Ask any physicist to name the top two theories of the 20th century, and you’ll almost always get the same automatic answer: Einstein’s relativity and quantum mechanics. But lately a few 21st century thinkers have hinted that maybe the third-place theory should move up a notch. In the wake of the computer revolution, information theory might deserve to displace relativity in the rankings.

That revisionist perspective reflects a late 20th century twist in the story of that century’s theories: the surreptitious merger of quantum theory with information science. Their origins had been entirely independent. Quantum mechanics arose in the 1920s as the math for describing the odd behavior of atoms and electrons; information theory came along two decades later, as formulas for quantifying communication over telephone lines. For decades the two theories led separate lives in fields of study far removed from one another. While physicists expended their intellectual energy on uniting quantum mechanics with relativity (a quest that continues, still without success), information scientists graduated from telephones to computers with only occasional concern for quanta. But then in the 1980s and ’90s, quantum and information science met, married and produced offspring — specifically, the intellectual enterprise known today as quantum information theory.

Initially, quantum information snared the attention of physics fans for its possible uses in sending secret codes and creating superfast computers. Those and other potential quantum technologies continue to occupy scientists’ time at research centers around the world, from Canada to Austria to Singapore. But theorists pursuing quantum information’s secrets are more motivated by the quest to acquire a deeper understanding of physical reality, and perhaps to grasp more fully the nature of quantum mechanics itself.

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“We always stress how interesting are the applications” of quantum information, says theorist J. Ignacio Cirac. “But actually if you work in this field, you don’t care so much about these things. What you see is that the science that is being done is really fantastic.”

**Computing reality**

At the heart of quantum information science is a novel representation for information known as the qubit. It’s the quantum analog of the 1s and 0s, or “bits,” processed by ordinary computers. But a qubit is infused with quantum magic, allowing it be both a 1 and a 0 at the same time. This “superposition” of identities gives quantum information extraordinary power.

Qubits can, for instance, transmit coded messages with absolute secrecy, typically in the form of particles of light, or photons (SN: 8/16/08, p. 24). Messages sent as a string of qubits, encoded in the orientations of photon vibrations, are secure because any attempt at eavesdropping would alter the message in a detectable way.

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Such quantum-secured systems are already commercially available, and may someday be a commercial necessity because of another quantum information application: quantum computing. For certain problems, computing with qubits held in a quantum memory can solve mathematical problems so hard that a standard supercomputer couldn’t find the answer within the lifetime of the universe. One such hard problem is finding the prime factors of very big numbers. Codes based on such problems, which now protect most electronic financial transactions, would be worthless in a world with full-scale quantum computers. (No need to worry about your credit card security just yet, though. Breaking today’s codes would require a quantum computer handling thousands or more qubits; the record for today’s prototypes is merely 14.)

Quantum computers wouldn’t be worth much for general number crunching. They would work only for problems that could be posed in the form of an algorithm amenable to the way quantum weirdness can cancel out wrong answers, allowing only the correct one to survive. But a sufficiently sophisticated quantum computer would be able to simulate molecular systems governed by quantum mechanics. You could harness that power to foretell the outcome of chemical reactions, for instance, without the bother of test tubes. Such computing ability could aid the design of new industrial materials or help develop powerful drugs with fewer side effects.

“Of the currently understood algorithms, the one that seems most promising for applications is simulation of a quantum system,” says Caltech physicist John Preskill. “We don’t currently envision that you’ll be sending your e-mail on a quantum computer. On the other hand, quantum games might be a blast.”

But for all these wonders, quantum computers and other quantum information processing systems are not mere investment speculation opportunities. They are tools for scientists to dig deeper into reality’s foundations. Quantum information could reveal nuances about the interface between mathematics and the physical world.

Cracking codes, for instance, involves solving the very hard math problem of finding the prime factors of a big number, hundreds of digits long. But the fact that a quantum-computing algorithm can solve such a problem — as discovered by the mathematician Peter Shor in 1994 — has deeper implications than just financial eavesdropping.

“Factoring is a hard problem classically,” says Preskill, “but Shor’s algorithm shows that it is an easy problem quantumly. And so it seems the boundary between what problems are hard and what problems are easy is different in our physical world — because it’s a quantum world — than it would be if the world were classical.” In other words, quantum information processing reveals something about math’s relation to physical reality that prequantum scientists couldn’t have imagined.

Or at least that’s what quantum scientists believe. Proving that the physical world really does allow quantum solutions to some hard math problems requires actually building a large-scale quantum computer. “We hope to be able to verify that these extraordinary computational resources in quantum systems really are part of the way nature behaves,” Preskill noted recently in Vancouver at the annual meeting of the American Association for the Advancement of Science. “We could do so by solving a problem that we think is hard classically … with a quantum computer, where we can easily verify with a classical computer that the quantum computer got the right answer.”

