杏吧原创

Whatever the u-bit is, it rotates quickly (Image: Natalie Nicklin)

Our best theory of nature has imaginary numbers at its heart. Making quantum physics more real conjures up a monstrous entity pulling the universe鈥檚 strings

Leader: 鈥�The u-bit may be omniscient, but it鈥檚 no God particle鈥�

IF YOU鈥橵E ever tried counting yourself to sleep, it鈥檚 unlikely you did it using the square roots of sheep. The square root of a sheep is not something that seems to make much sense. You could, in theory, perform all sorts of arithmetical operations with them: add them, subtract them, multiply them. But it is hard to see why you would want to.

All the odder, then, that this is exactly what physicists do to make sense of reality. Except not with sheep. Their basic numerical building block is a similarly nonsensical concept: the square root of minus 1.

This is not a 鈥渞eal鈥� number you can count and measure stuff with. You can鈥檛 work out whether it鈥檚 divisible by 2, or less than 10. Yet it is there, everywhere, in the mathematics of our most successful 鈥� and supremely bamboozling 鈥� theory of the world: quantum theory.

This is a problem, says respected theoretical physicist of Williams College in Williamstown, Massachusetts 鈥� a problem that might be preventing us getting to grips with quantum theory鈥檚 mysteries. And he has a solution, albeit one with a price. We can make quantum mechanics work with real numbers, but only if we propose the existence of an entity that makes even Wootters blanch: a universal 鈥渂it鈥� of information that interacts with everything else in reality, dictating its quantum behaviour.

What form this 鈥渦-bit鈥� might take physically, or where it resides, no one can yet tell. But if it exists, its meddling could not only bring a new understanding of quantum theory, but also explain why super-powerful quantum computers can never be made to work. It would be a truly revolutionary insight. Is it for real?

The square root of minus 1, also known as the imaginary unit, i, has been lurking in mathematics since the 16th century at least, when it popped up as geometers were solving equations such as those with an x² or x³ term in them. Since then, the imaginary unit and its offspring, two-dimensional 鈥渃omplex鈥� numbers incorporating both real and imaginary elements, have wormed their way into many parts of mathematics, despite their lack of an obvious connection to the numbers we conventionally use to describe things around us (see 鈥�Complex stuff鈥�). In geometry they appear in trigonometric equations, and in physics they provide a neat way to describe rotations and oscillations. Electrical engineers use them routinely in designing alternating-current circuits, and they are handy for describing light and sound waves, too.

But things suddenly got a lot more convoluted with the advent of quantum theory. 鈥淐omplex numbers had been used in physics before quantum mechanics, but always as a kind of algebraic trick to make the math easier,鈥� says of Kenyon College in Gambier, Ohio.

Quantum complications

Not so in quantum mechanics. This theory evolved a century ago from a hotchpotch of ideas about the subatomic world. Central to it is the idea that microscopic matter has characteristics of both a particle and a wave at the same time. This is the root of the theory鈥檚 infamous assaults on our intuition. It鈥檚 what allows, for example, a seemingly localised particle to be in two places at once.

And it turns out that two-dimensional complex numbers are exactly what you need to describe this fuzzy, smeared world. Within quantum theory, things like electrons and photons are represented by 鈥�wave functions鈥� that completely describe all the many possible states of a single particle. These multiple personalities are depicted by a series of complex numbers within the wave function that describe the probability that a particle has a particular property such as a certain location or momentum. Whereas alternative real-number descriptions for something like a light wave in the classical world are readily available, purely real mathematics simply does not supply the tools required to paint the dual wave-particle picture.

鈥淧urely real mathematics simply does not supply the tools to paint the complete quantum picture鈥�

Hidden complexity

The odd thing is, though, we never see all that quantum complexity directly. The quantum weirdness locked up in the wave function 鈥渃ollapses鈥� into a single real number when you attempt to measure something: a particle is always found at a single location, for example, or moving with a certain speed. Mathematically, the first thing you do when comparing a quantum prediction with reality is an operation akin to squaring the wave function, allowing you to get rid of all the i鈥榮 and arrive at a real-number probability. If there is more than one way for a thing to end up with, say, a particular location, you add up all the complex number representations for each different way, and then square the sum.

The fact that this was a rather odd way to go about things first struck Wootters over 30 years ago, when he was writing his PhD thesis. In the 鈥渞eal鈥� world, the probability of rolling a 10 from a pair of dice is 3/36, because there are three ways to produce a 10 from the total of 36 possible outcomes. We add probabilities, we don鈥檛 add them and square them. 鈥淭his procedure would be strange even if the square roots were real,鈥� says Wootters.

The involvement of complex numbers only makes things worse. Complex numbers are two-dimensional with real and imaginary parts, whereas the probabilities of observable reality are only one-dimensional 鈥� purely real. That implies some of the information stored in the complex numbers disappears every time we make a quantum measurement. This is very unlike the macroscopic, classical world, where there is at least in theory a perfect flow of information from the past to the future: given complete information about the properties and position of a pair of dice, for example, we can predict how they will fall.

Replace those complex square roots with real square roots, Wootters realised, and you can at least stem this troubling information haemorrhage. 鈥淩eal square roots would be comprehensible 鈥� not a strange thing 鈥� if nature is interested in a strong link between past and future,鈥� he says.

As it turned out, he wasn鈥檛 the first to try to make quantum theory real. In the 1960s, Swiss physicist had reformulated quantum mechanics using only real numbers. But when he tried to impose the fundamental quantum principle of uncertainty 鈥� the idea that we cannot determine quantities such as the position and momentum of a particle simultaneously with full accuracy 鈥� he found that he still needed something to play the role of the imaginary unit.

