The experimental test of Bell's theorem
which the French physicist Alain Aspect conducted in 1882 attracted
the attention of those who were interested in philosophical problems
of quantum physics.(1) This experiment manifested one
of the most paradoxical characteristics of quantum system, namely
the non-separability of two contingent events, concerning the
correlation of polarized photon pairs at a distance. Both philosophers
and physicists were reminded of the celebrated debate between
Bohr and Einstein about the completeness of quantum mechanics
in the 1930s.(2) The imaginary experiment, which Einstein
used in his polemics against the alleged completeness of quantum
mechanics, became a real one through the progress of technology.
The combination of conceptual analysis and experimental tests
revived the controversy about the philosophical status of quantum
physics in the new light. The test of Bell's theorem became a
starting point for refreshed research into the nature of quantum
phenomena for those who ventured on a new cosmology beyond a positivistic
or pragmatic interpretation of quantum formulae.(3)
As philosophers and physicists do not seem to appreciate the meta-theoretical
significance of Einstein's criticism of the Copenhagen interpretation
of quantum mechanics, I shall first reconsider the so-called EPR
argument which Einstein presented with his collaborators, Podolsky
and Rosen, and then evaluate this argument in the light of experimental
tests of Bell's theorem.
The original form of the EPR argument was summed up by Einstein and his coauthors as follows(4):
In a complete theory there is an
element corresponding to each element of reality. A sufficient
condition for the reality of a physical quantity is the possibility
of predicting it with certainty, without disturbing the system.
In quantum mechanics in the case of two physical quantities described
by non-commuting operators, the knowledge of one precludes the
knowledge of the other. Then either (1) the description of reality
given by the wave function is not complete or (2) these two quantities
cannot have simultaneous reality. Consideration of the problem
of making predictions concerning a system on the basis of measurements
made on another system that had previously interacted with it
leads to the result that if (1) is false then (2) is also false.
One is thus led to conclude that the description of reality as
given by a wave function is not complete.
We can write the above argument in the form of syllogism which contains two propositions.
Proposition C (the completeness of quantum mechanics): Quantum mechanics is complete in the sense that there are no hidden parameters which explain the statistical data in a deterministic way.
Proposition S (the simultaneous reality of complementary physical quantities):
The complementary physical quantities,
to which the canonical conjugate operators correspond in the standard
formulation of quantum physics, have simultaneous reality in the
sense that we can predict with certainty their values without
disturbing the system.
The formal structure of the EPR argument is as follows:
| The Major Premise | |
| The Minor Premise | |
| The Conclusion | |
This argument is sometimes called
the EPR paradox, for it says that if we admit the completeness
of quantum physics, then we are necessarily led into the contradiction
(
), for the major premise is equivalent
to
It is noteworthy that the semantic structure of the EPR argument against the completeness of quantum mechanics is similar to Goedel's argument against the completeness of formalized arithmetic, for Goedel proved that if a formalized system of arithmetic is consistent, then it cannot be complete. As the criterions of completeness are different between formalized arithmetic and physics, this similarity only holds in an analogous sense, but it helps us to understand the meta-physical aspect of the EPR argument. Bohr seemed to understand this aspect of the argument, for he once said that he could see no reason why the prefix "meta" should be reserved for logic and mathematics and why it was anathema in physics.(5) The Bohr-Einstein debate was essentially meta-physical in the sense that they tackled the aporias of quantum physics at and beyond the boundary of human observation.
The EPR argument was not generally accepted as valid by his contemporary physicists, because it was interpreted as an argument against the indeterminacy principle established by Heisenberg. Though Einstein's earlier arguments against the Copenhagen interpretation aimed at pointing out a possibility of measuring two complementary physical quantities beyond the limit of exactitude imposed by the indeterminacy principle, the purpose of the EPR argument was not the refutation of this principle, but essentially the semantic claim that if we accept the completeness of quantum physics, then we are, through considering a suitable imaginary experiment, necessarily led to the contradiction of both accepting and not accepting the indeterminacy principle.
