“NOBODY understands quantum mechanics,” lamented Richard Feynman. But Anton
Zeilinger at the University of Vienna aims to prove him wrong. His research
group has demonstrated the futuristic phenomena of quantum teleportation and
quantum encryption, and these successes have encouraged Zeilinger to search for
the essence of quantum mechanics—the irreducible kernel from which
everything else flows. He believes that he has found it. If he is right, all the
mysteries of the quantum world will turn out to be inescapable consequences of a
single, simple idea.
Quantum theory describes the world with astonishing precision, whether
applied to elementary particles a hundred thousand times smaller than atoms or
to currents in superconducting rings a billion times larger. And yet it seems to
present a catalogue of intertwined conundrums. The most fundamental is
quantisation, the notion that energy, spin and other quantities only come in
discrete steps. Another enigma is the probabilistic nature of the quantum world,
at odds with the classical world of definite physical properties. Then there is
entanglement, the profound connectedness of objects and processes across large
distances, and superposition, the astonishing proposition that an electron can
be both here and there, a current can flow simultaneously clockwise and
anticlockwise, and a cat can be both dead and alive, until you look to see
which.
Physicists have anxiously devised one philosophical interpretation of quantum
mechanics after another. In the Copenhagen interpretation, the outcome of an
experiment is only revealed when the quantum system interacts with a macroscopic
apparatus in the laboratory, which eliminates all possibilities but one. The
many-worlds interpretation insists that all possible outcomes of an experiment
actually occur in as many parallel universes, but as we only occupy a single
branch of the hydra-headed multiverse, we experience only one outcome. Or, if
you prefer, there’s the guiding wave interpretation, which assigns an
undetectable “pilot wave” to each particle to steer it along a perfectly
determined path. Altogether there are at least eight serious and reputable
interpretations of the theory, which implies that no single one is
convincing.
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Zeilinger thinks that before we can truly understand quantum theory, it must
be connected in some way to what we know and feel. The problem, he says, is the
lack of a simple underlying principle, an Urprinzip. All the other
major theories of physics are based on such principles—pithy,
comprehensible maxims that anchor the formulae in the everyday world.
Take the science of heat. Though highly mathematical and abstract,
thermodynamics is based on two basic principles that can be described in
colloquial terms. The first law of thermodynamics is just the conservation of
energy: it means that there are no perpetual motion machines. The second law of
thermodynamics is simply the statement that heat tends to flow from warm objects
to cooler ones. When the stuff called energy was invented to quantify these
laws, it was strange and undefinable, and even today we don’t know what energy
is. Yet energy quickly became a robust term in daily conversation and government
policy.
The special theory of relativity is also based on two principles, namely,
“Inside a speeding transatlantic jet, you have no way of knowing how fast you
are going,” and “The speed of light shone from this jet is the same as the speed
of light from a stationary source.” That second statement is counterintuitive,
but it is simple to understand and turns out to be a stubborn experimental fact.
And general relativity, Einstein’s theory of gravity, is based on the thought
that a freely falling person feels no weight. None of these theories suffer from
the confusions of quantum mechanics.
Now Zeilinger proposes to rebuild quantum mechanics on a similar basis, to
put it in terms that need no debatable philosophy.
Perhaps it is no surprise that the terms he uses are those of information. We
live in the age of information. We depend increasingly on information
technology, our schools teach information processing and information science,
and our industry and commerce are information based. But until now, the concept
of information has only hovered on the edge of physics.
About a decade ago, John Archibald Wheeler urged that information should take
centre stage. What we call reality, he thinks, arises from the questions we ask
about it and the responses we receive. “Tomorrow, we will have learned to
understand and express all of physics in the language of information,” he
said.
The atom of information is the bit—the quantity contained in the answer
to a yes or no question. If experiments are questions we ask of nature, then the
simplest of them have yes or no answers: “Did the photon arrive here, or not?”,
“Did the counter click, or not?” We can also ask more complex questions, but
they can always be built up from simpler yes or no questions like these.
Zeilinger’s conceptual leap is to associate bits with the building blocks of
the material world. In quantum mechanics, these building blocks are called
elementary systems, and the archetypal elementary system is the spin of an
electron. The only possible outcomes of measuring an electron’s spin are “up”
and “down”. You can choose any axis to measure the spin along—vertical,
horizontal or tilted—but once that axis is chosen, only the two results
are possible, as if the electron were a spinning top that can be one way up or
the other, but can’t point to any intermediate direction. These outcomes could
just as well be labelled “yes” and “no”, or, in the fashion of digital
computers, “1” and “0”.
