Over at Michael Nielsen’s blog, Michael has a post telling us that he won’t be posting again until August. Personally Michael’s lack of posting scares the bejebus out of me: if he’s not posting, he must be working on some grand research which will make everything I do look even more trivial than before. Michael, you’re scaring me!
Anyway, along with the post Michael posts a comment by UW’s John Sidles trying to stir up some debate by asking about a paper by Conway and Kochen, “The Free Will Theorem”, quant-ph/0604069. Actually I had heard about this paper a while ago, via some non-arxiv channel (where I can’t remember, exactly) and had basically guessed from the brief description I had heard what the paper was about. This is how you know that you are getting old and curmudgeony when you can hear a title to a paper and a description of the results and can guess the way in which those contents were prove (There are rumors, which I myself have never verified, that at a certain well known quantum computing research group, the days starts as follows. A little before lunch, the researchers wander in, check their email and look at the day’s postings on the arxiv. Now they don’t do anything more than read the titles. The research group then proceeds to go to lunch. At the lunch they discuss, with great debate, the most interesting papers posted that day. Having never ever even read the papers! There is a similar story about a certain researcher in quantum computing, who, if you tell that researcher a new result, (s)he will, within a day, almost always be able to rederive the result for you. Of course, my personal nickname for this person is “The Oracle” and it is tempting to tell “The Oracle” that a certain open problem has been solved, when it has not been solved, and see if (s)he can come up with the answer!)
(A note: throughout this post I will use the words “free will” to describe something which, you may or may not agree is related to “free will” as you imagine it. In particular if an object is said to not have free will if its future evolution can be predicted from information in the past lightcone of the object. If it cannot be so predicted with certainty it is then said to possess free will. In fact, I find this definition already interesting and troublesome: can we ever predict anything by only knowing information in our past light cone? How do we know that in the next instance of our evolution a light ray will hit us and burn us up? Certainly we cannot see such a light ray coming, can we? We can, of course, use physics to explain what happened: but can we use it to predict our future behavior? Of course for the electromagnetic field, we could sheild ourselves from such radiation and reasonably assume that we can predict what is occuring. But what about gravity, which can’t be sheilded? For an account of this type of argument I recommend Wolfgang’s comments
Okay, back to the story at hand. What is Conway and Kochen’s free will theorem? The basic idea is quite simple. I will explain it in the context of Bell’s theorem and the Kochen-Specker theorem, since the author’s don’t describe it in this manner. Bell’s theorem, we known, tells us that there is no local hidden variable theory explaining what quantum theory predicts. The Kochen-Specker theorem is less well known (which leads, in my opinion, the proponents of this different result to suffer a severe inferiority complex in which they constantly try to argue that the KS theorem is more important than Bell’s theorem.) What the Kochen-Specker theorem says is that if there is a hidden variable theory of quantum theory, it must be contextual, i.e. the Kochen-Specker theorem rules out non-contextual hidden variable theories. The way I like to think about the Kochen-Specker theorem is as follows: suppose that there are some hidden variables associated with some quantum system. Now if you make a measurement on this system you will get some outcomes with differing probabilities. Now sometimes you get outcomes with certainty. You’d like to say that when you perform this measurement, this outcome is actually associated with the value of some real hidden variable. But what the KS theorem tells you is that this is not possible: there is no way that those measurement outcomes are actually associated with the hidden variables in a nice one to one manner. What does this have to with contextuality/non-contextuality? Well the “context” here is what other measurement outcomes you are measuring when you measure along with the outcome associated with a particular hidden variable. In non-contextual hidden variable theories, what those other measurement results are doesn’t matter: it is those types of theories that the KS theorem rules out.
(Note: From my personal perspective, I find the KS theorem fascinating, but not as disturbing at Bell’s theorem: that “what you measure” determines “what you can learn” is a deep insight, and one that tells us something about the way reality can be described. However it is not that difficult to imagine the universe as a computer in which accessing the memory of the computer depends on the context of your input: i.e. to get ahold of memory location which holds the value 01001010, you need to query the machine and it seems perfectly reasonable to me that the machine is set up in a manner such that I can’t get all of those bits, since my measurement will only get some of them and the context of the measurement will change some of the other bits. This was basically John Bell’s reaction to the Kochen-Specker theorem. Interestingly there is a claim in this Conway and Kochen theorem that this loophole has been filled! I have a bit to say about this below. Of course no matter where you come out in this arguement, there is no doubt that the KS is DEEP: it tells us that the universe is not a computer whose memory we can gain total access to. And if we can’t gain access to this memory, then does the memory have any “reality”?!!)
Well I’m rambling on. Back to the subject at hand, the free will theorem. In the free will theorem, Conway and Kochen set up an experiment in which you take two spin-1 particles and perform measurement on these spins. (Now for those of you in the know you will already be suspicious that a spin-1 particle was used (the 3 dimensional irrep of SU(2)) as well as an entangled quantum state…sounds like both KS and Bell doesn’t it?)) The free will theorem is then:
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
In other words if we have free will, then particles have free will! How does the theorem get proven? Well basically the proof uses the KS theorem as well as the perfect correlations arising from maximally entangle spin-1 systems. First recall that the KS theorem says that hidden variable theories must be contextual, i.e. if I give you just the measurement directions involved in a measurement, there is no way to map this onto yes/no outcomes in a manner consistent across a set of possible measurements. But suppose, however, that your map to yes/no outcomes (i.e. the particles response) also depends on a hidden variable representing information in the particles past light cone, i.e. that the particles have no free will (contray hypothesis.) Now because we are dealing with a maximally entangled spin-1 system, two spacelike separated parties, A and B, will obtain the same outcomes for their measurement results for measurement directions for which they measure along the same direction. So for fixed values of the information in the past of both parties, the particle response should be identical and can only depend on local measurement direction. But this is not possible when one chooses an appropriate set of directions corresponding to the Kochen Specker proof. One can thus conclude that we cannot freely choose the measurements directions, i.e. that not all choices of measruements are possible: there must be hidden variables associated with the measurement choice as well. Thus we have shown that particles having dependence on information in the past light cone implies that the measurement choice must have dependence on information in the past light cone. Having shown the contrapostive, we have shown the free will theorem.
Now the interesting thing about the free will theorem is that doesn’t tell us whether the universe alows us to have free will or not. It simply says that if we assume some form of free will, then the particles we describe will also have free will. Of course the “free will” we describe here is “independence of (classical) information in the past light cone,” so some would object to this definition of “free will.” In particular, by this definition, a system which is totally random has free will. But is seems to me that the interesting question about free will is not whether one can have such random systems, but whether one can have a mixture of determined and undetermined evolutions. I mean the fundamental paradox of free will seems to me to be that free will involves a lack of cause for an action, but we want this action to itself have causes. In this respect, the above theorem suffers a bit, in my opinion, for a simplistic version of free will which is too absolutist for my tastes. What I find fascinating is whether we can “quantify” different versions of free will and what such quantifications would tell us about our real world.
Well it seems that I’ve had the free will to ramble on quite a bit in this post. Hopefully you might decide that the subject is interesting enough to choose to read the paper on your own 😉