τ keeps on slipping...

Apologies: this post is somewhat more specialized than my normal fare, and probably will be boring as hell without a some mathematical knowledge.

Electrons have a negative charge. When you think about it, this really doesn't make much sense; after all, electrons are the charge carriers for electric charge, and so we would hope to assign their charge a positive value. That electrons carry a negative charge isn't a fundamental statement about reality, though, but rather an unfortunate consequence of an arbitrary decision made early on when electricity was being studied, but when electrons were still undiscovered.

Similar cases of unfortunate arbitrary conventions can be seen in other areas of mathematics and science. Recently, for instance, Michael Hartl has argued that $\pi$ is not the right constant to use in the equations governing such things as circles, frequencies and angles. Rather, Hartl argues that $\tau = 2\pi$ is a much more natural choice. Using this convention, the circumference $c$ of a circle is $c=\tau\ r$, eliminating the awkward factor of 2 in $c=2\pi r$. It may seem that we lose something when considering the area $A = \frac12 \tau\ r^2$ of a circle in this notation, but in fact this is much more natural for expressing as an integral, as those familiar with calculus will be happy to note.

Today, I'd like to show you somewhere else in physics where changing notation makes things much more natural. Concretely, I'd like to argue that using a different $\tau$ makes quite a lot of sense, when we using $\tau = it$ as a replacement for the time $t$ in equations. In fact, I take it as a lesson of quantum mechanics that we should consider time to lie along an imaginary axis and not along the real axis. This notational trick, known as Wick rotation, simplifies many physical equations, such as Schrödinger's equation. I find that \(\frac{d}{dt} \left\vert\psi\right\rangle = i \hat{H} \left\vert\psi\right\rangle\)  makes much more sense expressed in imaginary time:
\[\frac{d}{d\tau} \left\vert\psi\right\rangle = \hat{H} \left\vert\psi\right\rangle\] Adopting this convention also makes it manifestly clear why complex conjugation is intimately related to time reversal, since $\tau^* = -\tau$.

It is, in fact, quite rare for $t$ to appear in quantum mechanics without a factor of $i$ attached. Even when describing a classical object interacting with a quantum mechanical system, such as an oscillating field introducing a time-varying term to a system's Hamiltonian (that is, the operator which describes the energy of a system--- if that makes no sense, don't worry), we write something like \[\hat H(t) = \cos(\omega t)\ \hat\sigma_x + \sin(\omega t)\ \hat\sigma_y.\] But wait!, you say! There's no $it$ in that equation! As it turns out, there actually is, but we've hidden it by using trigonometric functions where an exponential function is more natural: \[ \hat{H}(\tau) = e^{-\omega\tau\hat\sigma_z/2}\hat\sigma_x e^{\omega\tau\hat\sigma_z/2} \] This form also has the advantage of making it manifest that the oscillation of the classical field can be thought of as a coordinate rotation of a time-independent field.

Other key results of quantum mechanics become much cleaner with the imaginary-time convention. For instance, this convention along with the natural units convention that $\hbar = 1$ makes Ehrenfest's theorem much less awkward to write: \[\frac{d}{d\tau}\left\langle \hat A\right\rangle = \left\langle \frac{d\hat A}{d\tau}\right\rangle + \left\langle[\hat H, \hat A]\right\rangle\]

At the end of the day, such notational choices as the sign of an electron's charge, the choice of circle constant, or the axis which we use to represent time are all arbitrary. We can do physics quite well even when a choice lacks something in mathematical beauty. My point, then, in exploring the fun of $\tau$ is to show that even though our choice of notation is an arbitrary choice made for the convenience of the humans that work with it, by making our notational choices carefully, we can coax out and make manifest deep truths.

What are vectors?

As I've said before, science is social-- oops. Wrong mantra. What I meant to say is that vectors are an abstract way of describing a pattern. Specifically, the vectorspace axioms formally describe a kind of mathematical object, the vector, that encapsulates the geometric and algebraic properties of a large class of seemingly disparate objects. By using the vectorspace axioms, we will be able see that lists of numbers such as are vectors, as are arrows on the 2D plane.

Rather than describe how to do so myself, though, I will try something different. Vectors are important in much of physics, and so lots of people have already written much about them. Thus, for the bulk of the work in describing vectors, I will defer to these other writings. A very physics-oriented approach can be found over at Dot Physics, starting with a trig-based introduction to vectors, followed by a discussion of how to represent vectors. An alternate physics-motivated discussion of vectors can be found at HyperPhysics.

For the more mathematically motivated amongst us, Wikipedia has a good page describing a very special family of vector spaces called ℝⁿ that is used to describe points in Euclidean space. MathWorld has a few good articles on vectors, including a technical definition and listing of properties and a more concise listing of the vectorspace axioms. Finally, the Unapologetic Mathematician derives vectorspaces from a more general construction called a module (warning: not for the feint of math).

To understand why we care about vectors in quantum information and computation, however, takes one more observation. A quantum state can be written as a linear combination of some set of basis states. For example, an arbitrary qubit state can be written as . This important property means that quantum states are a kind of vector in what we call a Hilbert space. This has some profound implications for how we think of and manipulate quantum states, as we shall explore in forthcoming posts.

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

[source]

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?