Montag, 31. Oktober 2011

The Origin of the Quantum, Part II: Incomplete Evidence

In the previous post, we have had a first look at the connections between incompleteness, or logical independence -- roughly, the fact that for any mathematical system, there exist propositions that that system can neither prove false nor true -- and quantumness. In particular, we saw how quantum mechanics emerges if we consider a quantum system as a system only able to answer finitely many questions about its own state; i.e., as a system that contains a finite amount of information. The state of such a system can be mapped to a special, random number, an Ω-number or halting probability, which has the property that any formal system can only derive finitely many bits of its binary expansion; this is a statement of incompleteness, known as Chaitin's incompleteness theorem, equivalent to the more familiar Gödelian version.
In this post, we will exhibit this analogy between incompleteness and quantumness in a more concrete way, explicitly showcasing two remarkable results connecting both notions.
The first example is taken from the paper 'Logical Independence and Quantum Randomness' by Tomasz Paterek et al. Discussing the results obtained therein will comprise the greater part of this post.
The second example can be found in the paper 'Measurement-Based Quantum Computation and Undecidable Logic' by M. Van den Nest and H. J. Briegel; the paper is very interesting and deep, but unfortunately, somewhat more abstract, so I will content myself with just presenting the result, without attempting to explain it very much in-depth.

Dienstag, 11. Oktober 2011

The Origin Of The Quantum, Part I: An Incomplete Phase Space Picture

In the last post, we have familiarized ourselves with some basic notions of algorithmic information theory. Most notably, we have seen how randomness emerges when formal systems or computers are pushed to the edges of incompleteness and uncomputability.
In this post, we'll take a look at what happens if we apply these results to the idea that, like computers or formal systems, the physical world is just another example of a universal system -- i.e. a system in which universal computation can be implemented (at least in the limit).
First, recall the idea that information enters the description of the physical world through viewing it as a question-answering process: any physical object can be uniquely identified by the properties it has (and those it doesn't have); any two physical objects that have all the same, and only the same, properties are indistinguishable, and thus identified. We can thus imagine any object as being described by the string of bits giving the answers to the set of questions 'Does the object have property x?' for all properties x; note that absent an enumeration of all possible properties an object may have, this is a rather ill-defined set, but it'll serve as a conceptual guide.
In particular, this means that we can view any 'large' object as being composed of a certain number of 'microscopic', elementary objects, which are those systems that are completely described by the presence or absence of one single property, that may be in either of two states -- having or not having that particular property. Such a system might, for instance, be a ball that may be either red or green, or, perhaps more to the point, either red or not-red. These are the systems that can be used to represent exactly one bit of information, say red = 1, not-red = 0. Call such a system a two-level system or, for short, and putting up with a little ontological inaccuracy, simply a bit.