Achilles in the Quantum Universe: The Definitive History of Infinity by
Richard Morris, Henry Holt, New York, $25, ISBN 0805047794

RICHARD MORRIS has given us an engaging and accessible book about the role
that infinite quantities play, and have played, in physics. The Achilles of the
title is, of course, the one who overtakes the tortoise in the ancient paradox
proposed by Zeno of Elea: the tortoise has a head start, so to catch up,
Achilles must first reach the point where the tortoise was when the race began.
But by then the tortoise has moved on and reached a slightly more distant point,
so to catch up, Achilles must next reach that point—and so on ad
infinitum. Hence Achilles must perform an infinite number of physical actions
before catching up with the tortoise.

The traditional perplexed conclusion attributed to Zeno is that motion is
impossible. A better conclusion is that, when the laws of physics allow it, we
can and do perform infinite numbers of certain types of actions. Therefore we
should not exclude infinite quantities on principle from our conception of the
physical world. We must take them seriously, and that is what Achilles in
the Quantum Universe does.

It is a pity that the book confines itself to physics, barely touching upon
the mind-boggling mathematical theory of transfinite numbers, with their
infinite hierarchy of higher-order infinities. Admittedly—this being only
a finite book—the author had to draw the line somewhere, but I disagree
with him that transfinite numbers are not relevant here because they have “no
applications in the natural sciences”. Many of the underlying philosophical
issues are the same and actually, transfinite numbers have even crept into
physics on rare occasions.

The great German mathematician David Hilbert once wrote that “the literature
of mathematics is glutted with inanities and absurdities which have had their
source in the infinite”. The story of the bitter disputes surrounding
transfinite numbers, which still have not been fully resolved, would have
complemented and illuminated the story that Morris tells.

Never mind. There is excitement enough in the infinities of physics, such as
the infinite distances in an open universe, the infinite forces that exist
inside black holes and at the Big Bang, the infinite energies that cancel each
other out in quantum field theory, or the infinity of universes postulated by
quantum cosmology. In all these cases, we have to keep asking ourselves: does
infinity make sense? Does it make sense to deny that infinity makes sense? If
you are in any doubt that the answer to these questions are yes and no
respectively, read this book.

I wonder whether the subtitle “The Definitive History of Infinity” was the
author’s idea or merely a reverberation in the publisher’s mind of the magical
phrase “A Brief History of Time”. [Hawking’s book still appears in bestseller
lists around the world years after publication.] The author must surely know
that his book isn’t the definitive history of anything. Indeed, although it is
full of interesting historical snippets, it is not really a history at all but
an exploration of ideas—and probably all the better for that.

The most economical way to send a message may be for the receiver to
send himself half of it, according to two American physicists. Charles
Bennett of IBM’s Thomas J. Watson Research Center, York-town Heights, New
York, and Stephen Wiesner hit on their bizarre communications system after
thinking about sending messages as streams of individual photons. They concluded
that it is possible to use a photon to carry two bits of information instead
of just one.

But intuitively, if you want to send a message in the form of an object
which can be put into one of N indistinguishable states, the maximum number
of different messages you can send is N. For example, a photon can have
two distinguishable polarisation states: ‘left-handed’ or ‘right-handed’.
So if you were to send a message by altering the polarisation of a single
photon and transmitting it, it seems obvious that at most you could send
two distinguishable messages: one by sending a left-handed photon and the
other by sending a right-handed one.

But Bennett and Wiesner say this intuition is false. They have found
that if your communications medium can be put into one of N distinguishable
states then, by using a special pre-transmission set-up, you can transmit
up to N₂ different messages. This is equivalent to cramming a
two-bit message into each physical bit used for transmission. So the polarisation
of a photon can be used to send up to four different messages instead of
the two you might expect.

