Bell Tests and Biased Samples
When people first hear about Bell's inequalities for the results
of separate quantum interactions they naturally try to think of a
conventional realistic model that will reproduce the predictions
of quantum mechanics and/or the results of actual experiments.
The cannonical "realistic" model is one in which each particle is
considered to possess a definite spin (or polarization) axis, and
measurement along a given axis will return a positive or negative
result corresponding to the SIGN of cos(theta) where theta is the
difference between the spin axis and the measurement axis. Assuming
two "coupled" spin-1/2 particles in an EPR-type experiment have
exactly opposite spin directions, this model agrees with the QM
predictions when the difference "alpha" between the two measurement
angles is 0, pi/2, and pi (i.e. the measurement angles are equal,
perpindicular, or opposite) with correlations are 0, 1/2, and 1,
respectively. Of course, for intermediate values of alpha this
basic model predicts that the correlation varies linearly, thereby
differing from the predictions of QM as shown below
1 |-----------------*-*/-
| * /
| * /
correlation | * /
| * /
1/2|----------/----------
| / * ----- basic realistic
| / * model
| / *
| / * ***** QM predictions
0 |/*_*_________________
0 pi/2 pi
alpha
This simple model is described in all elementary introductions to
quantum mechanics, along with an explanation of why this model
does not accurately represent what is observed. At this point the
typical introductory text goes on to describe the effects of
measurement inefficiency, noting that all EPR tests to date have
relied on relatively low detection efficiencies, leaving open the
theoretical possibility that the data is systematically biased in
just such a way as to mimic the predictions of quantum mechanics.
The most simplistic approach along these lines is to modify the basic
realistic model so that it gives results approximating the predictions
of QM for all values of alpha by supposing that a measurement of a
particle's spin has THREE possible outcomes: positive, negative, or
null. The "null" outcome represents cases in which the measurement
fails to resolve the spin of the particle. Further, we suppose that
a measurement is effective only if the positive or negative spin axis
is within an arc subtended by a fixed angle "beta" centered on the
measurement axis.
Note that the basic model assumed beta equals pi, from which it follows
that a measurement is ALWAYS effective. On the other hand, if beta is
some fixed value LESS than pi, and if we neglect all particle-pairs in
an EPR-type experiment when either of the spin measurements return
"null", then the correlations of the remaining pairs can approximate
the correlations predicted by QM.
For example, suppose beta equals just pi/2. This implies that each
individual spin measurement is effective (i.e., returns a non-null
result) over only half the range of possible spin axes. Now consider
an EPR-type experiment in which we measure the spins of two coupled
particles, with measurement axes that differ by alpha. Clearly if
alpha equals pi/2 the effective ranges of the two measurements are
mutually exclusive, so we will be unable to gather any correlation
data at all.
|
M1 = null | M1 = positive
M2 = positive | M1 axis M2 = null
* | * /
* | */
* | / *
* |/ *
--------------------------------
* |\ *
* | \ *
* | *\
* | * \
M1 = negative | M2 axis M1 = null
M2 = null | M2 = negative
However, for any other value of alpha the "effective quadrants" for
measurements 1 and 2 will overlap. Also, notice that if alpha is less
than pi/2 the overlap will consist entirely of disagreements, whereas
if alpha is GREATER than pi/2 the overlap will consist entirely of
agreements. Thus, assuming beta=pi/2, the correlation is 0.0 for
alpha less than pi/2, and 1.0 for alpha greater than pi/2. The pair-
measurement effectiveness is 0.0 at alpha=pi/2 and ramps up to 0.5
at alpha=0 and alpha=pi.
On the other hand, if we expand our assumed effectiveness arcs to
beta = 3pi/4 we will record only disagreements when alpha is less
than pi/4, and only agreements when alpha is greater than 3pi/4, and
the correlation will vary linearly in between (because now we can get
some overlap "on both sides" of the spin). Plots of correlation the
and pair-measurement effectiveness for beta = pi, 3pi/4, and pi/2 are
shown below.
