Wednesday, July 28, 2010

What is a State?

OK. Enough politics. I am a scientist, after all. Part of why I spend so long on politics, however, is that it makes for an accessible topic insofar as that I can contribute with little special expertise. As my science career progresses, however, I am more confident of my ability to contribute usefully to the world of science blogging.

Since I am in quantum information research, much of my science blogging will necessarily be talking about quantum states. As such, it's worth starting out by discussing the notion of a state more generally. I apologize to the quantum foundations people in the room, as I will likely butcher this horribly, but I've got to start somewhere, eh?
1 a : mode or condition of being b (1) : condition of mind or temperament (2) : a condition of abnormal tension or excitement
2 a : a condition or stage in the physical being of something b : any of various conditions characterized by definite quantities (as of energy, angular momentum, or magnetic moment) in which an atomic system may exist
Rather than start with a physical definition, I'll start with the computer science notion of a state-- at least, one notion. In computer science, we often think of the state of a machine as being that set of information which is required to predict (that is, to simulate) the future states of that machine. This is by necessity somewhat recursive, but we can disentangle it somewhat. If you're lucky enough to have a laptop where hibernation works properly, then you're already somewhat familiar with a state, as it is the state of the computer which gets written to and read from the disk during the hibernation and resuming processes. The contents of the computer's memory completely describe what it means for the computer to resume its execution, so that we may discover the contents of the computer's memory in the future.

In a very real sense, this is exactly what physicists mean when they use the word "state." If one knows the full state of a physical system, then they can predict as much of the future of that system as is allowed by the laws of nature. If, as Newton and others thought, those laws are deterministic, then that means that one can predict all future states of the system. The state of the system, then, is a description of the system so complete that it is for all intents and purposes the identity of that system. To take a materialistic view of myself for a moment, I am then equivalent to a full and complete state of my physical body. In fact, we can be recursive again and define my physical body as that whose state is necessary to describe me. (If you find this kind of recursive definition of self as satisfying as I do, you may also like reading Scott Aaronson's notes on a complexity theoretic approach to free will.)

For a specific example, consider a pool ball on a table-- we will presume for now that it cannot go up or down, despite whatever trick shots one tries. Then, if we wish to simulate the trajectory of this pool ball, we must know for at least one given moment exactly where it is, how quickly it is moving, in which direction it is moving, and the axis and magnitude of its spin. That is, we must know x, v and ω. If we know all this, then we may as well dispense with the table and simply run a computer simulation, as the state given by these three vectors completely describes the entire dynamics of that system. If one of those three vectors changes, then the pool ball is no longer in that same state. To put it yet another way, if we have two tables with one ball each, and if their states are identical, then the balls themselves are indistinguishable (not in the sense of indistinguishable particles, mind you, but in the sense of state discrimination).

The astute reader will note here that I have pulled a bit of a fast one on them. This notion of state is not the only kind of state that gets bandied about in physics. Rather, it is a special kind of state called an ontic state-- that is, one corresponding to reality. Statistics allows us to also speak of an epistemic state, which describes not reality itself, but our knowledge of it. Thus, an epistemic state is not in general sufficient to describe or simulate a system, but is a complete characterization of a given agent's interactions with that system. Unlike ontic states, which we assume to be objective in order to have a reality consistent with multiple observers, epistemic states are subjective. Two observers may validly have different epistemic states for a system in some fixed ontic state.

One may, in fact, go as far as to say that all states actually discussed in physics are epistemic, since we cannot even in principle have complete knowledge of a system. I do not subscribe to this view myself, but I find it helps to remind me that ontic states such as those discussed in the pool example are often states not of real systems, but of toy models we make to approximate real systems. A physical pool ball is much more complicated than a list of three vectors, and a true physical ontic state would reflect this.

Understanding these somewhat orthogonal views of a state helps clarify many counterintuitive aspects of physics, such as quantum teleportation. If, as is true in quantum mechanics, a state cannot be copied, then there is no physical difference between transmitting a state and transmitting an object with that state-- both lead to exactly the same state of reality after the fact. Thus, quantum teleportation can be seen not just as some sci-fi-esque "beaming" of an object, but something much more interesting: a clever way of communicating. Of course, fully exploring this is a subject for a future post.

To close out this discussion of a state, I wish to be so bold as to assign a bit of homework until my next post. As you go about your day, think of what the states of objects around you might be like-- what information would you need to reproduce or to simulate those objects perfectly?

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