This magic is performed by harnessing a unique property of quantum particles
such as photons. The property is known as ‘non-locality’, and was proved
theoretically by the late John Bell of CERN, the European centre for particle
physics, and later corroborated experimentally by a group of physicists
led by Alain Aspect in France.

In classical physics, any physical system is completely described by
a description of its subsystems and how they interact. But this not true
in quantum physics. There are other properties of a system which, though
they are accessible to experiments on the system as a whole, are not reflected
in the outcome of any combination of separate experiments on its subsystems.
These properties are called quantum correlations.

Quantum correlations can exist between objects that are any distance
apart, which is why they are called non-local. Although the term ‘correlation’
is borrowed from probability theory, quantum correlations need not be probabilistic.
In some situations, they can carry information that, in principle, is retrievable
with certainty.

Bennett and Wiesner have devised a hypothetical quantum communications
system in which the receiver reads a two-bit message from two physical bits.
But, unusually, only one of those bits – the transmitted bit – has come
from the sender of the message. The other – the reference bit – never leaves
the receiver.

The diagram shows the sequence of events. Curiously, it is the receiver
who makes the first move. He prepares two bits in a state that contains
certain non-local quantum correlations, stores one as a ‘reference’ bit
and transmits the other to the sender. When the sender gets this pre-transmission
bit, he stores it.

To ensure that the non-local correlation between the two bits is maintained,
each bit must be kept isolated from its surroundings. When it is time to
send the message, the sender then performs one of four physical operations
on the stored bit before transmitting it back to the receiver (Physical
Review Letters, vol 69, p 2881).

These operations, which are specified by Bennett and Wiesner, affect
the quantum correlations in one of four ways. These are not distinguishable
by measurements on the transmitted bit and the reference bit separately.
But by measuring both bits jointly, the receiver can determine which of
the four operations the sender performed, and so receive one of four messages.

Because of the pre-transmission, the total number of bit journeys is
the same as it would be classically, so in that respect Bennett and Wiesner’s
communications system gains us nothing. But the fact that half the journeys
are in the opposite direction could be useful in situations where transmission
in the desired direction is more expensive than in the opposite direction.

Most importantly, the pre-transmission can take place before the sender
has even thought of the message. So it may be possible to devise a communications
system that hoard bits during off-peak periods and then used them during
peak periods, so doubling its peak capacity.

As a bonus, the transmitted message is automatically ‘quantum encrypted’
– the Heisenberg uncertainty principle prevents anyone who does not possess
the reference bit from reading the message (see ‘Quantum keys for keeping
secret’, żìĂš¶ÌÊÓÆ”, 16 January). If they try to measure the polarisation
of the transmitted bit, they will get a purely random answer, regardless
of the values of the two bits the polarisation effectively carries. However,
this method of quantum cryptography is vulnerable to an eavesdropper who
intercepts and replaces the pre-transmission as well as the actual transmission.

After the pre-transmission, the parties can, alternatively, choose to
send a message in the opposite direction to that shown in the diagram. To
do this, the ‘receiver’ performs one of four operations on the reference
bit and transmits it to the ‘sender’.

Like most practical applications of the quantum theory of computation,
this one is not yet technologically feasible. Bennett and Wiesner, and Anton
Zeilinger, suggest that this scheme could be implemented using nonlinear
optical crystals. But whether or not it turns out to be of value in transmitting
messages, such an unexpected upset to ideas about communications is of undoubted
theoretical significance.

Many candidates have been proposed for the invisible, or ‘dark’, matter,
which astronomers believe makes up 90 per cent of the mass of the Universe.
But the existence of most of the proposed forms of matter is difficult to
confirm or deny. However, if the dark matter is made of planet-sized black
holes, there is a way of telling definitively, according to two theoretical
physicists in the US.

Jeff Peterson of Princeton University and Toby Falk of the University
of California, Santa Barbara, believe it may be possible to detect an effect
which combines beautifully one of the largest-scale phenomena known to cosmology
– gravitational lensing – and one of the most delicate of miscroscopic phenomena
– quantum interference. The technique has the potential to reveal objects
as small as 10 million tonnes at the distance of nearby galaxies, says the
physicists.

