Friday, April 11, 2008

The Bloch Sphere

Turns out there’s a sphere with my name on it.

No, it’s not really named after me, but the Bloch sphere does illustrate the power of animation for visualizing complex concepts.

Okay, here’s the Bloch Sphere:

And here’s the description from the Wikipedia article:

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system named after the physicist Felix Bloch. Alternately, it is the pure state space of a 1 qubit quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and pure states can be explicitly given. In generalized form, the Bloch sphere may also refer to the analogous space of an n-level quantum system.

Quantum mechanics is mathematically formulated in Hilbert space or Projective Hilbert space. The space of pure states of a quantum system is given by the rays in the Hilbert space (the “points” of projective Hilbert space). The space of rays in any vector space is a projective space, and in particular, the space of rays in a two dimensional Hilbert space is the complex projective line, which is isomorphic to a sphere. Each pair of antipodal points on the Bloch sphere corresponds to a mutually exclusive pair of states of the particle, namely, spin up and spin down for a Stern-Gerlach experiment oriented along a particular axis in physical space.

Got all that?

Ironically enough, the discussion page for this article includes the following comment.

Does anyone know what the _significance_ of the Bloch sphere is? Why does it come up so often in discussions, given how simple it is?

Simple? Doesn’t look so simple to me. I’d fail the exam, relatively certain that the professor wouldn’t give me extra credit just because my name is Bloch.

While I don’t understand the math—and probably never will—an animated version on this page created by Alain Michaud gives me a lot more perspective:

According to Alain, the black line is called the “Bloch Vector,” and it seems to trace a sphere that corresponds to the wave. While I’d still fail the exam, the animation at least clues me in that there’s something “going on,” so to speak. To most people, neither image is going to provide much clarity, but I’ll bet most math students would find the animation very helpful.

It turns out that this representation has something to do with quantum computing – and the qubit.

While the computers we use rely on bits of information that are represented by one of two states (on or off, represented by 1 or 0 in binary math), a qubit can store more information by representing 1, 0, or a “superposition” of both states because of that wave thing going on.

If quantum computation really takes off, I’m going to suggest that the units of information be called “blochs” (as in “I’m going to upgrade to a 500 gigabloch server.”)

That’s not without precedent. Frequencies, or the ratio of cycles per second used to be called, well, “cycles per second.” Now they’re called “hertz” in honor of physicist Heinrich Hertz.

Megablochs, gigablochs, and terablochs. Has a nice ring to it, don’t you think?

Posted by Richard Bloch

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