The EPR Paradox and Bell's Inequality Principle    updated 31-AUG-1993 by
-----------------------------------------------    original by John Blanton

        In 1935 Albert Einstein and two colleagues, Boris Podolsky and
Nathan Rosen (EPR) developed a thought experiment to demonstrate what they
felt was k of completeness in quantum mechanics.  This so-called "EPR
paradox" has lead to much subsequent, and still on-going, research. This
article is an introduction to EPR, Bell's inequality, and the real
experiments which have attempted to address the interesting issues raised
by this discussion.

        One of the principle features of quantum mechanics is that not all
the classical physical observables of a system can be simultaneously known,
either in practice or in principle.  Instead, there may be several sets of
observables which give qualitatively different, but nonetheless complete
(maximal possible) descriptions of a quantum mechanical system.  These sets
are sets of "good quantum numbers," and are also known as "maximal sets of
commuting observables."  Observables from different sets are "noncommuting

        A well known example of noncommuting observables are position and
momentum.  You can put a subatomic particle into a state of well-defined
momentum, but then you cannot know where it is - it is, in fact, everywhere
at once.  It's not just a matter of your inability to measure, but rather,
an intrinsic property of the particle.  Conversely, you can put a particle
in a definite position, but then it's momentum is completely ill-defined.
You can also create states of intermediate knowledge of both observables:
If you confine the particle to some arbitrarily large region of space,
you can define the momentum more and more precisely.  But you can never
know both, exactly, at the same time.

        Position and momentum are continuous observables.  But the same
situation can arise for discrete observables such as spin.  The quantum
mechanical spin of a particle along each of the three space axes are a set
of mutually noncommuting observables.  You can only know the spin along one
axis at a time.  A proton with spin "up" along the x-axis has undefined
spin along the y and z axes.  You cannot simultaneously measure the x and y
spin projections of a proton. EPR sought to demonstrate that this
phenomenon could be exploited to construct an experiment which would
demonstrate a paradox which they believed was inherent in the
quantum-mechanical description of the world.

        They imagined two physical systems that are allowed to interact
initially so that they subsequently will be defined by a single Schrodinger
wave equation (SWE).   [For simplicity, imagine a simple physical
realization of this idea - a neutral pion at rest in your lab, which decays
into a pair of back-to-back photons.  The pair of photons is described
by a single two-particle wave function.]  Once separated, the two systems
[read: photons] are still described by the same SWE, and a measurement of
one observable of the first system will determine the measurement of the
corresponding observable of the second system.  [Example:  The neutral pion
is a scalar particle - it has zero angular momentum.  So the two photons
must speed off in opposite directions with opposite spin. If photon 1
is found to have spin up along the x-axis, then photon 2 *must* have spin
down along the x-axis, since the total angular momentum of the final-state,
two-photon, system must be the same as the angular momentum of the intial
state, a single neutral pion.  You know the spin of photon 2 even without
measuring it.] Likewise, the measurement of another observable of the first
system will determine the measurement of the corresponding observable of
second system, even though the systems are no longer physically linked in
the traditional sense of local coupling.

        However, QM prohibits the simultaneous knowledge of more than one
mutually noncommuting observable of either system.  The paradox of EPR is
the following contradiction:  For our coupled systems, we can measure
observable A of system I [for example, photon 1 has spin up along the
x-axis; photon 2 must therefore have x-spin down.] and observable B of
system II [for example, photon 2 has spin down along the y-axis; therefore
the y-spin of photon 1 must be up.] thereby revealing both observables for
both systems, contrary to QM.

