3/PH/SB Quantum Theory - Week 8 - Dr. P.A. Mulheran
BELL’S INEQUALITY AND THE NATURE OF REALITY
8.1 Introduction

The nature of reality can be tested through the correlations between results of
measurements performed on pairs of particles that are part of a coherent system.
This was first suggested by Einstein, Podolsky and Rosen (EPR) in 1935 who
hoped to demonstrate the inadequacies of quantum theory and the Copenhagen
Interpretation.

Consider first the story told by John Bell about Bertlmann’s socks as an example
of everyday correlation. Bertlmann always wore odd socks so that if you see that
one of his socks is red than you know immediately that the other is blue.

Of course there is absolutely no mystery here because we believe in local realism
in classical physics. On that particular day Bertlmann’s first sock was always red
and the other always blue, regardless of when or how we observe them.

However were the observation of socks a quantum phenomenon then we might
start to get worried. Before seeing his feet, our knowledge of the socks is
completely uncertain and the wave function describes a linear superposition of
the possible outcomes of the experiment. Furthermore upon observing that the
first sock is red, the wave function instantaneously collapses to the appropriate
eigenfunction of the sock-operator (!) which means the second sock is now
definitely blue. In other words observing sock number one determines
instantaneously the nature of sock number two, despite the fact that they are
spatially separated onto different legs!

This is precisely the sort of “spooky action at a distance” that EPR were worried
about. Surely the Copenhagen Interpretation must be wrong; real socks do not
behave like that and so neither should the quantum equivalent. Perhaps what is
wrong is that there is something missing from quantum theory. Could there be
hidden variables that we cannot detect but which nevertheless determine from
the outset the results of all experiments?

Surprisingly Bell in the 1960’s showed how experiments could be devised to
rule out the possibility of local hidden variables dictating the results of these
experiments. These experiments are based on correlated results on the spin of
particles or the polarisation of photons rather than the colour of socks (of
course!). Subsequently many experimental tests, including those by Aspect in
the early 1980’s, confirm the predictions of the standard quantum theory and
rule out local hidden variable theories.

Therefore it seems that there is no local realism in real world experiments
performed on systems governed by quantum mechanics. It implies that “spooky
action at a distance” which violates relativity is real, and that the classical view
of physics as an objective science of local reality is fundamentally wrong.
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3/PH/SB Quantum Theory - Week 8 - Dr. P.A. Mulheran
8.2 Measuring the spin of coherent pairs: the correlation coefficient

Consider pair of spin-half particles (e.g. protons) moving apart (along the y-axis as
usual) after a scattering experiment where the total angular momentum of the
system is strictly zero (these conditions can be constructed in real experiments).
Then if one particle is measured to have a z-component of spin up, the other must
be z-down and vice versa.

Now imagine that we measure the spin of the first particle with an SGZ apparatus,
so finding Z1-UP or Z1-DOWN at random. Immediately afterwards we measure
the spin of the second particle with a Stern-Gerlach apparatus SG which lies in
the x-z plane and is oriented at an angle  to the z-axis. Using quantum theory we
can easily calculate the probability that if the first result is Z1-UP the second result
is 2-UP:
PUU  sin 2   2  .

Similarly we can calculate the probabilities PUD for observing Z1-UP and 2DOWN, PDU for Z1-DOWN and 2-UP, and PDD for Z1-DOWN and 2-DOWN:
PUD  cos 2   2  , PDU  cos 2   2  , PDD  sin 2   2  .

We can now calculate the correlation coefficient between the two results, that is
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S Z 1 S 2 , using these probabilities:
2
1

 PUD    PDD  PDU    cos  .
2

the average value of the product
1
CQT      PUU
2
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3/PH/SB Quantum Theory - Week 8 - Dr. P.A. Mulheran
8.3 A Local Hidden Variable Theory

Let us construct a simple (but ultimately wrong) local hidden variable theory
for the spins which preserves classical notion of realism. We assume that the
spin of a particle is a real vector oriented in space at random, and if its zcomponent is positive then a measurement of Sz results in +1 (in units of /2).
Similarly if it is measured by an SG apparatus it gives a result +1 if its spin
vector makes an angle between    and    with the z-axis. Then for this
hidden variable theory
PUU 
and so





, PUD  1  , PDU  1  , PDD  .