Factoring would be one example of such a problem — very hard to solve, but easy to check to see if the answer you get is right. But other hard math problems probably exceed even a quantum computer’s capabilities. “Some problems are quantumly hard,” says Preskill. Knowing which problems are hard for quantum computers would offer insight into what kinds of mathematical computations are actually possible in the physical universe.

One computational claim challenged by quantum computing, MIT computer scientist Scott Aaronson points out, is a long-held belief called the extended Church-Turing thesis. It basically states that anything feasibly computable by a physical device is also computable by an idealized “universal” computer known as a Turing machine.

“This is a falsifiable claim about the laws of physics,” Aaronson said at the AAAS meeting. It expresses the belief that if physical laws are like computer code, any reasonable programming language for nature’s laws could emulate another one.

“This would say that any reasonable laws of physics that someone could write down, they can all reasonably simulate any other laws,” Aaronson said.

But Shor’s factoring algorithm contends that quantum computers could do things a Turing machine couldn’t. So either quantum computing is actually impossible (not likely, as that would imply some flaw in quantum mechanics itself), or the Church-Turing thesis is incorrect as a statement about the physical world. Unless somehow there exists an unknown way for an ordinary computer to simulate quantum physics. “No one has proved that there isn’t one,” said Aaronson. “But this would be an astounding mathematical discovery.”

**Quantum roots**

An equally or more profound discovery would be identifying the physical principle that requires reality to obey the rules of quantum mechanics in the first place. In the beginning, the quantum pioneers simply figured out the math that works — a math that requires the weirdness of multiple possible realities (SN: 11/20/10, p. 15). Inquiring into the underlying reason why such weird math worked so well was long considered foolish. A standard response to junior physicists who raised such questions was “shut up and calculate.”

But in recent decades, the quest to find a physical principle from which quantum mechanics can be built has become more popular, and quantum information has been at the heart of many such efforts. Much of the work along these lines was inspired by the late physicist John Archibald Wheeler, who believed that quantum physics — and existence itself — might have its roots in aspects of information theory. His slogan “it from bit” summarized the view that reality somehow emerges from cosmic information processing.

Several efforts to derive quantum math from basic principles have echoed the it-from-bit philosophy. Last year, for instance, Giulio Chiribella of the Perimeter Institute for Theoretical Physics in Waterloo, Canada, and collaborators from Italy showed a way to derive quantum mechanics from a set of five axioms plus one postulate, all rooted in information theory terms (SN: 8/13/11, p. 12).

“In this approach the rules by which information can be processed determine the physical theory, in accordance with Wheeler’s program ‘it from bit,’ ” Chiribella and Italian colleagues Giacomo Mauro D’Ariano and Paolo Perinotti wrote in *Physical Review A*.

Their system is built on axioms such as “causality” — in essence, the notion that signals from the future cannot affect the present. In other words, the odds of an experiment turning out one way or the other do not depend on a future choice of which measurements to perform. Another axiom, called “ideal compression,” asserts that the information in a system always can be condensed into an encoded form that could then be decoded to reproduce all of the original information.

All five axioms in this approach reflect basic aspects of ordinary information, no quantum weirdness necessary. “These axioms represent standard features of information processing that everyone would, more or less implicitly, assume,” Chiribella and colleagues wrote.

Their postulate, however, departs from the nonquantum classical world by introducing what they call the “purification” principle. That principle is something like a law of conservation of information, contending that all the information a quantum system contains can be recovered by reversing all its interactions with its environment. In practice, of course, keeping track of every interaction of a system with its environment isn’t possible. But as a principle, it implies the ability to know all the information there is to know about a whole system yet remain ignorant of some of its parts. Therein lies the key to quantum weirdness, and that postulate plus the other information-theoretic axioms yields the quantum math that has long perplexed anyone who has tried to understand where it comes from.

**Relatively simple**

Chiribella and colleagues’ axiomatic system is not the only one to reproduce the formulas of quantum physics, though (but they claim theirs is the first to do so using only principles that can be expressed in terms of actual physical operations). And none of the various approaches satisfy everybody, anyway. Some physicists (Wheeler was among them) believe that the physical principle underlying quantum physics should be crisp and clear and even in retrospect, perhaps, obvious.

Christopher Fuchs, a quantum physicist at the Perimeter Institute, argues that the secret principle (or principles) at the core of quantum mechanics ought to be similarly simple to the two pillars of Einstein’s theory of relativity: The speed of light is constant, and the laws of physics do not depend on how you are moving. Axioms that need paragraphs of explanation do not meet this crisp-and-clear test. Fuchs still appreciates information’s role in quantum theory, but he does not believe that Wheeler’s “it from bit” is the right story.

“I suspect quantum theory is mostly about information,” Fuchs said in a recent interview. “But there is going to be some piece remaining behind that you really can’t pin down as being a statement about information. Instead I expect it to be a statement about the character of the world.… The distillate that remains, the part that can’t be given an information-theoretic reason, will be our first glimpse into what quantum reality is all about.” ** **