The curse of the attractive observer

Imagine you have a magnet and you want to measure its orientation. You design an apparatus with two polarizations along a solid rod. At any instant it can be in state + with a probability p and in state - with a probability 1-p. Picture yourself tossing a weighted coin labeled + and - on each side every second. A probability p means after an hour or so, having tossed the coin N times, you will have obtained close to p*N times + and (1-p)*N times  - .
When left alone, the detector relaxes to p = 0.5, i.e., it's positively and negatively polarized an equal number of times during any large enough interval of time. If the polarization is something like a electric tension and we read it from a voltmeter (which measures the average voltage over some short time), we'll view on its screen the value V = (+1)*p + (-1)*(1-p) = 2 * 0.5 -1 = 0.
Let's now say the detector is placed in such a way that it makes an angle x with the direction along which the magnet is pointing. Then p will tend to p = 1 (V = 1) if the detector is aligned with the magnet and to p = 0 (V = -1) if they are anti-aligned, and to p = (cos x +1) / 2 in general, so that

  V = (cos x +1) / 2 - [1 - (cos x +1) / 2] = cos x

which is the component of the magnet along the rod. Thanks to this, our detector allows us to fully measure the spatial orientation of the magnet: you first measure V along one direction, then V along its perpendicular and you combine the results. This figure sums it up:

We have plotted the evolution of V and p over time, so you can track what's going on in the detector as it performs a measurement at an angle of 45º. As you can see, V tends to cos 45º = 0.71 :

Now consider that you're not measuring a big magnet but a tiny one. So tiny, that actually the influence of your detector on the magnet is no longer negligible. Whenever you turn on your detector and it polarizes positively or negatively, the magnet will try to point in the same direction. Then what can we expect from this complicated interaction? Well, if you measure the direction of the magnet long enough so that you can tell its value with certainty (that is, V has reached a stable value) then so will the magnet have aligned itself with your detector. That means only two well defined outcomes are possible: V =1 and V = -1.

We ran a simulation with a very simple set of equations satisfying the properties described above. For each angle we ran the experiment a 1000 times and plotted the percentage of times the detector converged to a positive polarization. Here's our result:

The plotted curve is (cos x +1) / 2 . The result may appear at first as being the same as before but after close examination it is profoundly different. An observer trying to determine the orientation of a large magnet will indeed measure at the end of each experiment a value V that follows the curve above. However, an observer trying to measure a very small magnet will always measure a polarization V = 1 or -1 at the end of each experiment, even though an average of V over many experiments performed in the same conditions will result in the drawn curve. A very important consequence is that if you try to determine the spatial direction of your magnet by doing two measurements along two perpendicular directions you won't come to any conclusion at all, since the first measurement will ruin it for the second one.

A subject using such a tool to perceive the world around him will come to the puzzling conclusion that microscopic magnets have an inherent randomness to them, and that they can only be in two possible states. It is a curious fact that the peculiar properties described above happen to be exactly the ones found when you try to measure the spin of an electron. In this case, the weird quantum mechanical behaviour of particles can be fully understood using this classic analog we just studied. One might even wonder if the microscopic behaviour we see in electrons has something to do with the limits of how very delicate things can create discernible structures in very bulky ones.