Quantum Nonlocality (Bell Experiments)

This is an attempt at explaining a very interesting property of quantum physics – nonlocality.  Nonlocality means objects can influence one another “instantaneously” even when separated by large distances.  This is a concept that even Einstein had trouble accepting.

What’s explained here is a real experiment that provides evidence of nonlocality.  We’ll try to keep it as simple as possible but include enough information to allow you to see the key idea.   

We’ll talk about this result in terms of measuring the so-called spin of a certain object.  All you really need to know about spin for now is that you can measure the spin of this object in any direction (up, down, left, right, north, north-east, etc.) and the result will always be either +1 or -1.

Now, let’s talk about the experiment.  In this experiment we’ll have 2 of these objects, call them A and B.  These objects have been “prepared” together in a special way (don’t worry about this now, more to come), and then the two objects are separated from one another.  Let’s say object A is in front of you and B is on the moon.  You measure the spin of A in some direction and someone on the moon will measure the spin of B in another direction.  The experiment is repeated several times and each time we record 3 things: the spin of A (+1 or -1), the spin of B, and the angle between the measurement directions.  For example, if you measured +1, the man on the moon measured -1, and you both measured along the same direction, then it would be recorded as +1, -1, 0degrees.  That’s all there is to the experiment!  The interesting things show up when you analyze the results after repeating this experiment many times, so let’s look at those results.

This experiment has been repeated many times, in many places, by different people and as long as the objects were prepared together in the special way mentioned we find the following patterns in the results. 

1.       No matter what direction is chosen, the spin of A is +1 half of the time (-1 half of the time), like a coin flip.  The same is true for B.

2.       Every time the measurements are made in the same direction, the resulting spins are opposite:  if you measure the spin in the same direction then whenever A is +1 B is -1, and whenever A is -1 B is +1.

3.       When there is a 90 degree difference in the measurement directions, then B is +1 half the time when A is +1.  Similarly B is +1 half the time when A is -1.

4.       When there is a 45 degree difference in the measurement directions, then B is -1 85% of the time when A is +1.  

So there is a clear relationship between the measurement results on A and those on B, but what makes this relationship happen?  One explanation is that when A was measured it affected B, causing the combined results to be related.  This at first seems absurd because A and B are so far away from one another, and any message sent from one to the other would need to travel “instantaneously,” faster than the speed of light.  A seemingly better explanation is that, when A and B were together at the start of the experiment the results were already specified for any particular measurement direction, and those specifications give the relationships/probabilities listed above.  However, the latter option just doesn’t work out.  Let’s investigate why.

Let’s assume the results were pre specified and then see what goes wrong.  We’ll take 1000 pieces of paper and write specifications on each one so that they give the probabilities we described above.  For simplicity we’ll only specify the results in three directions:  your “up” direction, your “right” direction (90 degrees from your up direction), and the direction midway between those two (45 degrees from each) (we’ll call the latter “up-right”).  We’ll specify the results for both A and B in each of these three directions, i.e. each sheet of paper will say something like A=+1up, B=-1up, A=+1up-right, B=-1up-right, A=-1right, B=+1right.  It turns out that it’s impossible to do this while respecting the relationships 1-4 listed above!  Let’s see why.

We’ll organize these papers into a square area, and put all the ones in with A=+1up in the top half of the square.  We know all the remaining ones have A=-1up (and B=+1up from bullet 2 above).  Furthermore, we know half the papers in the top half (250) have B=+1right and half have B=-1right (from bullet 3 above).  Similarly for the specifications in the bottom half of the square.  We’ll put all the specifications with B=-1right (and A=+1right) in the left half of the square.  Now, let’s think about how many specifications have A=+1up-right.  From the fourth bullet we know B=-1up-right for 85% of the specifications in which A=+1up (i.e. 425 in the top half of the square).  Similarly, we know B=-1up-right for 85% of the specifications in which A=+1right (i.e. 425 in the left half of the square).  Since at most 250 of the B=-1up-right specifications can be in the top left corner, there must be at least 425-250=175 in the top right corner.  Similarly, there must be at least 175 in the bottom left corner.  Hence, there must be at least 250+175+175 = 600 specifications with B=-1up-right.  This is a contradiction to rule 1, which said that half of the specification were +1 (hence half -1) in any given direction.  So pre-specifying the results doesn’t work!!!

		Figure 1.  Picture of the specifications organized into a square.