The imaginary experiment in the EPR argument involved a system of two particles with the wave function
As 
and
are
commuting operators, the above wave function can have the determinate
values of 
and p=0. So measuring
enables the calculation of 
without in
any way disturbing the system. There is an element of reality
corresponding to this determinate value. If we measure
instead of
, then we can also calculate
the determinate value 
which also corresponds
to another element of reality. If quantum mechanics represents
all elements of reality (the proposition C), then the position
and momentum of the second particle have simultaneous reality,
which contradicts the major premise, i.e. the principle of indeterminacy.
This principle would not be violated if we insisted that two or
more physical quantities can be regarded as simultaneous elements
of reality only when they can be simultaneously measured or predicted
with certainty. So the EPR argument presupposes that the measurement
of the second particle is independent of that of the first particle
because the distance between two particles are so great that they
may be considered as causally separable elements of reality. Making
this assumption explicit, we can reformulate the EPR argument
as follows:
The Assumption L (the separability of local elements of reality):
The physical system are separable
into two or more parts which are causally independent of each
other at an given instant. The observation of the one cannot causally
influence that of the other in so far as the four-dimensional
distance between them is space-like (dx2+dy2+dz2-c2dt2
0).
The reformed EPR argument shows that the alleged incompleteness
is proved only under the assumption of L.
| The Major Premise |  |
| The Minor Premise |  |
| The Conclusion |  |
Though Einstein mentioned this assumption toward the end of his paper, he took it for granted because the breakdown of L was so unreasonable for him. If the principle of local causality did not hold, then the partial description of the whole universe would be, strictly speaking, impossible on account of the dubious concept of a closed system in the level of quantum phenomena.
It was Bohm(6)who first explicitly stated that the assumption L was incompatible with the current theoretical structure of quantum mechanics. He even said that the name "quantum mechanics" might be a misnomer because "mechanics" is necessarily associated with L, i.e. the separability of local elements of reality.(7)
The local hidden variable theory
is formally expressed as (
), for it must
presuppose the locality thesis L and the incompleteness of quantum
mechanics. There have been many trials of hidden variable theories,
and so many counter arguments which aimed at excluding such a
possibility from the peculiarity of quantum statistics. The most
famous criticism against the hidden variable theory was given
by von Neumann who mathematically proved that the basic axioms
of formalized quantum mechanics exclude hidden variables.(8)
His proof was, however, not conclusive in the case of the hidden
variable theory which does not share the axioms of quantum mechanics.(9)
Bell proposed a crucial experiment between quantum mechanics and a local hidden variable theory.(10)He proved that there is a limit on the extent of correlation of statistical results that can be expected for any type of local hidden variable theory. The limit is expressed in the form of inequality which is now called the Bell inequality. Bell showed that quantum mechanics sometimes violates this inequality, especially in the correlation at a distance in the imaginary experiment of the EPR argument. This experiment became realizable when we replace the original version of the EPR experiment with the measurement of spin-components of two spin-1/2 particles or with the measurement of polarization of two photons. Real tests of the Bell inequality have been carried out by many groups of investigators.(11) The most conclusive was done by Aspect (1982) and the result was that the Bell inequality was really violated.
Aspect utilized the correlated photon
pairs 
and
which
counter-propagate along Oz and impinge on the linear polarization
analyzers I and II. The result
+1 and -1 are assigned to linear polarizations parallel or perpendicular
to the orientation of the polarizer, and this orientation is characterized
by a unit vector a and b.
The quantum state of the whole system can be expressed as the following superposition:
Then we can calculate the probability of each photon's polarization along a given direction. P+(a)=P-(a)=P+(b)=P-(b)=1/2
P++(a,b)=P--(a,b)=(1/2)cos2(
ab)
P+-(a,b)=P-+(a,b)=(1/2)sin2(
ab)
Let EQM be the coefficient of correlation between two quantum events at the polarizers a and b:
EQM=P++(a,b)+P--(a,b)-P+-(a,b)-P-+(a,b)=cos2(
ab)
If 
ab=0,
then EQM=P, which means perfect
positive correlation.
If 
ab=
/2,
then EQM= -1, which means perfect negative correlation.
This kind of perfect correlation
seems miraculous if we assume the completeness of quantum mechanics
and admit the coincidence between two contingent events, for we
may wonder how 
gknows"
which channel was chosen at the last moment for
.