The idea that dark matter is in the form of objects of planetary size
or even less – stars that are too small to shine, or asteroids, or black
holes formed in the big bang – has become increasingly attractive as other
candidates have been ruled out experimentally. Such tiny objects would have
to exist in vast numbers, so some would lie close to the lines of sight
between the Earth and compact sources of light, such as stars or quasars.

This opens up the possibility of observing an effect known as gravitational
lens interference, says Peterson and Falk. The effect occurs when light
passing a massive object is bent towards the object by its gravity. This
bending is minuscule but twice that predicted by Newton’s theory of gravity.
The effect was first measured by Arthur Eddington in 1919 and was the first
experimental test of Einstein’s general theory of relativity. Eddington
found that the apparent positions of stars in the sky changed if their light
passed close to the Sun on its journey to the Earth.

In the most dramatic form, an entire galaxy acts as the lens. Astronomers
have observed several instances in which the light of a distant quasar is
‘lensed’ by the gravity of a foreground galaxy. The quasar’s light comes
to us along more than one route, creating multiple images.

Gravitational lensing is at the heart of Peterson and Falk’s method
for detecting tiny black holes and similar bodies. They suggest looking
at light from distant stars that follows two paths round an intervening
black hole. The path difference will be only a few wavelengths of the light,
which may cause optical interference between the two beams.

Optical interference, which is a so-called quantum phenomenon, can be
observed when light from a point source falls on a pair of parallel slits
in a screen. If the slits are very narrow and closely spaced, a pattern
consisting of alternating bright and dark strips form on a second screen
placed beyond the slits.

At any point on the screen, light waves arrive via alternative paths
of different lengths and so ‘interfere’ with each other. If they are exactly
out of phase, they cancel each other out; if they are exactly in phase,
they reinforce each other. Bright and dark rgions appear where the lenghts
of the two paths differ by an even and odd number of half-wavelengths respectively.

The ‘quantumness’ of this phenomenon arises because light actually travels
as particles called photons, which are far smaller than the spacing between
the slits. Contrary to the predictions of classical physics, photons will
continue to arrive and build up the same pattern of light and dark regions
no matter how weak the beam is. This happens even if the photons pass through
the apparatus one at a time, and so cannot ‘cancel each other out’, or ‘interfere’
with each other in any way.

John Wheeler was the first to suggest that optical interference and
gravitational lensing might be combined. Wheeler, then at the University
of Texas at Austin, considered his suggestion merely a ‘thought experiment’
to illustrate the fact that quantum phenomena are not always confined to
the small scales at which they are most familiar.

Wheeler noted that just as light from a point source in the laboratory
can reach a detector by passing through either of two slits, so light from
a distant start can reach a given point on Earth by two possible routes
passing either side of the massive object. The intensity of light which
an experimenter measures will vary periodically as the position of the Earth
changes.

Wheeler couched his idea as a thought experiment because he believed
that the geometry of all known gravitational lenses made it impractical.
The reason is that each type of light source has a characteristic ‘coherence
time’, during which the waves that it emits are closely in phase. Over periods
longer than the coherence time, phases become radomised, and this stops
any interference effects.

The typical different in lengths between two paths round a galaxy is
of the order of light-weeks, which means that the difference in travel time
round the two paths is very much longer than the coherence time of starlight.
This makes it hopeless to try to detect intereference due to light travelling
along alternative paths round something as big as a galaxy.

Peterson and Falk’s idea of searching for much smaller gravitational
lenses might make Wheeler’s though experiment both practicable and useful.
If the dark objects are there, they say, and we watch distant sources with
detectors that are sufficiently sensitive and which can respond rapidly
enough, sometimes – when the Earth and one of the lenses happen to move
into the right configuration – we should see the light intensity vary periodically.
We will observe optical interference.

