ISIS Report 18/03/04
How Not to Collapse the Wave Function
Quantum systems are more robust than previously thought, especially
when weakly measured, with startling results.
Dr. MaeWan Ho reports
A complete version of this article with illustration and sources is
posted on ISIS members’ website. Full details here.
Collapse of the wave function
In the standard quantum theory, a quantum system is in a superposition
of states or in quantum entanglement (see Box), which is invariably destroyed
by measurement. This is referred to as the ‘collapse’ of the wave
function  of probability amplitudes  that defines the system, so the system
ends up in one definite state and no other. In case of Schrödinger’s
cat, this collapse of the wave function amounts to being found definitely dead
or definitely alive.
Quantum superposition and quantum entanglement
‘Superposition’ and ‘entanglement’ are two
of the strange properties of a quantum system. ‘Superposition’ refers
to a quantum system existing simultaneously in multiple states, even states
that, common sense tells us, are mutually exclusive. This is usually told as
the parable of Schrödinger’s cat, shut up in a box with a vial of
cyanide that at any moment might be triggered to release its deadly gas by a
radioactive decay reaction. If the cat in the box were a quantum system, then
before anyone raises the lid to look in, the cat would be described by a
quantum ‘wavefunction’ of complex quantum amplitudes, which have to
be squared to give the usual probabilities (see "The quantum information
revolution", this series). This puts the cat in a superposition of
(classically) mutually exclusive states, simultaneously dead and alive,
and all states in between.
As soon as it is ‘observed’ by opening the lid,
however, the quantum superposition is destroyed, and the system’s wave
function ‘collapses’ into a classical state. The cat is found either
dead or alive.
‘Entanglement’ describes the way different particles
can become correlated no matter how far apart they are, so that a measurement
performed on one of them instantaneously determines the state of both the
measured particle as well as the entangled particle. This too, collapses the
wavefunction that previously describes the complex probability amplitudes of
the two particles together, no matter how far apart they are. 
But could this interpretation be overly simplistic? First of all, the
‘collapse’ of the wave function only applies to the measured
property of entangled states, while the unmeasured properties remain
indeterminate. For example, a measurement to decide whether a particle is spin
up or down leaves it and its entangled partner indeterminate as to whether the
spin is oriented left or right.
Then, there are other recent indications that observation need not
destroy entanglement.
Survival of the entangled
Altewischer and coworkers in Leiden University in the Netherlands showed
that entanglement between pairs of photons can survive even when one (or both)
of the entangled photons is converted into a surface ‘plasmon’ and
then back again into a photon.
Surface plasmons are oscillating electromagnetic fields strongly
localized at the surface of a metal and associated with the collective motion
of a large number of electrons. Surface plasmons are formed on metal films
perforated by an array of holes smaller than the wavelength of the photons. The
plasmon tunnels through the holes to form a similar plasmon on the other side,
and eventually reconverts back into a photon.
Altewisher’s team sent many entangled photon pairs onto such metal
films, and although many photons are lost as the result of absorption, the
surviving photons remain entangled.
The state of a twoparticle system is ‘entangled’ when its
quantummechanical wave function cannot be factorised into two singleparticle
wave functions. This leads to one of the strangest features of quantum
mechanics, nonlocality. In other words, when one particle is measured
to have a certain property, then the corresponding property of the other
particle is instantaneously determined.
It is quite easy to get entangled photons, a starting photon can
spontaneously split into a pair of entangled photons inside a nonlinear
crystal.
Altewischer’s team wanted to find out what happens when entangled
photons are sent through opaque metal films perforated with a periodic array of
holes smaller than the wavelength of the photons. Will entangled photons
survive such treatment and remain entangled? The answer is, surprisingly,
yes.
First, they launched photons of one energy (one wavelength) into a
crystal whose particular properties cause each photon to be split into two new
photons, a process called downconversion. By the conservation of energy, each
downconverted photon has twice the wavelength and half the energy. Not only
that, spin is conserved as well as energy, so the polarization of the two
downconverted photons are always opposite to each other. For example, if one
photon is measured to be linearly polarized in the vertical direction, the
other photon will always be linearly polarized in the horizontal direction, and
so on. It is as though the two photons know about each other instantaneously.
