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.

What is Quantum Mechanics?

So far, I've told you a little about where I believe quantum theory comes from. To briefly recap, information-theoretic incompleteness, a feature of every universal system (where 'universal' is to be understood in the sense of 'computationally universal'), introduces the notion of complementarity. This can be interpreted as the impossibility for any physical system to answer more than finitely many questions about its state -- i.e. it furnishes an absolute restriction on the amount of information contained within any given system. From this, one gets to quantum theory via either a deformation of statistical mechanics (more accurately, Liouville mechanics, i.e. statistical mechanics in phase space), or, more abstractly, via introducing the possibility of complementary propositions into logic. In both cases, quantum mechanics emerges as a generalization of ordinary probability theory. Both points of view have their advantages -- the former is more intuitive, relying on little more than an understanding of the notions of position and momentum; while the abstractness of the latter, and especially its independence from the concepts of classical mechanics, highlights the fundamental nature of the theory: it is not merely an empirically adequate description of nature, but a necessary consequence of dealing with arbitrary systems of limited information content. For a third way of telling the story of quantum mechanics as a generalized probability theory see this lecture by Scott Aaronson, writer of the always-interesting Shtetl-Optimized.
But now, it's high time I tell you a little something about what, actually, this generalized theory of probability is, how it works, and what it tells us about the world we're living in. First, however, I'll tell you a little about the mathematics of waves, the concept of phase, and the phenomenon of interference.

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.