Is the wavefunction ontological or epistemological?

Part 3: Bell Theorem

There are several ways to understand Bell’s 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 Bell 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 sets.

In particular, one Bell 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 Bell 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 Bell’s papers 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 reality:

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 inequality above:

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 EPR-B pair 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 particular direction.

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 Bell inequality yields:

½ sin(22.5)^2 + ½ sin(22.5)^2 >= ½ sin(45)^2

or

0.1464 >= 0.2500  IMPOSSIBLE!!!

And experiments confirm the violation of Bell’s 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).