correct. If you knew the momentum of an electron you could calculate its wavelength. De Broglie's thesis supervisor, Paul Langevin, didn't know what to make of this and showed the work to Einstein, who said, “I believe that it involves more than a mere analogy.” De Broglie got his PhD, and within three years experiments had been carried out which showed conclusively that electrons do indeed behave as waves exactly in accordance with his description. De Broglie's thesis work, for which he received the Nobel Prize in 1929, was also the inspiration for Erwin Schrödinger's development of the wave version of quantum mechanics.

So by 1927, soon after Schrödinger's wave mechanics and the particle version of quantum mechanics developed by Heisenberg and others had become established, de Broglie should have been a man whose ideas were taken seriously. Instead, his next big idea was first derided and then largely ignored.

DROPPING THE PILOT

The idea was as brilliantly simple as his earlier insight into the nature of wave-particle duality. While other researchers were struggling to choose between the wave version of quantum mechanics and the particle version of quantum mechanics, de Broglie said, in effect, why not have both? He suggested that the wave and the particle were equally real, and that what goes on in experiments such as the experiment with two holes is that a real wave (which became known, for obvious reasons, as a “pilot” wave) spreads through the apparatus and is responsible both for the interference and for guiding an equally real particle along trajectories determined by the interference, like a surfer riding the waves in the sea. The key point on this picture is that the particle always exists at a definite place even when it is not being measured or observed. Equally, the apparatus (in principle, the whole world) is occupied by a physically real field, like the electromagnetic field, which we are only aware of because of its influence on the statistical behavior of particles. Electrons (or photons) fired one after another through the experiment with two holes do not all follow exactly the same trajectory because of tiny differences in the speed or direction with which they start out. We can never measure the field, but we can measure the particles. This is known as a “hidden variables” theory, although as a result of what John Bell called “historical silliness”
³
it is the things we
can
measure, such as the positions of particles, that are regarded as the “hidden” variables, while the thing we
can't
measure, the field, is not. As Bell also said, “this idea seems so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.”
⁴

But ignored it was. De Broglie presented his ideas (in a much more fully worked out form than the sketch I have provided here) to the same Solvay Congress in 1927 that heard Schrödinger point out that there is nothing in the equations to suggest that the wave function “collapses,” as proposed by Niels Bohr and followers of his “Copenhagen Interpretation.” De Broglie's pilot wave idea was savagely attacked on mathematical grounds by Wolfgang Pauli, a mathematical physicist with a (largely justified) high opinion of his own ability and a (not always justified) tendency to dismiss the efforts of those he regarded as lesser minds. Although de Broglie did respond to these criticisms, and a modern reading of the proceedings of the meeting suggests that he successfully answered them, his diffident style left the impression that he had lost the argument. In addition, Pauli pointed out that the hidden variables idea led to some peculiar implications when applied to more than one particle. Crushed, de Broglie gave up promoting his idea. Those peculiar implications concern entanglement and what Einstein would later call “spooky action at a distance,” and are at the heart of the modern understanding of quantum physics and quantum computation. But what seemed at the time the final nail in the coffin of de Broglie's idea came in 1932, when Johnny von Neumann published his great book on quantum mechanics, in which, among other things, he “proved” that no hidden variables theory could be an accurate description of reality. He was wrong.

VON NEUMANN GETS IT WRONG

In order to appreciate just how wrong von Neumann was, we need to be clear about what hidden variables theories are—
and a good way to see that is by comparison with the all-conquering (for half a century) Copenhagen Interpretation. There are three significant points of conflict between these two views of the world:

1.        The Copenhagen Interpretation says that the wave function is a complete description of a system and contains all the information about it. Hidden variables theory says that the wave function is only part of the story, and that there are also particles with real positions and momenta.

2.       The Copenhagen Interpretation says that the wave mostly spreads out in line with Schrödinger's equation, but sometimes “collapses” to (more or less) a point. The mechanism for this collapse has never been satisfactorily explained. Hidden variables theory says that the wave always develops in line with Schrödinger's equation; there is no collapse.

3.        The Copenhagen Interpretation says that even though the evolution of the wave is deterministic, the process of collapse introduces an element of probability into the outcomes of experiments—that is, that quantum physics is stochastic. Hidden variables theory says that everything is deterministic, and that the only reason we cannot predict everything perfectly is that we can never know perfectly all of the starting conditions.

