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.

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.

The Universal Universe, Part I: Turing machines

In the mid-1930s, English mathematician Alan Turing concerned himself with the question: Can mathematical reasoning be subsumed by a mechanical process? In other words, is it possible to build a device which, if any human mathematician can carry out some computation, can carry out that computation as well?
To this end, he proposed the concept of the automated or a-machine, now known more widely as the Turing machine. A Turing machine is an abstract device consisting of an infinite tape partitioned into distinct cells, and a read/write head. On the tape, certain symbols may be stored, which the head may read, erase, or write. The last symbol read is called the scanned symbol; it determines (at least partially) the machine's behaviour. The tape can be moved back and forth through the machine.
It is at first not obvious that any interesting mathematics at all can be carried out by such a simplistic device. However, it can be shown that at least all mathematics that can be carried out using symbol manipulation can be carried out by a Turing machine. Here, by 'symbol manipulation' I mean roughly the following: a mathematical problem is presented as some string of symbols, like (a + b)². Now, one can invoke certain rules to act on these symbols, transform the string into a new one; it is important to realize that the meaning of the symbols does not play any role at all.
One rule, invoked by the superscript 2, might be that anything that is 'under' it can be rewritten as follows: x² = x·x, where the symbol '=' just means 'can be replaced by'; another rule says that anything within brackets is to be regarded as a single entity. This allows to rewrite the original string in the form (a + b)·(a + b), proving the identity (a + b)² = (a + b)·(a + b).
You can see where this goes: a new rule pertaining to products of brackets -- or, on the symbol level, to things written within '(' and ')', separated by '·' -- comes into effect, allowing a re-write to a·a + b·a + a·b + b·b, then rules saying that 'x·y = y·x', 'x + x = 2·x', and the first rule (x² = x·x) applied in reverse allow to rewrite to a² + 2·a·b + b², proving finally the identity (a + b)² = a² + 2·a·b + b², known far and wide as the first binomial formula.

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?