The Free Will Theorem

Today we prove Conway-Kochen free will theorem:

SPIN + TWIN + MIN + (very limited) free choice of the experimenters 

= free choice of the particles

SPIN is just a basic statement about the behavior of a spin-1 particle. It states that angular momentum operators exist and they behave in such a way to make the 101 property be true (see below).

TWIN states that it's possible to entangle two particles, such that their spins, when measured in the same direction, are opposite.

MIN is stated rather obscurely. I believe it means that it is impossible for an event to depend on another event outside of its past light cone. This is basically what special relativity states.

Free choice is defined as "non-functional", that is, not described by a function. In this interpretation, to say I have no free choice in going left or right, is to say that there is a function $f$, such that the direction I am going is $f($everything in my past lightcone$)$.

This proof goes in two steps. The first step is the Kochen-Specker Theorme uses only SPIN. The second step uses TWIN and MIN to construct an entanglement separated by a very long distance (like all those Bell-inequality experiments), and then assume (a very limited amount of) free choice of the experimenters, but not the particles, to get a contradiction.

Step 1: Kochen-Specker Theorem (1966)

This is well-known and I will direct you to plus magazine's proof. First read this, then read this. For those who know a bit more quantum mechanics, here's what 101 property means: Consider a spin-1 particle. Let $S_x$ be the operator of the angular momentum along vector $x$ for the particle, then $S_x$ has three possible eigenvalues: $\hbar, 0, -\hbar$. Normalize by setting $\hbar = 1$, we find that $S_x^2$ has two possible eigenvalues: $0, 1$. Then, it can be shown that for any triple of orthogonal vectors $x, y, z$, we have $S_x^2+S_y^2+S_z^2 = 2$, and so the measurement results must be one of $(1, 1, 0), (1, 0, 1), (0, 1, 1)$.

Notice that since $S_x^2 = S_{-x}^2$, we can safely consider a direction as defined by a line through the origin, rather than a vector.

Another note: sometimes, the configuration of 33 lines is called the Peres configuration. It's easy to verify that, if we represent each line as a vertex, and connect two vertices iff they represent orthogonal lines, then we obtain a graph with 72 edges, making up 16 triangles (corresponding to triple-orthogonal-lines) and 24 edges that do not make up any triangle.

The Stanford Encyclopedia contains more variations and ways to escape the conclusion of the Kochen-Specker theorem.

Step 2: Add entanglement (2009)

This step is a cleaned-up version of The Strong Free Will Theorem (Conway and Kochen, 2009).

Take two experimenters A, B, and separate them far enough so that during the entirety of the experiment, they cannot affect each other. Say, one is in the Milky Way, and the other in Andromeda. They are at rest relative to each other. 

Now we entangle two spin-1 particles a, b, and send a to A and b to B, and they perform the following experiment:

A measures on a the spin-square along three orthogonal lines $x, y, z$. Simultaneously (in their inertial frame), B measures on b the spin-square on one direction $w$. They complete their experiments fast enough so that there is no way they could affect each other.

Now, suppose the experimenters have enough free will to choose $x, y, z, w$, but the particles do not, then we have two well-defined functions, describing the particles $\theta_a(x, y, z), \theta_b(w)$, such that $\theta_a(x, y, z)$ ranges over $\{(1, 1, 0), (1, 0, 1), (0, 1, 1)\}$, and $ \theta_b(w)$ ranges over $\{0, 1\}$. 

We can make a further restriction on the experimenters' free choice, by limiting $x, y, z$ to only one of the 16 triple-orthogonal-lines in the 33 lines of the Peres configuration, and limiting $w$ to only one of the 33 lines. What does it mean to limit their free choice? It means that we could make it so that $\theta_a(x, y, z)$ is undefined when $x, y, z$ falls outside these 16 possibilities. In other words, we make it so the experimenter A is forced to choose from these 16 possibilities, and experimenter B is forced to choose from these 33 possibilities. (9 bits of free will... and that's enough.)

This is what it means for the experimenters to be free, but the particles not: 

• If the experimenters do not have this amount of "free choice", then $\theta_a(x, y, z)$ does not have to be well-defined for all 16 possibilities, any more than $WhatISee($I go at the speed of light$)$ is well-defined. We are only asking a little freedom: A only needs to be able to choose freely out of 16 options, and B only 33.
• If the particles were free, then the functions $\theta_a, \theta_b $ won't exist, by our current definition of "free choice".

Now, since the two particles are entangled, we have $\theta_a(x, y, z) = (\theta_b(x), \theta_b(y), \theta_b(z))$. Since as $x, y, z$ ranges over the 16 triple-orthogonal-lines, they also range over the 33 lines, and so $\theta_b$ satisfies the 101 rule on the 33-line Peres configuration, which is impossible. 

This proof-by-contradiction shows that particles also have some (very limited) free will, in the sense that we can't have both $\theta_a$ and $\theta_b$ as defined previously, but must "improvise" in some sense to satisfy the constraints imposed by SPIN+TWIN.