Altewischer’s team showed that the entanglement survives even when
one (or both) of the entangled photons is converted into a surface
‘plasmon’ and then back again into a photon. Although many photons
are lost as the result of absorption, the surviving photons are still
entangled.
Given the collective nature of the surface plasmon, which involves some
10^{10} electrons, it seems remarkable that entanglement should
survive. More surprisingly still, surface plasmon modes are shortlived,
lasting for just a few femtoseconds (10^{15}s).
The key to surviving entangled, it seems, is to remain coherent by
bouncing off other coherent systems.
In the commentary on the report, William Barnes of the University of
Exeter in the UK pointed out that if we simply use a metallic mirror to reflect
an entangled photon, we would expect entanglement to survive even though we
would still be making use of the collective motion of many electrons to provide
the reflection.
In both the surfaceplasmon and simple reflection processes, we rely on
the electronelectron scattering rate being low enough to allow the electron
motion to remain coherent.
But there are even stranger quantum encounters in store.
Survival of stranger encounters
Lucien Hardy, then at Oxford University in Britain, provided a beautiful
illustration of the sort of paradox that arises in connection with quantum
entanglement through a thoughtexperiment in 1992.
An entangled particle and antiparticle pair  an electron and a
positron  is created and each flies off in the opposite direction. But their
paths are made to cross at some point by strategically placed mirrors, and
instead of annihilating each other as particle and antiparticle are supposed to
do, they somehow manage to "be" and "not to be" at the same time and location.
Hardy considered a MachZender Interferometer (MZI), a setup containing
a halfsilvered mirror, which sends a quantum particle into a superposition of
states, in that it travels down two separate arms at once: the transmitted and
reflected paths.
The interferometer later reunites the two paths to meet at another
halfsilvered mirror, which is arranged so that if the particle has had an
undisturbed journey  no encounter with any other particles or fields  it hits
detector C. But if something disturbs the particle, it may hit a second
detector, D.
What happens when two such interferometers positioned so that one arm of
the first overlaps with one arm of the second (see Fig. 1)?
Figure 1. The overlapping paths of the electron and
positron in an entangled pair
If a positron  the antiparticle of an electron – is sent through
one interferometer, and an electron through the other at the same time, the two
particles travelling along the overlapping arms should meet in an
‘annihilation region’ and destroy one another. Hardy showed that
something much stranger happens: quantum theory predicts that both D
detectors could click simultaneously. Somehow both particle and antiparticle
could disturb each other, yet fail to annihilate, in the overlapping arms. This
is Hardy’s paradox.
Most people tend to ‘resolve’ the paradox by pointing out that
such paradoxes arise only because we make inferences that do not refer to
results of actual experiments. And, if the actual measurements were performed,
then standard measurement theory predicts that the system would have been
disrupted in such a way that no paradoxical implications would arise.
But an international team of scientists led by Yakir Aharonov in Tel
Aviv University, Israel, showed that Hardy’s thought experiment could be
carried out, and that it could give new observable results, provided that
weak measurements are made.
What would a weak measurement entail? It is one that does not disturb
the system significantly, so it remains quantum coherent, and also
sacrifices accuracy. According to Heisenberg’s uncertainty relations, an
absolutely precise measurement of position reduces the uncertainty in position
to zero, but produces an infinite uncertainty in momentum. But if one measures
the position up to some finite precision, then one can limit the uncertainty in
momentum to a finite amount.
Aharonov and his colleagues imagine such weak measurements are indeed
possible, and ask what happens to Hardy’s paradox. They find the paradox
far from disappearing. The results of their theoretical measurements turn out
to be most surprising and to reveal a "deeper structure" in quantum mechanics,
which makes it "even more paradoxical".
In the double MZI setup, it is arranged that if each MZI is considered
separately, the electron can only be detected at C and the positron only at
C+. However, because there is a region where the two particles overlap, there
is also the possibility that they will annihilate each other.
But the clicking of D and D+ would be paradoxical. If D clicked, that
means the positron must have gone through the overlapping arm, otherwise
nothing would have disturbed the electron, which would have gone to C. The
same logic applies if D+ clicked, that means the electron must have gone
through the overlapping arm to disturb the positron. But, there has been no
annihilation in either case, which is paradoxical.