This system is far more general than it seems. The formulae that describe it
apply, unchanged, to every conceivable quantum-mechanical system characterised
by just two states—from polarised light and molecules with just two energy
levels to counterrotating currents in a superconducting ring. Not forgetting
that touchstone of quantum mechanics, the two-slit experiment (see “Two becomes
DzԱ”).
Zeilinger avoids the question “What is an elementary system?” and asks
instead, “What can be said about an elementary system?” His conclusion is simply
stated: an elementary system carries one bit of information.
It sounds innocuous. But the consequences of Zeilinger’s principle promise to
be breathtaking. In the first place, it contains the fact that the world is
quantised—the very starting point of quantum mechanics. Because we can
only interrogate nature the way a lawyer interrogates a witness, by means of
simple yes-or-no questions, we should not be surprised that the answers come in
discrete chunks. Because there is a finest grain to information there has to be
a finest grain to our experience of nature. This is why electrons are restricted
to fixed energy levels in atoms, why light comes in pieces we call photons, and
perhaps, ultimately, why the Universe seems to be made out of discrete
particles. To the question, “Why does the world appear to be quantised?”
Zeilinger replies, “Because information about the world is quantised.”
Less obviously, Zeilinger’s principle leads to the intrinsic randomness found
in the quantum world. Consider the spin of an electron. Say it is measured along
a vertical axis (call it the z axis) and found to be pointing up.
Because one bit of information has been used to make that statement, no more
information can be carried by the electron’s spin. Consequently, no information
is available to predict the amounts of spin in the two horizontal directions
(x and y axes), so they are of necessity entirely random. If
you then measure the spin in one of these directions, there is an equal chance
of its pointing right or left, forward or back. This fundamental randomness is
what we call Heisenberg’s uncertainty principle.
In order to progress beyond a single elementary system, Zeilinger’s principle
has to be generalised. He proposes simply that two elementary systems carry
exactly two bits of information, and N systems carry N bits. This gives us a
natural explanation for one of the most fundamental and puzzling features of
quantum mechanics—entanglement.
When, say, two electrons are entangled, it is impossible even in principle to
describe one without the other. They have no independent existence. This seems
bizarre until you use Zeilinger’s principle. Concentrating on their spins, a
two-electron system contains two bits. For example, they might be “The spins in
the z direction are parallel,” and “The spins in the x
direction are antiparallel”. The two bits are thereby used up, and the state is
completely described—yet no statement is made about the direction of spin
of one electron or the other. The entire description consists of relative
statements, or correlations. This means that as soon as one spin is measured
along a certain direction, the other one is fixed, even if it happens to be far
away.
Zeilinger’s single, simple principle leads to these three cornerstones of
quantum mechanics: quantisation, uncertainty and entanglement. What, then, of
the more formal elements of quantum mechanics such as wave functions and
Schrödinger’s equation—the bread and butter of atomic physicists? The
road promises to be long and steep, but Zeilinger and his student Caslav
Brukner, have now begun the ascent.
Physicists use Schrödinger’s equation to work out how a particle will
behave in a given situation. It governs the evolution of things called wave
functions, inside a bizarre abstract arena called Hilbert space. Because Hilbert
space makes use of imaginary numbers, based on the square root of minus one,
these numbers—the amplitudes of the wave functions—have to be
squared to produce a real, observable quantity, such as the probability of a
particle being in a given place. It is not an intuitively obvious way of
describing things.
Zeilinger and Brukner discard it. Instead, they introduce a three-dimensional
space they call information space. The relationship between 2D Hilbert space and
3D information space is a bit like the relationship between an accurate
perspective drawing and a real, three-dimensional object. This new space is much
closer to our reality, as its axes correspond to the answers of yes or no
questions about an elementary system. An electron’s spin can be measured, or
quantised, along the x, y, or z axes of real space,
which gives the three dimensions of information space a clear correspondence
with reality. In other two-state systems the connections are not so obvious, but
three independent propositions will always exhaust the possibilities.
