Quantum Bayesian Networks

January 10, 2011

The Continents of Quantum Mechanics and Shannon Information Theory Are Now Colliding and Merging

Filed under: Uncategorized — rrtucci @ 4:22 pm

When continents collide (suture between India and Asia in the Himalayas)

How can one explain the intricate emission spectra of atoms and molecules? Why are atoms stable in the first place? Classical physics cannot answer these questions in a satisfactory way. One needs quantum mechanics to do so.

Quantum Mechanics underlies most discoveries in physics and chemistry of the 20th century. The following areas of physics use quantum mechanics intensively:

statistical mechanics, solid state physics (semiconductors, superconductors, superfluids), laser physics, quantum field theory, particle physics, nuclear physics, string theory, etc., etc.

Quantum mechanics explains laboratory observations in these areas, often predicting the data correctly to many decimal places.

Most of non-quantum physics (classical mechanics, electrodynamics, classical waves and optics, classical thermodynamics) was invented prior to the 20th century. Einstein’s theories of special and general relativity are the main exception to this rule. They are a mayor piece of non-quantum physics invented in the 20th century. Special relativity and quantum mechanics were merged long ago (into what we now call relativistic quantum field theory), but we still don’t fully understand how to merge general relativity and quantum mechanics. We believe that this is possible and that even general relativity will eventually be assimilated by the Borg of quantum mechanics.

Shannon’s information theory (SIT) is amazing for its generality, and because it was born almost fully mature, in a single, remarkable 1948 paper by Claude Shannon. SIT has so far been used mostly in communication, coding and error correction theory. It has had some impact on physics, but I think it is fair to describe that impact as minor. So minor that SIT is rarely taught in physics courses (undergraduate or graduate). The main reason for this is, I believe, because, as Feynman once reminded us, “nature isn’t classical, dammit. And if you want to make a simulation of nature, you’d better make it quantum mechanical”. Since the original SIT is classical, it is not totally unexpected that its impact on physics has been minor so far.

But the game has now changed. In the past 20 years, remarkable advances have been made in the quantum version of SIT (Shannon Information Theory). I think quantum SIT is revolutionizing how quantum mechanics is used and understood. I fully expect that quantum SIT will become, in the near future, part of the standard physics curriculum. It may even some day play a nontrivial role in solving the puzzle of quantum gravity and string theory. It has already been used to elucidate the escape of information from a black hole.

To understand quantum SIT, you will first need to understand classical SIT. In my opinion, the following two books are excellent and they complement each other well:

  • David MacKay’s “Information Theory, Inference, and Learning Algorithms”. This book is available for free in pdf form, although it is well worth purchasing in paper form. (David’s website here)
  • Cover and Thomas’s “Elements of Information Theory, 2nd Edition (2006)”. A very well written classic. Amazon link here.

You only need to learn the basics of classical SIT before plunging into quantum SIT. These 2 books contain much more than the basics, so it is not necessary or even advisable to try to absorb them initially in their entirety. Furthermore, note that many pedagogical expositions of quantum SIT start with a brief review of classical SIT.

The pedagogical expositions of quantum SIT that are currently available are all still very new and not as polished as the Cover&Thomas and MacKay books. I found the following lectures very helpful.

I also like this master thesis by Ivan Savov (currently a PhD candidate under Patrick Hayden). Of course, you should also look at some of the original papers in ArXiv.

The latest craze in the quantum SIT community are the so called “resource inequalities”, invented by Devetak, Harrow and Winter. I’ve just started learning about them. I don’t fully understand them yet, but they appear to be a very deep result that ties together much previous work. They look like this: \alpha[c\rightarrow c] + \beta[q\rightarrow q] + \gamma[qq] \geq 0, where \alpha,\beta,\gamma are real numbers and

  • [c\rightarrow c]= one use of a noiseless c-bit (classical bit) channel
  • [q\rightarrow q]= one use of a noiseless q-bit (quantum bit) channel
  • [qq]= one e-bit (entanglement bit) shared between two parties (Alice and Bob)

These inequalities apply only in the limit of a large number of copies of the resource. Exploring such a limit is typical of what is done in SIT theories. Devetak, Harrow and Winter have proven their inequalities in a mathematically rigorous way.

Some questions to ponder:

  • What are the best applications of quantum SIT?

  • What does quantum SIT tell us about quantum computation? Can quantum SIT be reduced to simple statements about qubit circuits and quantum computer programs?

I’m not sure what are the best answers to these questions. Those answers will probably evolve as this young field continues to grow.


1 Comment »

  1. I would like to correct Dr. Feynman:
    Nature isn’t classical, dammit, it’s quantum informational!

    Comment by illuminaughty — January 14, 2011 @ 1:18 pm

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