        QM dictates that this should be impossible, creating the
paradoxical implication that measuring one system should "poison" any
measurement of the other system, no matter what the distance between
them. [In one commonly studied interpretation, the mechanism by which
this proceeds is 'instantaneous collapse of the wavefunction'.  But
the rules of QM do not require this interpretation, and several
other perfectly valid interpretations exist.]  The second system
would instantaneously be put into a state of well-defined observable A,
and, consequently, ill-defined observable B, spoiling the measurement.
Yet, one could imagine the two measurements were so far apart in
space that special relativity would prohibit any influence of one
measurement over the other.  [After the neutral-pion decay, we can wait
the two photons are a light-year apart, and then "simultaneously" measure
the x-spin of photon 1 and the y-spin of photon 2.  QM suggests that if,
for example, the measurement of the photon 1 x-spin happens first, this
measurement must instantaneously force photon 2 into a state of ill-defined
y-spin, even though it is light-years away from photon 1.

        How do we reconcile the fact that photon 2 "knows" that the x-spin
of photon 1 has been measured, even though they are separated by
light-years of space and far too little time has passed for information
to have travelled to it according to the rules of Special Relativity?
There are basically two choices.  You can accept the postulates of QM"
as a fact of life, in spite of its seemingly uncomfortable coexistence
with special relativity, or you can postulate that QM is not complete,
that there *was* more information available for the description of the
two-particle system at the time it was created, carried away by both
photons, and that you just didn't know it because QM does not properly
account for it.

        So, EPR postulated the existence of hidden variables, some so-far
unknown properties, of the systems should account for the discrepancy.
Their claim was that QM theory is incomplete; it does not completely
describe the physical reality.  System II knows all about System I
long before the scientist measures any of the observables, and thereby
supposedly consigning the other noncommuting observables to obscurity.
No instantaneous action-at-a-distance is necessary in this picture,
which postulates that each System has more parameters than are
accounted by QM. Niels Bohr, one of the founders of QM, held the opposite
view and defended a strict interpretation, the Copenhagen Interpretation,
of QM.

        In 1964 John S. Bell proposed a mechanism to test for the existence
of these hidden parameters, and he developed his inequality principle as
the basis for such a test.

        Use the example of two photons configured in the singlet state,
consider this:  After separation, each photon will have spin values for
each of the three axes of space, and each spin can have one of two values;
call them up and down.  Call the axes A, B and C and call the spin in the A
axis A+ if it is up in that axis, otherwise call it A-.  Use similar
definitions for the other two axes.

       Nw perform the experiment.  Measure the spin in one axis of one
particle and the spin in another axis of the other photon.  If EPR were
correct, each photon will simultaneously have properties for spin in each
of axes A, B and C.

        Look at the statistics.  Perform the measurements with a number of
sets of photons.  Use the symbol N(A+, B-) to designate the words "the
number of photons with A+ and B-."  Similarly for N(A+, B+), N(B-, C+),
etc.  Also use the designation N(A+, B-, C+) to mean "the number of photons
with A+, B- and C+," and so on.  It's easy to demonstrate that for a set of

(1)    N(A+, B-) = N(A+, B-, C+) + N(A+, B-, C-)

because all of the (A+, B-, C+) and all of the (A+, B-, C-) photons are
included in the designation (A+, B-), and nothing else is included in N(A+,
B-).  You can make this claim if these measurements are connected to some
real properties of the photons.

        Let n[A+, B+] be the designation for "the number of measurements of
pairs of photons in which the first photon measured A+, and the second
photon measured B+."  Use a similar designation for the other possible
results.  This is necessary because this is all it is possible to measure.
You can't measure both A and B of the same photon.  Bell demonstrated that
in an actual experiment, if (1) is true (indicating real properties), then
the following must be true:

(2)    n[A+, B+] <= n[A+, C+] + n[B+, C-].

        Additional inequality relations can be written by just making the
appropriate permutations of the letters A, B and C and the two signs.  This
is Bell's inequality principle, and it is proved to be true if there are
real (perhaps hidden) parameters to account for the measurements.

        At the time Bell's result first became known, the experimental
record was reviewed to see if any known results provided evidence against
locality. None did. Thus an effort began to develop tests of Bell's
inequality. A series of experiments was conducted by Aspect ending with one
in which polarizer angles were changed while the photons were `in flight'.
This was widely regarded at the time as being a reasonably conclusive
experiment confirming the predictions of QM.