1
1
 2
C LHVT      PUU  PUD    PDD  PDU  
 1.
2
2
 
This result is very different to that of the standard quantum theory,
CQT  CLHVT ,
and an experiment could in principle be performed to rule out this particular
hidden variable theory. However Bell realised that by considering the correlation
between sets of experiments one can rule out all possible local hidden variable
theories!
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3/PH/SB Quantum Theory - Week 8 - Dr. P.A. Mulheran
8.4 Bell’s Inequality

Let us suppose that there is some local hidden variable(s)  that determine
beforehand what the result of the spin measurements will be. In this way we can
preserve the classical idea of realism. Then the result of measuring Sz for particle
number one is merely a function of the hidden variable, S Z1    , and likewise
the result of measuring S for particle number two is another function of the
hidden variable S 2    .

The hidden variables can take a range of possible values which follows a
normalised frequency distribution p(). We are free to choose what this
distribution is and furthermore what functions S Z1    and S 2    we use.
They can be made to reproduce the results of individual measurements, but as
we shall see they cannot be made to reproduce the results of certain correlation
experiments!

Under these rules the correlation coefficient becomes
CB     SZ 1   . S 2   . p  . d ,
where SZ1 and S2 take values of 1 (in units of /2).

We can now derive Bell’s inequality which concerns the correlation coefficients
between three separate experiments performed with (SGZ and SG), (SGZ and
SG), and (SGZ and SG-):
CB    CB    CB      1 .
This means that the correlation coefficients calculated using all possible local
hidden variable theories must obey the above inequality.

Recalling now the correlation coefficient calculated using quantum theory,
CQT      cos  ,
we see that if we choose  = 2 the left hand side of Bell’s inequality becomes
cos   cos 2   cos   1, 0   

2
.
In other words Bell’s inequality is clearly violated for many experimental set-ups
according to standard quantum theory.

Therefore careful experiments can in principle be used to distinguish between
the standard quantum theory and any local hidden variable theory. In fact
experimental tests clearly confirm the predictions of quantum theory and thus
rule out all local hidden variable theories!
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3/PH/SB Quantum Theory - Week 8 - Dr. P.A. Mulheran
8.5 The Aspect Experiment

Experimentalists typically perform the correlation tests suggested by Bell’s
inequality and its variants using pairs of photons with perpendicular polarisation,
because photon detectors are more efficient than the Stern-Gerlach apparatus.

The Aspect version from 1982 switches the photons between pairs of polarisers
on both sides of the experiment, so in total four different correlation coefficients
are measured. The experiments clearly show that the appropriate version of
Bell’s inequality for this series of experiments is violated by 10%, and the
results agree with the quantum theory predictions to within the experimental
error.

The Aspect experiment also changes the polarisers at such a high rate that no
possible signal travelling with a speed below that of light could pass the
information about the precise set-up between the photons at either side of the
apparatus before they are measured.

The conclusion that quantum systems cannot possibly be described by a local
hidden variable theory seems inescapable. However the experimental results are
consistent with the Copenhagen Interpretation: our knowledge changes after the
measurement of the first photon’s polarisation and so the wave function
instantaneously collapses thereby dictating the result of second experiment.

This action at a distance violates the relativistic speed limit, and can therefore
conflict with causality. Event 1 does not necessarily precede event 2 to an
observer in an inertial reference frame that is moving at a constant speed with
respect to the experiment. However no-one has yet managed to communicate
information faster than light using these correlations.

We conclude that Einstein’s “spooky action at a distance” appears to be real, and
our classical notion of physics as an objective science describing a universe that
“really exists”, with every object really located in a small region of space-time,
appears to be false. Clearly this is not the end of the story, and we must wait for
further experimental and theoretical progress to clarify what physics is really
about!
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