Stated like that, it is hard to see how any sensible person could choose the Copenhagen Interpretation over hidden variables theory unless there were very powerful evidence against the latter. But the power of von Neumann's “proof” lay
as much in his name as in the mathematics; not only that, it was refuted almost as soon as it was published, but the refutation was ignored. It is probably significant that the flaw in von Neumann's argument was found by a young researcher who came from outside the quantum mechanical fold, and was unlikely to be overawed by von Neumann's (or Bohr's) reputation; but the fact that she was a young outsider is also one factor in explaining why her refutation was ignored.

Grete Hermann was born in Bremen in 1901. She qualified as a secondary school teacher in 1921, then studied mathematics and philosophy, working under Emmy Noether in Göttingen and receiving a PhD in 1926 for research into what would later become known as computer algebra. For some years she stayed on in Göttingen, developing her interest in philosophy and the foundations of science, but she was also a committed socialist, editing a newspaper called
The Spark
and actively opposing the increasingly powerful Nazis. In 1936 she had to flee Germany to escape the regime, traveling via Denmark and France to England. In 1946 she returned to Germany and worked in education, helping to rebuild post-war society; she died in 1984, having lived long enough to see her early insight come in from the cold.

In 1934, Hermann had visited Werner Heisenberg's group in Leipzig to discuss the foundations of quantum physics in relation to her favored Kantian philosophy. It was there that she confronted von Neumann's “proof” of the impossibility of hidden variables theories, and pointed the flaws out to Heisenberg and his colleagues. The problem was not with the mathematics—von Neumann was not prone to making mathematical mistakes—but with the assumptions that went into the mathematics. Fortunately, this means that
we do not need to go through the math to understand what went wrong; but, surprisingly, it means that the mistake ought to have been obvious to anyone who looked carefully, as Hermann did, at what von Neumann was saying. His mistake involved an assumption about the way averages are taken in quantum mechanics. There is no perfect way of explaining the error in everyday language, but one that may help is to look at the way we take averages in everyday life. Imagine that we have two groups of people. In one group there are ten people, and their average height is 1.8 meters. In the other group there are twenty people, and their average height is 1.5 meters. Von Neumann's mistake was equivalent to saying that therefore the average height of the thirty people is 1.65 meters, the average of 1.8 and 1.5. But because there are twice as many people in the second group, the correct average is, of course, 1.6 meters.

Hermann's discovery of the failure of von Neumann's “proof” was published in 1935, but only in German and in a journal devoted more to philosophy than to physics, which physicists did not read. That also partly explains why it faded into obscurity.
⁵
But Heisenberg knew of it, directly from Hermann, and it is a mystery that will now never be resolved why he did not at least draw it to the attention of a wider circle of quantum physicists. Hermann herself was never in a position to promote her finding, even if she had wanted to, because of the dramatic changes in her life and career brought about by the political developments in Germany; in any case, though, she does not seem to have been particularly enamored of hidden variables theories, and had simply been concerned as a mathematician with setting the record straight about von Neumann's claim. To all intents and purposes, in
the quantum physics community it was as if Hermann had not existed, and from 1932 onwards “everybody knew” that hidden variables theories would not work, because Johnny von Neumann said so. Even people who never read his book believed this, because they had been told it was so. But while hidden variables theories languished, Einstein, along with a couple of colleagues, pointed out another problem with the Copenhagen Interpretation—in fact, an aspect of the “problem” with de Broglie's pilot wave pointed out by Pauli.

SPOOKY ACTION AT A DISTANCE

Einstein first expressed the idea behind what became known as the “EPR paradox,” although I prefer the term “puzzle,” in 1933, at another Solvay Congress. Leon Rosenfeld, one of Niels Bohr's collaborators, recalled that after a lecture by Bohr Einstein pointed out to Rosenfeld that if two particles interacted with one another and then flew apart,

an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum…[will] from the conditions of the experiment,…obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able to tell where the other particle is…is it not paradoxical? How can the final state of the second particle be influenced by a measurement performed on the first, after all physical connection has ceased between them?
⁶