If nature does provide situations in which gravitationl lens intereference
is detectable, it will be an exquisitely sensitive probe. Peterson and Falk
point out that the technique could reveal information about the sources
as well as the lensing objects. For example, it could detect bright patches
within sources such as quasars with a much greater resolution than any other
known method.

On a more philosophical level, the observation of this effect would
once again illustrate the power and robustness of physical theories. Imagine:
a proton sets out from a quasar by two alternative routes. A billion years
and 10 ²² kilometres later, the two routes converge at our telescope.
Not only can be detect the fact that the routes differ in length by a few
mircrons, we can reliably use this difference to tell us that the photon
passed by a small asteroid a million years ago!

Of course, it all depends on whether there really are vast numbers of
dark objects out there. We may soon know.

The device, built by Charles Bennett and John Smolin at IBM, uses very
faint flashes of light to transmit messages over an unprotected communication
channel. It allows the sender and receiver to agree on a code without ever
meeting in person. If an eavesdropper tries to monitor the signal in transit,
the uncertainty principle, an inescapable property of the quantum world,
ensures that the signal is disturbed in such a way that the sender and receiver
are alerted.

In building the device, which uses the polarisation of photons to encode
information, Bennett and Smolin have created the first information processing
device with capabilities that exceed those of the Universal Turing Machine.
This is a theoretical model proposed by the English mathematician Alan Turing
in 1936. All existing computers are, in effect, based on the Turing Machine.

Although both quantum theory and the modern theory of computation are
more than 50 years old, it is only recently that scientists have studied
in detail the implications of one for the other. Quantum computers should
be faster at some tasks and also able to tackle tasks that classical computers
cannot.

Building general-purpose quantum computers, however, is far beyond the
reach of present technology. The main problem is maintaining ‘quantum coherence’
during the intricate self-interactions of a typical computation. Loosely,
a coherent quantum system is one that is ‘isolated’ from the outside world.
Coherence is destroyed by any interaction in which the environment, or any
variable not participating in the computation, effectively ‘measures’ or
acquires any information about the system.

At present, physicists are able to maintain coherence only in very simple
systems, such as photons or electrons travelling from a diffraction grating
to a detector. Bennett and Smolin have nevertheless managed to use such
a system to carry out a new type of computation.

Suppose that Alice and Bob want to exchange messages in such a way that
an eavesdropper, Eve, who might be listening, cannot understand what they
are saying. If Alice and Bob already have some secret information in common,
they can use thisas a ‘key’ in a so-called ‘cryptographicalgorithm’. This
will scramble theirmessages so that Eve, who may knowthe algorithm but does
not know the key, cannot decipher it in a reasonable time.

The larger the amount of shared secret information that Alice and Bob
start with, the harder they can make Eve’s task. In fact, if the messages
they send are shorter in total than their secret key, they can make eavesdropping
impossible. But if they have no shared secret information to begin with,
they will not be able to agree on a key without letting Eve know it, too.
Can they still communicate securely? This is where ‘public key’ cryptography
comes in.

Classical public key cryptography was first proposed in 1976 by Diffle
and Hellman, and shortly afterwards Rivest, Shamir and Adelman published
the first practical version. Each participant secretly choosesa private
key, which is never transmitted to anyone else, and calculates from it a
public key.

Using Bob’s public key, Alice, or anyone else, can transmit messages
which only Bob can easily decipher in normal circumstances. Obviously, the
private and public keys are related mathematically, and in principle the
‘private’ key could be deduced from the public one. The trick is to make
computing the public key from the private key relatively simple, but the
reverse computation prohibitively time-consuming.

No classical public-key cryptographic system has yet been proved to
be secure. There is the danger that some ingenious mathematician will come
up with a quick way of calculating the function in both directions. Another
problem is the increasing speed of computers. A computation that would occupy
a present-day computer for a million years might be done in an hour on a
home computer in 20 years’ time.