This gmiracle" would
disappear if we succeeded in making a locally deterministic model
for the above simultaneous perfect correlation. Such a model has
to assume the hidden causal mechanism which predetermines both
results of measurements. This causal mechanism can be represented
by local hidden variable 
which has a
probability distribution
For simplicity we assume one hidden
variable 
, but we may use many hidden
variables in the following consideration by using multiple integrals.
The problem is whether or not such a hidden deterministic model
can represent the strong correlation signified by EQM in the case
that a and b
are neither parallel nor perpendicular.
Let the function A(
,a)
and B(
,b)
determine the measured values of
polarization at the polarizer a and b respectively:
A(
,a)=1
or -1 ; B(
,b)=1
or -1
Then the coefficient of correlation
d(a,b)
given by the statistical expectation value with respect to
.
With respect to four different directions a,a',b,b', we define the quantity S=E(a,b)-E(a,b')+E(a',b){E(a',b').
Then we can prove the inequality
-2
S
2
This is the Bell inequality which was tested by Aspect's experiment.(12)
Quantum physics shows that SQM based
on EQM does not satisfy this inequality. In the experimental situation
in which 
ab=
ab'=a'b'=22.50,
aa'=
bb'=450,
ab'=67.50,
we get SQM=
,
which invalidates the Bell inequality.
As quantum mechanics and any kind of local hidden variable theory predict different statistical results, the above experiment may well be called a crucial experiment. As the result was for quantum mechanics, we must conclude that we cannot make quantum mechanics complete by introducing local hidden variables.
The violation of the Bell inequality
means that the combined proposition (
)
is false,
for such a local hidden variable
theory cannot explain the correlation at a distance in the system
of two particles which have previously interacted with each other.
If we admit both the validity of the EPR argument and the experimental
test of Bell's theorem, we have to abandon L, i.e. the separability
of local elements of reality.
| The Conclusion of the Reformed EPR Argument |  ( )
|
| The Experimental Test of Bell's Theorem | 
|
| The Final Conclusion |  L
|
We must notice that the final conclusion is independent of our attitude toward the completeness versus incompleteness problem of quantum mechanics.
We cannot prove , as Einstein intended
to do, the incompleteness of quantum mechanics as a result of
the falsified premise L, but this falsification itself depends
logically upon the validity of the EPR argument and empirically
upon the violation of the Bell inequality, for the validity of
the argument is one thing and the truth of its conclusion is quite
another.
Moreover, the EPR argument makes us reconsider the nature of the indeterminacy principle, for there seems to exist no mechanical interference between the observer and the observed in the imaginary experiment concerned. This principle was originally interpreted by Heisenberg as the inevitable inexactitude of measurement due to uncontrollable mechanical interactions between the observer and the observed, but once we have verified the simultaneous correlation between distant events and admit the non-separability of local element of reality, we must amend Heisenberg's interpretation in such a way that the indeterminacy principle holds primarily on the level of the definition of quantum phenomena where the observer and the observed are not separable from each other. Bohr seemed to anticipate this view in his reply to the EPR argument(13):
Of course there is in a case like that just considered no question of a mechanical disturbance of the system under investigation during the last critical stage of the measuring procedure. But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.
Bohr, however, rejected the semantic
criterion of completeness and reality in the EPR argument, and
chose to talk only about quantum phenomena which we can define
through the macroscopic apparatus of observation. Bohr's standpoint
was that quantum mechanics did not require the depth structure
under quantum phenomena , but certainly not that it was ontologically
self-sufficient for the world-description. Rather, classical physics
was to Bohr indispensable for the definition of quantum phenomena.
It was important for Bohr to recognize that ghowever
far the phenomena transcend the scope of classical physical explanation,
the account of all evidence must be expressed in classical terms".(14)
So quantum phenomena need classical physics for their definition
in terms of experimental apparatus, whereas any single classical
model of reality cannot exhaust the varieties of quantum phenomena.
Bohr's philosophy of complementarity was pragmatic and provisional
in the sense that it sidestepped the difficult problem of quantum
measurement, i.e. how to describe "the
collapse of the wave function"
within the framework of quantum mechanics.(15) If we
want to get a unified picture of macroscopic and microscopic reality,
we must present a suitable framework of ontology which can assimilate
the main characteristics of quantum physics, especially the non-separability
of local elements of reality.