____3pi/4___ 1 _______________________pi
1 | | / / |
| | / / |
| pi/2| / /pi | 3pi/4
corr | | / / eff |\ /
| | / / | \ /
1/2| |// 1/2| \ ___ / pi/2
| // |\ /
| / /| | \ /
| / / | | \ /
| / / | | \ /
| / / | | \ /
0 |/_____/____| 0 | \ /
-------------------------- --------------------------
0 pi/2 pi 0 pi/2 pi
alpha alpha
The correlations based on an assumption of beta = 3pi/4 clearly resemble
the cosine curve predicted by QM, and if we refine our "effectiveness"
model so that the chances of a "null" result vary smoothly with theta
we can certainly tailor the correlation curve to agree exactly with
the QM predictions.
Of course, all of this is quite familiar to anyone who has ever
thought about tests of Bell's inequalities. It's understood that
if our measurement efficiencies are low AND there is a systematic
effect such that the recorded pairs are a biased sample, then the
simple Bell inequalities are no longer relevant. Furthermore, it's
obvious that we can imagine realistic mechanisms that yield biases
such that the simple QM correlations are reproduced within the
biased sample. The question is (and has always been) whether the
hypothesis of such a bias is consistent with our observations.
Needless to say, if our measurements were affected by the type of
bias described above we would expect to see the measurement efficiency
reach a minimum at alpha = pi/2. No such minimum appears in the
data. (Aspect reported that although he found slight variations in
his measurement efficiencies [versus alpha], the variations were not
significant.)
Now, if we were really desperate, we might try to explain the absence
of a minimum in the efficiency at alpha=pi/2 by assuming that the
value of beta is pi for one of the measurements but something less
than pi for the other, in which case our measurement efficiency would
be constant. True enough, but this shows that in order to match a
particular set of QM result we not only need to invoke a systematic
bias in the measurement efficiencies, we need an *asymmetric*
systematic bias, i.e., a bias that works on only one of the two
particles in each coupled pair.
Thus, we must imagine that when two particles emerge from a singlet
state in opposite directions and interact with spin-measuring devices,
one of them exhibits a definite spin when measured at any angle, but
the other sometimes doesn't exhibit spin, if the measurement angle
happens to be nearly perpindicular to its spin axis. Admittedly most
actual experiments have involved physically asymmetric emissions, but
we would require not just any old asymmetry, but a very specific
asymmetry, i.e., absolute perfection on one arm of the experiment
and just the right amount of imperfection on the other arm to match
the predictions of QM. As John Gribbin nicely summarized in his
popular book "In Search of Schrodinger's Cat",
"It is still possible to argue that only the photons that
violate Bell's inequality are detected, and that the others
would obey the inequality, if only we could detect them.
But ... it does seem like the height of desperation to
make that argument."
Notice that, according to the logic of our naive realistic model,
we need to claim that particles with spins sufficiently near the
UP/DOWN boundary must simply disappear, i.e., they must not be
counted at all, but this is really a non sequitur. It's conceivable
that proximity to the UP/DOWN boundary might cause the particle to
exhibit NEITHER spin UP or DOWN characteristics, but this does not
imply that we would fails to detect the particle. For example, in
a Stern-Gerlach apparatus if some of the particles exhibited no
spin UP or DOWN, they would arrive at the center of the screen,
but in fact when we perform such experiments (which have been down
countless times since the 1930's) we find that ALL the particles
are deflected either UP or DOWN. Similar comments apply to the
polarization of photons.
So, what becomes of the naive "realistic spin" model? It will
doubtless go on being re-discovered by every freshman Physics
student - most of whom quickly discern its inadequacy, especially
when they attempt to integrate it with particles of differing spins,
and with other quantum interference effects such as those involved
in two-slit experiments, tests of the Stern-Gerlach effect, the
photo-electric effect, black-body radiation, and so on, all of
which are in fantastically precise agreement with modern quantum
mechanics. However, the most compelling factor for many students
ends up being the realization that, of all possible dependancies
between spin correlations and relative measurement angles, there
is a unique dependancy that minimizes the net exhibited non-
conservation of spin for all possible interactions, and that
happens to be precisely the dependancy predicted by quantum
mechanics. Thus, if QM was wrong, it could be shown that the
workings of nature are, in a very intuitively palpable sense,
arbitrarily defective.
For a related discussion, see
On the Cumulative Results of Quantum Measurements
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