## Is quantum mechanics `it?’ Bell’s definition of `free choice.’

I haven’t blogged in awhile thanks to an insane amount of work, but now’s as good a time as any to toss another post up here since I just got back from the APS March Meeting in Baltimore (where I wish I still was — no snow and a good deal warmer). Yesterday I attended a talk by Renato Renner on some work he has done with Roger Colbeck. Before I comment on it, let me just say that I had an interesting lunchtime conversation during which we discussed how fractured the foundations community is: people have their pet theories and foundations meetings tend to end up making no progress (as a whole) because they are either devoted to a single, common viewpoint or, if they are more general, people just end up yelling at each other. Well, maybe `yelling’ is a little harsh. But, anyway, the point is that I can’t say I’ve ever known a foundations person to change `camps,’ so-to-speak. I will say that there are some foundations people who are refreshingly open-minded. Terry Rudolph comes to mind as does Max TegmarkÂ (speaking of which, it freaked me out when I discovered that Terry is about my age — Terry was already well-known before I finished my PhD…).

Anyway, Renato’s talk was built on Bell’s notion of `free choice.’ At the beginning, Renato said he didn’t explicitly rule out any other notion of free choice, but he claimed never to have seen one. I will say that I may have misinterpreted his aim with this talk as astutely and rationally pointed out to me by Mark Wilde, but I still think some of my argument holds. At any rate, I’ll come back to all of that later. First, let’s review Bell’s definition of `free choice’ and see what it implies.

Bell defined a `free choice’ to be one that was completely independent of any event in it’s *future* light cone. **UPDATE**: Mark Wilde noted that the official definition given in the various papers is that a ‘free choice’ is one in which the choice event is only correlated with events in its future light cone. I think they say the same thing, though. The idea is natural enough since it implies that nothing in the *future* can affect a *prior choice, *i.e something in the past. Note, however, that this definition does *not* say anything about events in the *past* light cone of the `choice’ event. In other words, imagine we have a single, six-sided die that we wish to use as a counter for something. So, for example, suppose I wish to mark an event with the number 5. I can turn the die to the side with 5 facing upwards and can mark it. If at some later time I lose the die entirely (maybe the dog swallowed it), that event (of losing it) does not affect the prior choice I made to turn it to the side showing 5. So that is Bell’s definition of `free choice’ (and is supposedly the only one Renato is aware of).

But suppose I wish to mark some event with the number 8. Somewhere along the line, there was an event prior to my `choice’ event in which I ended up with a six-sided die so I am unable (with the tools I have) to mark the 8. My choice in this case is certainly not free. One might take the view that it was the `future’ event of me ending up needing an 8 that limited my choice, but needing an 8 does not necessarily need to be in the future of me ending up with a six-sided die (though the event of me gaining the *knowledge* of that need, *is*). At any rate, the point is that the definition of `free choice’ is both a little fuzzy and certainly can be interpreted as not necessarily being completely free. Nevertheless, this definition does appear to allow for the existence of some classically `free choices.’

Now, nothing is necessarily wrong with that in the Colbeck-Renner theory because they argue that quantum theory, in this situation, is maximally informative. In other words, because quantum theory is contextual, I don’t have to worry about having the six-sided die when I need to measure an 8. I can just choose a basis (i.e. a die) that allows me to measure an 8. That is (very roughly) contextuality — the result of my measurement (the 8) — depends on my choice of basis. That, of course, is a truly free choice (and, in fact, in this case the `choice event’ is also independent of all events in the *past* light cone as well).

Now, it appears that, as a result of the fact that quantum theory is maximally informative under the more restrictive definition of `free choice’ (i.e. one that does *not* include the past light cone dependence), they assume that there is nothing more general. In other words, they ask whether, under this assumption of `free choice,’ there are `extensions’ of quantum theory. It should be obvious that if the definition of `free choice’ remains the same, then there really *are* no *useful* extensions of quantum theory since it is maximally informative. This, of course, implies that quantum mechanics is complete.

But here’s my problem: this takes a somewhat restrictive definition of `free choice’ and then proves that something that is ultimately less restrictive is thus maximally informative. That’s a bit like cherry-picking data in order to prove a point you already think is true. So, for example, suppose I have a deck of cards that only consists of the spades and clubs (so all the cards are black) but does not include the ace of spades and the ace of clubs. Suppose it is my intention to prove that I have a complete set of black cards. So I get another deck that now includes all four suits, but it still lacks the two aforementioned aces. Now I claim that since my new set includes everything I had before and nothing new, the old set is maximally informative. Of course, we know that this is not true because we know there are still two missing aces. It came across to me in Renato’s presentation that he was essentially claiming that his proof shows that those other aces don’t exist, i.e. quantum mechanics truly is `it.’ Now, as I said, Mark Wilde had a completely different impression and so perhaps this was never Renato’s intention. At any rate, if it is, then, to me, there are two specific objections to that claim (and the corresponding proof).

First, one can, of course, question the definition of `free choice.’ Why *not* define free choice in terms of the past light cone events as well? (This is where Mark noted that he thinks the point was to show the limitations of Bell’s definition, in which case this is actually pretty good work, though I wonder if it was necessary). In other words, why not use the quantum case as the definition? To me, that seems like it’s more `free’ than Bell’s definition. Sure, maybe it implies that there are no truly free choices in classical physics, but personally I don’t find that all that hard to believe. I want a Porsche, but that doesn’t mean I can run out and buy one.

The second objection is that because the argument works from the `ground up,’ so-to-speak, *and* `finds’ quantum theory to be maximally informative, it rules out more general theories *somewhat* by fiat. In other words, what if quantum theory is actually a more specific case of some more general theory? For example, what if quantum theory is actually a special case of a generalized probabilistic theory? Because Colbeck and Renner used Bell’s definition of `free choice’ it seems to me that they were *guaranteed* to find that quantum mechanics was maximally informative. It is worth noting that here is another point at which Mark disagreed — he didn’t think it was that obvious.

I suppose there’s nothing inherently wrong with that, unless one is then claiming that there’s no point in *looking* for a more general theory or one is arguing that quantum mechanics is `it.’ I kind of got the impression that they were saying exactly that — there’s no point in pursuing anything deeper or more fundamental. I apologize to Renato and Roger if I misconstrued their work, but they can think of it this way: at least it succeeded in getting people to discuss it!

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