        Three years later Franson published a paper showing that the timing
constraints in this experiment were not adequate to confirm that locality
was violated. Aspect measured the time delays between detections of photon
pairs. The critical time delay is that between when a polarizer angle is
changed and when this affects the statistics of detecting photon pairs.
Aspect estimated this time based on the speed of a photon and the distance
between the polarizers and the detectors. Quantum mechanics does not allow
making assumptions about *where* a particle is between detections. We
cannot know *when* a particle traverses a polarizer unless we detect the
particle *at* the polarizer.

        Experimental tests of Bell's inequality are ongoing but none has
yet fully addressed the issue raised by Franson. In addition there is an
issue of detector efficiency. By postulating new laws of physics one can
get the expected correlations without any nonlocal effects unless the
detectors are close to 90% efficient. The importance of these issues is a
matter of judgement.

        The subject is alive theoretically as well.  In the 1970's
Eberhard derived Bell's result without reference to local hidden variable
theories; it applies to all local theories.  Eberhard also showed that the
nonlocal effects that QM predicts cannot be used for superluminal
communication.  The subject is not yet closed, and may yet provide more
interesting insights into the subtleties of quantum mechanics.


1.  A. Einstein, B. Podolsky, N. Rosen:  "Can quantum-mechanical
description of physical reality be considered complete?"
Physical Review 41, 777 (15 May 1935).  (The original EPR paper)

2.  D. Bohm:  Quantum Theory, Dover, New York (1957).  (Bohm
discusses some of his ideas concerning hidden variables.)

3.  N. Herbert:  Quantum Reality, Doubleday.  (A very good
popular treatment of EPR and related issues)

4.  M. Gardner: Science - Good, Bad and Bogus, Prometheus Books.
(Martin Gardner gives a skeptics view of the fringe science
associated with EPR.)

5.  J. Gribbin:  In Search of Schrodinger's Cat, Bantam Books.
(A popular treatment of EPR and the paradox of "Schrodinger's
cat" that results from the Copenhagen interpretation)

6.  N. Bohr:  "Can quantum-mechanical description of physical
reality be considered  complete?" Physical Review 48, 696 (15 Oct
1935).  (Niels Bohr's response to EPR)

7.  J. Bell:  "On the Einstein Podolsky Rosen paradox" Physics 1
#3, 195 (1964).

8.  J. Bell:  "On the problem of hidden variables in quantum
mechanics" Reviews of  Modern Physics 38 #3, 447 (July 1966).

9.  D. Bohm, J. Bub:  "A proposed solution of the measurement
problem in quantum  mechanics by a hidden variable theory"
Reviews of Modern Physics 38  #3, 453 (July 1966).

10.  B. DeWitt:  "Quantum mechanics and reality" Physics Today p.
30 (Sept 1970).

11.  J. Clauser, A. Shimony:  "Bell's theorem: experimental
tests and implications" Rep.  Prog. Phys. 41, 1881 (1978).

12.  A. Aspect, Dalibard, Roger:  "Experimental test of Bell's
inequalities using time- varying analyzers" Physical Review
Letters 49 #25, 1804 (20 Dec 1982).

13.  A. Aspect, P. Grangier, G. Roger:  "Experimental realization
of Einstein-Podolsky-Rosen-Bohm gedankenexperiment; a new
violation of Bell's inequalities" Physical  Review Letters 49
#2, 91 (12 July 1982).

14.  A. Robinson: "Loophole closed in quantum mechanics test"
Science 219, 40 (7 Jan 1983).

15.  B. d'Espagnat:  "The quantum theory and reality" Scientific
American 241 #5 (November 1979).

16. "Bell's Theorem and Delayed Determinism", Franson, Physical Review D,
pgs. 2529-2532, Vol. 31, No. 10, May 1985.

17. "Bell's Theorem without Hidden Variables", P. H. Eberhard, Il Nuovo
Cimento, 38 B 1, pgs. 75-80, (1977).

18. "Bell's Theorem and the Different Concepts of Locality", P. H.

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