Tuesday, 29 March 2011

Does Bell's Inequality rule out local theories of quantum mechanics?

In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics.  This so-called "EPR Paradox" has led to much subsequent, and still ongoing, research.  This article is an introduction to EPR, Bell's Inequality, and the real experiments that have attempted to address the interesting issues raised by this discussion.
One of the principal features of quantum mechanics is that not all the classical physical observables of a system can be simultaneously well defined with unlimited precision, even in principle.  Instead, there may be several sets of observables that 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 observables".
A well known example is position and momentum.  You can put a subatomic particle into a state of well-defined momentum, but then the value of its position is completely ill defined.  This is not a matter of an inability to measure position to some accuracy; rather, it's an intrinsic property of the particle, no matter how good our measuring apparatus is.  Conversely, you can put a particle in a definite position, but then the value of its 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 value of its momentum more and more precisely.  But the particle can never have well-defined values of both position and momentum at the same time.  When physicists speak of how much "knowledge" they have of two noncommuting observables in quantum mechanics, they don't mean that those observables both have well-defined values that are not quite known; rather, they mean the two observables do not have completely well-defined values.
(Technically speaking, the situation is a little more complicated.  Even for observables that don't commute, it is sometimes possible for both to have well-defined values.  Such subtleties are very important to those who examine the derivation of Bell's Inequality in great detail in order to find hidden assumptions.  For the purposes of this short article, we'll overlook these finer points.)
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 is 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 an undefined spin value 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 that would demonstrate a paradox that 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 will subsequently be defined by a single quantum mechanical state.  (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 wave function, 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 initial 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 the second system, even though the systems are no longer physically linked in the traditional sense of local coupling.
(By "local" is meant that influences between the particles must travel in such a way that they pass through space continuously; i.e. the simultaneous disappearance of some quantity in one place cannot be balanced by its appearance somewhere else if that quantity didn't travel, in some sense, across the space in between.  In particular, this influence cannot travel faster than light, in order to preserve relativity theory.)
QM creates the puzzling situation in which the first measurement of one system should "poison" the first 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 wave function".  But the rules of QM do not require this interpretation, and several other perfectly valid interpretations exist.)  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.  For example, after the neutral-pion decay, we can wait until the two photons are light years apart, and then "simultaneously" measure the x-spin of the photons.  QM suggests that if say the measurement of photon 1's x-spin happens first, then this measurement must instantaneously force photon 2 into a state of well-defined x-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.  We can accept the postulates of QM as a fact of life, in spite of its seemingly uncomfortable coexistence with special relativity, or we 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, but that we didn't know that information, perhaps because it cannot be known in principle, or perhaps because QM is currently incomplete.
So, EPR postulated that the existence of such "hidden variables", some currently unknown properties, of the systems should account for the discrepancy.  Their claim was that QM theory is incomplete: it does not completely describe physical reality.  System II knows all about System I long before the scientist measures any of the observables, thereby supposedly consigning the other noncommuting observables to obscurity.  Furthermore, they claimed that the hidden variables would be local, so that no instantaneous action at a distance would be necessary.  Niels Bohr, one of the founders of QM, held the opposite view that there were no hidden variables.  (His interpretation is known as the "Copenhagen Interpretation" of QM.)
In 1964 John Bell proposed a mechanism to test for the existence of these hidden variables, and he developed his famous inequality as the basis for such a test.  He showed that if the inequality were ever not satisfied, then it would be impossible to have a local hidden variable theory that accounted for the spin experiment.
Using the example of two photons configured in the singlet state, consider this: in the hidden variable theory, after separation, each photon will have spin values for each of the three axes of space, and each spin will have one of two values; call them "+" and "−".  Call the axes x, y, z, and call the spin on the x-axis x+ if it is "+" on that axis; otherwise call it x−.  Use similar definitions for the other two axes.
Now perform the experiment.  Measure the spin on one axis of one photon 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 x, y and z.
Next, look at the statistics.  Perform the measurements with a number of sets of photons.  Use the symbol N(x+, y−) to designate the words "the number of photons with x+ and y−".  Similarly for N(x+, y+), N(y−, z+), etc.  Also use the designation N(x+, y−, z+) to mean "the number of photons with x+, y− and z+", and so on.  It's easy to demonstrate that for a set of photons
(1)   N(x+, y−) = N(x+, y−, z+) + N(x+, y−, z−)
because the z+ and z− exhaust all possibilities.  You can make this claim if these measurements are connected to some real properties of the photons.
Let n[x+, y+] be the designation for "the number of measurements of pairs of photons in which the first photon measured x+, and the second photon measured y+".  Use a similar designation for the other possible results.  This is necessary because this is all that it is possible to measure.  You can't measure both x and y for 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[x+, y+] <= n[x+, z+] + n[y−, z−].
Additional inequality relations can be written by just making the appropriate permutations of the letters x, y and z and the two signs.  This is Bell's Inequality, and it is proved to be true if there are real (perhaps hidden) variables 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 that confirmed 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 judgment.
The subject is alive theoretically as well.  Eberhard and later Fine uncovered further subtleties in Bell's argument.  Some physicists argue that it may be possible to construct a local theory that does not respect certain assumptions in the derivation of Bell's Inequality.  The subject is not yet closed, and may yet provide more interesting insights into the subtleties of quantum mechanics.



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