Using classical information technology, it is impossible to design a
public-key cryptographic system that is not vulnerable to advances in pure
mathematics or computer technology.

Eve can simply copy all the messages that Alice and Bob send each other.
Then, if Alice and Bob had no secret information in common, Eve can find
out just as well as Bob can exactly what operation Alice uses to scramble
her messages. To unscramble them, one need only undo all these operations.
Bob may have a secret quick way of doing this, but that is the only advantage
he can possibly have over Eve. Given enough time, Eve will be able to read
whatever Bob can in Alice’s messages.

Bennett’s idea, invented with Gilles Brassard of the University of Montreal,
is to use the quantum uncertainty principle to prevent any eavesdropper
from reading a message without being detected. The uncertainty principle
says that if certain physical variables are prepared or measured precisely,
then certain other variables become multi-valuedor ‘fuzzy’. In particular,
the values that those other variables used to have can become inaccessible
to any measurement, however careful.

This applies to the linear polarisation of a photon (whether its polarisation
is horizontal or vertical, say) and its circular polarisation (whether it
is spinning in a right- or left-handed sense about its direction of travel).
It is not difficult to prepare photons in states of pure linear or circular
polarisation, but never both simultaneously. It is also possible to measure
precisely either the linear or the circular polarisation. But any measurement
of the linear polarisation makes the former value of the circular polarisation,
whatever it may have been, inaccessible; measuring it will give a random
result. Similarly, measuring the circular polarisation randomises any subsequent
measurement of the linear.

If Alice sends a stream of photons to KK Bob, she could use their individual
linear polarisations to transmit a message in binary code, using ‘horizontal’
to mean ‘zero’ and ‘vertical’ to mean ‘one’. Bob could measure the linear
polarisation (by passing the photons through a vertical polaroid filter
and using a photomultiplier to detect which have passed through) and thus
read the message. But this method of communication is insecure. There is
nothingto prevent Eve from measuring the polarisations herself and transmitting
appropriately polarised photons on to Bob so that the eavesdropping will
remain undetected.

Similarly, the circular polarisations of the photons could be used to
send insecure messages, by using ‘left-handed’ to mean ‘zero’ and ‘right-handed’
to mean ‘one’. But now suppose that Alice sends a message using these conventions
but choosing randomly whether to use the linear or the circular polarisation
code. Eve cannot intercept such a message and pass it on unchanged. Because
of the quantum uncertainty principle she cannot measure both the linear
and the circular polarisation. And if, for a particular photon, she chooses
wrongly and measures the polarisation variable that was not used to encode
the message, the bit that was stored in that photon will be irretrievably
randomised. This is the effect that makes the QPKD system work.

When Bob receives each photon, he chooses randomly which of its polarisation
variable to measure. In this way, he randomises about half the bits. But
the other half, provided that they were not measured or tampered with en
route, will have the values that they had when transmitted by Alice. When
Bob has received a large number of bits, he publicly exchanges information
with Alice about some proportion of them. Bob says which variable he measured
and Alice tells him, in about half the cases, what result he should have
got.

Then they do a statistical test to find out whether there was any eavesdropping.
If there was none they compare notes, again, publicly, about the remaining
photons. The photons for which they happened to choose the same variable
contain secret information that cannot be known by any eavesdropper.

The prototype QPKD machine is capable of transmitting its secret keys
over a distance of some 50 centimetres. Plans are in hand for an improved
model using optical fibres that would have a range of tens or perhaps hundreds
of metres. Relay stations cannot be used because they would have to measure
the photons, lose quantum coherence, and so be vulnerable to eavesdropping.
So it may be some time before quantum cryptographic technology becomes widely
useful in real applications.

The lasting significance of QPKD will be for the foundations of computer
science. The Turing Machine is no longer a universal model for practical
computations.