Alternatively, if D has clicked, the positron must have gone through
the overlapping arm. But since there was no annihilation, the electron must
have gone through the nonoverlapping arm. The clicking of D+ indicates that
the electron must have gone through the overlapping arm, and the positron the
nonoverlapping arm. But these two statements are contradictory, which ends in
paradox again.
So far, these statements are based on no measurements being made as to
which arm the positron or electron actually went. If a standard detector is put
in the path of the electron in the overlapping arm, we can find the electron
there, but the measurement itself disturbs the path, so the electron could end
up in D detector even if no positron were present. The paradox disappears.
Weak measurements however, will give different predictions. The weak
measurement does not disturb the system significantly, but it will be
imprecise. To make up for this, the measurement has to be repeated many times
in order to get as close to the real answer as possible. (Alternatively, they
can do the experiment with a large number of electronpositron pairs and
measure the total number of electrons or positrons that go through each arm.)
And to simplify the measurements, they concentrate only on the results when D
and D+ both click. The measurements, being weak can be made simultaneously
without disturbing the system or each other.
The results – based on calculating the probabilities from the
complex quantum amplitudes  show that indeed, the electron and positron,
each has a probability 1 of being in the overlapping region, and a
probability of 0 of being in the nonoverlap region. But they could not
both be in the overlapping region; and quantum mechanics is consistent
with this too – the joint probability of both being in the overlapping
region is 0.
Intuitively, the positron must have been in the overlapping arm
otherwise the electron could not have ended at D and, further, the electron
must have gone through the nonoverlapping arm as there was no annihilation.
This is confirmed by the joint probability of positron being in the overlapping
arm and electron in the nonoverlapping arm equals to 1. Similarly, D+ clicking
means that the electron must have been in the overlapping arm otherwise the
positron could not have ended at D+, and further, the positron must have gone
through the nonoverlapping arm as there was no annihilation. The joint
probability of the positron being in the nonoverlap region and electron in the
overlap region, too, is equal to 1. But these two statements are at odds with
each other, as there was only one positronelectron pair. Quantum mechanics
solves this paradox by having the joint probability of electron and positron
being both in the nonoverlapping branch equal to –1! That is being
not merely absent, but negatively present.
Finally, "the electron did not go through the nonoverlapping arm as it
went through the overlapping arm" is also confirmed – a weak measurement
finds no electrons in the nonoverlapping arm, the probability of electron
being in the nonoverlapping arm equals to 0. But we know that there is an
electron in the nonoverlapping arm as part of a pair in which the positron is
in the overlapping arm, as the joint probability of that is equal to 1. How is
it possible to find no electrons in the nonoverlapping arm? The answer is
given by the existence of the –1 joint probability of both electron and
positron pair in the nonoverlapping arms, bringing the total number of
electrons in the nonoverlapping arm to zero! So the negative presence cancels
out the positive presence, resulting in absence.
Klaus Molmer of Aarhus University in Denmark, initially sceptical, now
thinks he knows how to do actual weak measurements. He suggests probing the
locations of a pair of ions that are first cooled down to their lowest energy
state, then hit with two carefully engineered laser pulses to send them into a
superposition, which would move them to positions they should never occupy. He
set up the ions so they will always fluoresce, except when they are in this
paradoxical superposition. As soon as the fluorescence vanishes, he carries out
a weak measurement on the ions’ position using another laser. The centre
of mass should lie somewhere between the pair.
But the weak measurements show that, in the paradoxical quantum state,
the ions’ centre of mass actually lies outside this region.
Molmer thinks that most of what has been done todate with quantum
systems employs weak measurement, only physicists haven’t realised it. And
it could have practical consequences. For example, they could expose flaws in
quantum cryoptography, in which it has always been supposed that disturbance
caused by measurement would prevent eaves droppers decoding messages (see "Quantum information secure?" this
series). But an eavesdropper who uses weak measurement would escape detection,
and hence succeed in breaking the code.
All in all, reality is stranger, much stranger than we can imagine. I
particularly like the idea of being positively present, absent, or negatively
present simultaneously.