Any quantum system has to describe how states change over time, so the point
in information space has to move. It seemed natural to Zeilinger and Brukner to
have the point move as if it were a real, classical object. So they used the
mechanical equation that governs the motion of bullets and billiard balls. When
translated back into its equivalent form in Hilbert space, it turns out to be
none other than Schrödinger’s equation.
Their next aims are to generalise this approach from one elementary system to
many, and to find a way to include continuous variables such as position and
speed. Eventually, they must find a way to account for the information contained
in quantities like mass and charge.
Even if this programme succeeds, physicists may dismiss it as old wine in new
bottles. Why should they adopt the principle if it doesn’t tell us anything new?
But in fact it already does. In October 1999, Zeilinger and Brukner turned their
theory around to propose a new measure of information.
A new theory of quantum information is needed if we are to handle the quantum
computers of the future. This technology promises one day to perform
calculations far faster than ordinary computers can, by exploiting the ability
of quantum systems to be in more than one state at a time.
Physicists call the building blocks of their planned quantum computers
“qubits”. A qubit is simply an elementary system such as an electron spin.
Because a qubit can be in a superposition of 1 and 0, it must hold not only
classical information, but some more elusive quantum kind of information too.
Many practitioners feel that ordinary information theory must be contained in
quantum information theory.
The number of classical bits in a system has traditionally been evaluated
using a formula derived by the American engineer Claude Shannon. Say your system
is a hand of cards. If you wanted to e-mail a friend to describe your hand,
Shannon’s formula gives the minimum amount of information you’d need to include.
But Zeilinger and Brukner noticed that it doesn’t take into account the order in
which different choices or measurements are made.
This is fine for a classical hand of cards. But in quantum mechanics,
information is created in each measurement—and the amount depends on what
is measured when—so the order in which different choices or measurements
are made does matter, and Shannon’s formula doesn’t hold. Zeilinger and Brukner
have devised an alternative measure that they call total information, which
includes the effects of measurement. For an entangled pair, the total
information content in the system always comes to two bits.
Without Shannon’s theory, progress in telecommunications during the second
half of the 20th century would have been far slower. Perhaps total information
will become as important in the 21st century.
Zeilinger’s principle is a newborn baby. If its fate is anything like that of
Planck’s century-old energy quantum, years will pass before it grows up and
gains acceptance in the mainstream of physics. But if it does, it will transform
physics as thoroughly as its venerable predecessor.
According to Richard Feynman, there is one experiment that exposes “the only
mystery” of quantum mechanics. Light from a single source is split into two
beams that travel along different paths. Where the beams recombine, two
detectors measure how the two waves interfere with each other: detector S fires
if the beams interfere constructively and detector D fires if they interfere
destructively, cancelling each other out. For a beam with no interference,
either S or D will fire, each with a 50-50 probability. When the two paths have
the same length, the waves will be in phase where they recombine and the beams
will interfere constructively, so detector S keeps firing and D never fires.
Now suppose the light is so feeble that photons can only travel through the
apparatus one at a time. That interference remains. The startling implication is
that each photon has to travel along both paths simultaneously. The same goes
for beams of electrons, neutrons, atoms, or molecules. Zeilinger has even seen
this happening to buckyballs—big, football-shaped carbon molecules.
But if instruments are installed to measure which path the photons travel
down, the detectors start firing randomly: interference is destroyed. How,
except by magic, can this be reconciled with the previous experiment on the same
apparatus? How do photons know whether to go down only one path, or both?
Zeilinger’s answer is that our choice of measurement is putting that
information into the photon. But it can only carry one bit. So if we arrange the
experiment so that the photon is destined to trigger detector S, that bit is
used up, and we can have no knowledge of which path it traversed. On the other
hand, if we decide to know which path it travelled, we cannot predict whether S
or D will fire.
This experiment highlights another troubling aspect of quantum mechanics,
called the measurement problem. Each photon seems to undergo a mysterious
metamorphosis from a quantum wave to a classical particle in the act of
measurement. But according to Zeilinger’s principle, we simply cannot know
enough about the photon to call it either wave or particle. Zeilinger’s
elementary system is no more than a carrier of information.
Two becomes one
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Further reading:
A Foundational Principle for Quantum Mechanics
by Anton Zeilinger, Foundations of Physics,
vol 29, p 631 (April 1999)
www.quantum.at