quantum mechanics - What does John Conway and Simon Kochen's "Free Will" Theorem mean?

The way it is sometimes stated is that

if we have a certain amount of "free will", then, subject to certain assumptions, so must some elementary particles."(Wikipedia)

That is confusing to me, but it seems to be an amazing theorem. It has been interpreted as ruling out hidden variable theories, but there is still some dissent. Lubos has a good discussion of it on his blog in the birthday blog for John Horton Conway. I assume it means that the outcomes of microscopic measurements are not deterministic.

What does the theorem assume and what does it prove?

Answer

I shall attempt here to give an explanation of the meaning of the theorem with a limited background. Issues such as the validity of the proof I shall leave aside.

The Free Will Theorem (assuming SPIN, TWIN, and FIN)]. "If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responses of the particles are equally not functions of the information accessible to them."

This theorem is a combination of ingredients which explain the hypotheses: The EPR setup of two particles (the SPIN and TWIN axioms); special relativity in a limited form (the FIN (later MIN) axiom); the Kochen-Specker "non-existence of a function" theorem/paradox.

So loosely we can imagine the typical EPR setup with two spacelike separated Observers making independent "choices" as to which of several axes to measure spin in two correlated particles: A and B, say. Let A's measurement be at time $t_A$. Then the conclusion is that no function of past properties (past light cone to $t_A$) predicts the outcome for A.

So the key phrase is "no function".

This theorem was primarily motivated to further exclude a fully deterministic interpretation of QM (obviously with hidden variables). Such hidden variables would give rise to a function of them - which here doesnt exist.

A secondary issue - where the "free will" comes from - is whether this just reconfirms an essential randomness in QM. Well the argument here, I believe, is that they interpret "random" to mean "a random function of" - but since no function exists even a random function doesnt exist (of any earlier properties). The responses of the particle A is determined at time $t_A$ based on no prior information (or information on B) - just the "free" choices of the experimenters as to what to measure.

A link to the Strong Free Will Paper: http://www.ams.org/notices/200902/rtx090200226p.pdf