Back in 1980, this was all a shrug-shoulders affair for Wootters. He had other fish to fry 鈥� namely, helping to found a new branch of physics, quantum information theory. This portrays quantum mechanics in terms of the information it can encode, and has allowed physicists to take advantage of fundamentally quantum processes such as teleportation and entanglement between particles to transport information more efficiently and securely than is possible by conventional, classical means. It also lies behind current attempts to build super-powerful quantum computers. Wootters鈥檚 role as one of the founding fathers of the field has made him one of the .

It was an invitation in 2009 to give a talk in Vienna, Austria, that prompted him to return to the problematic role of i in quantum theory. Could all the quantum information theory developed in the meantime provide any new resolution?

By October 2012, working with his students Antoniya Aleksandrova and Victoria Borish, Wootters had progressed far enough to think it could. Conventional quantum theory talks of information in terms of qubits, probabilistic versions of the traditional binary bits that classical computers crunch. Wootters and his colleagues were able to replace these qubits with real-number equivalents, and so capture all the weird correlations and uncertainties of conventional quantum theory without an i in sight (). While he was about it, Wootters also last year published his 30-year-old insight that a real-number quantum theory could solve the problem of imperfect information flow during quantum measurements ().

All this came with a huge sting in the tail, however. Like Stueckelberg鈥檚 earlier attempt, this theory also needs an extra element to play the part of i. It turns out to be a monster: a physical entity Wootters dubs the u-bit.

A u-bit is a master bit: an entity that interacts with all the other bits describing stuff in the universe. Mathematically, this omnipresent conduit of information is represented by a vector on an ordinary, real two-dimensional plane. What it represents physically, no one, least of all Wootters, can tell, but by entangling itself with everything else in the universe, it is sufficient to replace every single complex number in quantum theory. The mathematical description of the u-bit supplies one further, slightly whimsical clue to its physical identity: whatever and wherever it is, it must be rotating quite fast.

鈥淭here is one whimsical clue to the u-bit鈥檚 identity: whatever it is, it must be rotating quite fast鈥�

It all sounds rather like a mistimed April fool, but that might just be the idea鈥檚 strength, says Schumacher. 鈥淚 think a good paper in fundamental physics shares some characteristics with a good joke: it has an unexpected take on a familiar idea, and yet in retrospect it has a certain screwball inevitability. By that standard, Bill鈥檚 u-bit theory is a very good joke.鈥�

of the Perimeter Institute in Waterloo, Ontario, Canada, is more sceptical, pointing out that the nature of the u-bit makes it very difficult to test the theory. 鈥淚t鈥檚 not yet clear whether it is something that can be manifested at a particular location such that you could find it in a detector like we did the Higgs boson,鈥� he says.

But the u-bit鈥檚 influence might be felt indirectly. Since it interacts with everything in the universe, it can rough up even a supposedly isolated quantum system, making it 鈥渄ecohere鈥� and lose its vital quantum properties in a way not predicted by standard quantum theory. That could be bad news for the quantum computers built on the back of quantum information theory, which are very sensitive to environmental disturbance. With a u-bit lurking off-stage, physicists could try all they like to seal off such a device from the outside world, for example by putting it inside a huge fridge to cool down its interactions, but there would be nothing that could shield it from decoherence induced by the u-bit.

The length of time a conventional qubit could stave off decoherence is governed by the strength of the u-bit鈥檚 interaction and its rate of rotation. An experiment carried out in 2011 by researchers at the National Institute of Standards and Technology in Boulder, Colorado, showed that qubits made out of trapped ions , a result that would suggest the u-bit interaction is weak, if it exists at all. 鈥淭he theory is definitely sensible, but it is currently not clear whether the u-bit is really realised in nature,鈥� says of Heidelberg University in Germany.

For an information theorist like Wootters, it is easy to see the attraction of an entity that does away with the profligate waste of information seemingly hard-wired into the mathematics of quantum theory. But he is the first to admit that the u-bit is no panacea for quantum ills. It doesn鈥檛 itself explain what wave-particle duality is, or why a particle can be in two places at once. Something like entanglement is actually more pervasive in the new formulation, because the u-bit can be fully entangled with any number of objects. In conventional, complex-number quantum mechanics, full-on entanglement is limited to two objects.

But even if the real-number reformulation just turns out to be a new angle from which to view quantum theory鈥檚 mysteries, that could be valuable. Despite quantum theory鈥檚 enormous success in agreeing with experiment, its conflict with our intuitions has left us casting round for a 鈥渘arrative鈥� 鈥� a compelling explanation of why things should be the way quantum theory suggests they are. Often, those interpretations have been impeded by dwelling too much on one aspect of the theory, says Schumacher. 鈥淭here has been an amazing amount of nonsense written about quantum theory because people used to understand it only as a theory about waves, for instance.鈥� That makes one insight from Wootters鈥檚 work 鈥� that we should look at the information-carrying capacity of a quantum state to understand the physics behind it 鈥� an important one, says M眉ller.

As for the question of quantum theory鈥檚 irreality, perhaps we have just to learn to love i. After all, it is not just quantum mechanics where its influence is felt. Complex numbers are also increasingly vital in describing optical waveguides, transitions between different states of matter and many other aspects of classical physics. 鈥淧eople always thought of complex numbers just as a tool, but increasingly we are seeing that there is something more to them,鈥� says mathematician of Brunel University in London. History tells us we have come to accept the reality of other mathematical concepts such as zero and the negative numbers only after long tussles. Perhaps 鈥渋maginary鈥� was an unfortunate choice of words when we came to name i. Unless, of course, instead of an i, a u-bit lurks, ubiquitous 鈥� and unseen.

This article appeared in print under the headline 鈥淩eality bits鈥