Entangled Choices

The essence of quantum entanglement (as exhibited in things like 
EPR experiments and tests of Bell's inequalities) is that the joint
probabilities for combinations of spacelike-separated events depend 
on the choices of particular measurement operations that are applied
to entangled systems at those separate events, and moreover that 
this dependence is non-linear.  It is the non-linearity that makes 
it impossible to account for the dependence in conventional terms.

In order to convey any sense of quantum entanglement, we must first
introduce a CHOICE of measurements to be made on each of the two
coupled systems, and then we need to examine the correlations 
resulting from various combinations of choices.  For example, suppose
we produce a pair of objects in such a way that any one (and only one)
of three measurements, A, B, or C, can be performed on its contents.
Whichever measurement is performed, the result will be either 0 or 1.

Several pairs of objects are prepared in this way, and one member 
of each pair is sent off to Mars while the other stays here.  Then 
in both locations (here and Mars) we perform one of the three 
allowable measurements on each object and record the results.  Our 
choices of measurements (A, B, or C) may be arbitrary, by flipping 
coins or whatever.  We find that, regardless of which measurement we 
decide to make, the chances are 50% of getting "1", and the folks on 
Mars discover the same thing.  This is all that either of us can 
determine separately.

However, when we bring all the results together and compare them in 
matched pairs, we find the following correlations

                               Earth
                            A    B    C
            
                      A     0   3/4  3/4
                Mars  B    3/4   0   3/4
                      C    3/4  3/4   0

The numbers in this matrix indicate the fraction of times that the
Mars and Earth results agree (both 0 or both 1) when the indicated
measurements have been made on the two members of a matched pair
of objects.  Notice that if Earth and Mars happenned to choose to 
make the same measurement for a given pair of objects, the results 
NEVER agree.  In other words, they are always the opposite (1 and 0, 
or 0 and 1).  Also notice that the overall probability of agreement 
if both measurements are selected at random is 1/2.

This shows the true nature of quantum entanglement, which is of an
entirely different character from classical correlations that may
exist between spacelike separated events.  For example, classically
we could prepare each pair of objects in advance to always give 
the same results, and this correlation would persist even after 
the objects have become separated.  This is the kind of distant 
correlations that we deal with constantly in everyday life.  However,
in the case described above there is no way (classically) of preparing
the pairs of objects in advance of the measurements such that they 
will give the joint probabilities listed above.  To see why, notice 
that each object must be ready to respond to any one of the three 
measurements, and if it happens to be the same measurement as is 
selected on its matched partner, then it MUST give the opposite 
answer.  Hence if the Earth object will answer "0" for measurement
A, then the Mars object MUST answer "1" for measurement A.  Likewise 
for the other measurements, so there are only eight ways of preparing
a pair of envelopes

                      Earth   Mars
                      A B C   A B C
               
                  a   0 0 0   1 1 1
                  b   0 0 1   1 1 0
                  c   0 1 0   1 0 1
                  d   0 1 1   1 0 0
                  e   1 0 0   0 1 1
                  f   1 0 1   0 1 0
                  g   1 1 0   0 0 1
                  h   1 1 1   0 0 0

These preparations, and ONLY these, will yield the perfect anti-
correlation when the same measurement is applied to both objects.
Unfortunately, if we simply randomly select one of these eight 
preparations (with equal probabilities) for each pair of objects, 
we won't match the other correlations predicted by quantum mechanics.
Instead we get
                               Earth
                            A    B    C
    
                      A     0   1/2  1/2
                Mars  B    1/2   0   1/2
                      C    1/2  1/2   0

Notice that the overall probability of agreement if the two 
measurements are selected randomly is NOT 1/2 (as quantum mechanics
says it must be).  Instead, it is 1/3, so this isn't right.  

Still we might imagine that some other selection strategy for choosing
from the eight possible preparations might give the right overall 
results.  However, regardless of our strategy, the overall preparation
process must result in some linear convex combination of the eight 
cases, i.e., there must be positive constants a,b,..,h whose sum is 
1 and such that the "agreement probabilities" in the first table are
satisfied.  This implies, among other things, that

    c+d+e+f = 3/4        b+d+e+g = 3/4        b+c+f+g = 3/4

Adding these up gives  2(b+c+d+e+f+g) = 9/4, and so the sum of the
coefficients b through g is 9/8, which exceeds 1.  Hence there is
NO linear combination of those eight preparations that can yield
the joint correlations predicted by quantum mechanics.  This is a
typical "Bell inequality", and its violation illustrates that the 
questions presented by the phenomena of quantum mechanics are deeper
and more profound than is sometimes realized.

Nevertheless, it remains true that no superluminal transfer of
information is implied.  This is actually one of the most interesting
features of this analysis, because it shows that in some circumstances
our classical understanding demands that superluminal communication
"must have occurred" in order to give the observed results, even
though in fact there is no *effective* transfer of information at 
all.  See Quantum Entanglement and Bell's Theorem for a more 
detailed discussion of these issues.