The Interpretation of Quantum Mechanics

So far on this blog, I have argued that quantum mechanics should be most aptly seen as a generalization of probability theory, necessary to account for complementary propositions (propositions which can't jointly be known exactly). Quantum mechanics can then be seen to emerge either as a generalization (more accurately, a deformation) of statistical mechanics on phase space, or, more abstractly (but cleaner in a conceptual sense) as deriving from quantum logic in the same way classical probability derives from classical, i.e. Boolean, logic.

Using this picture, we've had a look at how it helps explain two of quantum mechanics' most prominent, and at the same time, most mysterious consequences -- the phenomena of interference and entanglement, both of which are often thought to lie at the heart of quantum mechanics.

In this post, I want to have a look at the interpretation of quantum mechanics, and how the previously developed picture helps to make sense of the theory. But first, we need to take a look at what, exactly, an interpretation of quantum mechanics is supposed to accomplish -- and whether we in fact need one (because if we find that we don't, I could save myself a lot of writing).

Untangling Entanglement

What to Feynman was interference (see the previous post), to Erwin Schrödinger (he of the cat) was the phenomenon known as entanglement: the 'essence' of quantum mechanics. Entanglement is often portrayed as one of the most outlandish features of quantum mechanics: the seemingly preposterous notion that the outcome of a measurement conducted over here can instantaneously influence the outcome of a measurement carried out way over there.
Indeed, Albert Einstein himself was so taken aback by this consequence of quantum mechanics (a theory which, after all, he helped to create), that he derided it as 'spooky' action at a distance, and never fully accepted it in his lifetime.
However, viewing quantum mechanics as a simple generalization of probability theory, which we adopt in order to deal with complementary propositions that arise when not all possible properties of a system are simultaneously decidable, quantum entanglement may be unmasked as not really that strange after all, but in fact a natural consequence of the limited information content of quantum systems. In brief, quantum entanglement does not qualitatively differ from classical correlation; however, the amount of information carried by the correlation exceeds the bounds imposed by classical probability theory.

The Emergence of Law

For many scientists, the notion of a lawful, physical universe is a very attractive one -- it implies that in principle, everything is explicable through appeal to notions (more or less) directly accessible to us via scientific investigation. If the universe were not lawful, then it seems that any attempt at explanation would be futile; if it were not (just) physical, then elements necessary to its explanation may lie in a 'supernatural' realm that is not accessible to us by reliable means. Of course, the universe may be physical and lawful, but just too damn complicated for us to explain -- this is a possibility, but it's not something we can really do anything about.
(I have previously given a plausibility argument that if the universe is computable, then it is in principle also understandable, human minds being capable of universal computation at least in the limit; however, the feasibility of this understanding, of undertaking the necessary computations, is an entirely different question. There are arguments one can make that if the universe is computable, one should expect it to be relatively simple, see for instance this paper by Jürgen Schmidhuber, but a detailed discussion would take us too far afield.)
But first, I want to take a moment to address a (in my opinion, misplaced) concern some may have in proposing 'explanations' for the universe, or perhaps in the desirability thereof: isn't such a thing terribly reductionist? Is it desirable to reduce the universe, and moreover, human experience within the universe, to some cold scientific theory? Doesn't such an explanation miss everything that makes life worth living?
I have already said some words about the apparent divide between those who want to find an explanation for the world, and those who prefer, for lack of a better word, some mystery and magic to sterile facts, in this previous post. Suffice it to say that I believe both groups' wishes can be granted: the world may be fully explicable, and yet full of mystery. The reason for that is that even if some fundamental law is known, it does not fix all facts about the world, or more appropriately, not all facts can be deduced from it: for any sufficiently complex system, there exist undecidable questions about its evolution. Thus, there will always be novelty, always be mystery, and always be a need for creativity. That an underlying explanation for a system's behaviour is known does not cheapen the phenomena it gives rise to; in particular, the value of human experiences lies in the experiences themselves, not in the question of whether they are generated by some algorithmic rule, or are the result of an irreducible mystery.

Maxwell's Demon, Physical Information, and Hypercomputation

The second law of thermodynamics is one of the cornerstones of physics. Indeed, even among the most well-tested fundamental scientific principles, it enjoys a somewhat special status, prompting Arthur Eddington to write in his 1929 book The Nature of the Physical World rather famously:

The Law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

But what, exactly, is the second law? And what about it justifies Eddington's belief that it holds 'the supreme position among the laws of nature'?
In order to answer these questions, we need to re-examine the concept of entropy. Unfortunately, one often encounters, at least in the popular literature, quite muddled accounts of this elementary (and actually, quite simple) notion. Sometimes, one sees entropy equated with disorder; other times, a more technical route is taken, and entropy is described as a measure of some thermodynamic system's ability to do useful work. It is wholly unclear, at least at first, how one is supposed to relate to the other.
I have tackled this issue in some detail in a previous post; nevertheless, it is an important enough concept to briefly go over again.

A Difference to Make a Difference, Part II: Information and Physics

The way I have introduced it, information is carried by distinguishing properties, i.e. properties that enable you to tell one thing from another. Thus, whenever you have two things you can tell apart by one characteristic, you can use this difference to represent one bit of information. Consequently, objects different in more than one way can be used to represent correspondingly more information. Think spheres that can be red, blue, green, big, small, smooth, coarse, heavy, light, and so on. One can in this way define a set of properties for any given object, the complete list of which determines the object uniquely. And similar to how messages can be viewed as a question-answering game (see the previous post), this list of properties, and hence, an object's identity, can be, too. Again, think of the game 'twenty questions'.
Consider drawing up a list of possible properties an object can have, and marking each with 1 or 0 -- yes or no -- depending on whether or not the object actually has it. This defines the two sides of a code -- on one side, a set of properties, the characterisation of an object; on the other side, a bit string representing information this object contains. (I should point out, however, that in principle a bit string is not any more related to the abstract notion of information than the list of properties is; in other words, it's wrong to think of something like '11001001' as 'being' information -- rather, it represents information, and since one side of a code represents the other, so does the list of properties, or any entry on it.)

A Difference to Make a Difference, Part I: Introducing Information

Picture a world in which there are only two things, and they're both identical -- let's say two uniform spheres of the same size and color, with no other distinguishable properties.
Now, ask yourself: How do you know there are two of them? (Apart from me telling you there are, that is.)
Most people will probably answer that they can just count the spheres, or perhaps that there's one 'over there', while the other's 'right here' -- but that already depends on the introduction of extra structure, something that allows you to say: "This is sphere number 1, while that is sphere number 2". Spatial separation, or the notion of position, is such extra structure: each sphere, additionally to being of some size and color, now also has a definite position -- a new property. But we said previously that the spheres don't have any properties additionally to size and color. So, obeying this, can you tell how many spheres there are?
The answer is, somewhat surprisingly, that you can't. In fact, you can't even distinguish between universes in which there is only one sphere, two identical ones, three identical ones etc. There is no fact of the matter differentiating between the cases where there are one, two, three, etc. spheres -- all identical spheres are thus essentially one and the same sphere.
This is what Leibniz (him again!) calls the identity of indiscernibles: whenever two objects hold all the same properties, they are in fact the same object.
Now consider the same two-sphere universe, but one sphere has been painted black. Suddenly, the task of determining how many spheres there are becomes trivial! There's two: the one that's been painted black, and the one that hasn't. But how has this simple trick upgraded the solution of this problem from impossible to child's play?