Is the wavefunction ontological or epistemological?
There are several ways to understand
theorem. Let’s start with a simple but relevant coin game. Suppose two people,
say Alice and Bob want to play the following game: each of them has a coin and
they go in separate rooms. Then each of them flip their coin and record the
outcome. Also they try to guess their partner’s experimental outcome. They win
if both of them guessed right. Suppose this
game is played N times and suppose Alice and Bob agree ahead of time on a
common strategy to maximize their odds of winning this games, but they cannot
communicate during the actual games. What would be the best possible odds of
winning the game? A natural first guess would be 25% because the coin flips are
independent and each can guess right 50% of the time. However, there are two
strategies of reaching 50% overall (try to figure out the solution). The 50% correlation limit is Bell’s limit
(or the local hidden variable
limit) in this particular case.
There are many
inequalities and they establish the maximum possible correlations of classical
separated systems unable to send signals. However, using quantum mechanics resources one can achieve higher
correlations. This can be checked experimentally and the major merit of Bell’s
theorem is that it provides a way to settle the issue of the existence of local
hidden variable theories by experimental means. One way to understand why
quantum mechanics can achieve higher correlations is by invoking quantum
superposition. But the actual root cause of superposition is the existence of
Hilbert spaces which demands that the logic of nature is the logic of
projections, and not the usual Boolean logic. In particular the logical
operators “OR” and “NOT” are behaving strangely in quantum mechanics. To see
how this happens, we need to start with the usual Venn diagrams of classical
In particular, one
inequality (based on an idea by d’Espagnat) can be nicely explained pictorially
using three overlapping circles corresponding to three sets: A, B, and C:
From the picture one can easily see that [A and not B] + [B and not C] >= [A and not C]
This is my favorite example of
inequality and the interesting fact is that in a particular case, Nature (and
quantum mechanics) does violate it.
So how can we translate this in an actual physical system? We’ll follow one of
from his famous book: Speakable and
Unspeakable in Quantum Mechanics (http://www.amazon.com/Speakable-Unspeakable-Mechanics-Collected-philosophy/dp/0521368693):
Bertlmann’s socks and the nature of
Suppose we have a source of pair particles in total spin zero as in the
EPR-B experimental setting.
Also suppose we measure the spin on 0, 45, and 90 degrees angles. From the
The number of particles that could pass at 0 and not at 45
The number of particles that could pass at 45 and not at 90
The number of particles that could pass at 0 and not at 90
Since in an
the spins are always anticorrelated, it is impossible to detect the left and
the right particles spin oriented in the same direction. Hence we can switch
the “not part” for one particle with measuring the other particle on that
Now the inequality becomes:
The probability of one particle passing at 0 and the other at 45
The probability of one particle passing at 45 and the other at 90
The probability of one particle passing at 0 and the other at 90
It is time to plug in quantum mechanics’ mathematical formalism and compute those probabilities. For an angle alpha between the two measurement directions, the probability is ½ [sin(alpha/2)] ^2
Plugging this in the
½ sin(22.5)^2 + ½ sin(22.5)^2 >= ½ sin(45)^2
0.1464 >= 0.2500 IMPOSSIBLE!!!
And experiments confirm the violation of
inequalities precisely as quantum mechanics predicts. The explanation for
nature’s strange correlations cannot be explained by any local causal mechanism.
Regarding correlations, nature is nonlocal. For some time people got spooked by
this, but now the emphasis is on engineering: exploiting this correlation nonlocality
for practical applications.
In the next post I’ll attempt to introduce an intuition on all this and illustrate in a geometric way the Hilbert space (responsible for superposition and strange correlations).