# Quantum Bayesian Networks

## September 30, 2012

### Parlez-vous Thermodynamique?

Filed under: Uncategorized — rrtucci @ 6:22 am

I’ve preached a lot in this blog about SIT (Shannon Information Theory). The problem with pure SIT is that is says nothing about energies. And living without being able to talk about energies is an unbearably cruel fate for a physicist. But, what’s this…Ta,tan…Thermodynamics to the rescue. According to Thermodynamics, temperature is defined as $T = \frac{dQ}{dS}$, a change in energy divided by a change in information (information = entropy, probability spread).

One of my future goals is to make QB nets (quantum Bayesian networks) speak the language of thermodynamics. What is the connection between the two?, you might ask.

In my previous papers, I showed how QB nets can be used to represent quantum density matrices. Suppose that we consider a protocol described by a density matrix $\rho(t)$ that evolves as function of time $t$. Suppose at time zero, $\rho(0)$ is a tensor product of several density matrices, one of which is the thermal (aka Gibbs or Boltzmann, or canonical ensemble) density matrix $\exp(-\beta H)/Z$, where $H$ is a Hamiltonian operator, $\beta$ is a positive real number and $Z= tr(e^{-\beta H})$. Also $\beta = \frac{1}{k_B T}$ where $k_B$ is Boltzmann’s constant and $T$ is the temperature of the system whose Hamiltonian is $H$. Since there is a “thermal density matrix” in such a protocol, the “thermal” part beckons Thermodynamics and the “density matrix” part beckons QB nets.

QB nets are also an ideal vehicle for representing measurement theory in quantum mechanics. If you combine Thermodynamics, QB nets and outlandish measurements, you immediately find yourself wandering into the territory of Maxwell demons.

I think Maxwell demonology is a great gedanken experiment on which to improve my knowledge about the nether world of QB nets and thermo. So next month (appropriately enough, Halloween month), I will dedicate myself to trying to translate some of the already existing Maxwell demonology protocols to the language of QB nets. And thus make Reverend Bayes preach about Maxwell demons, hell, brimstone and thermodynamics. Maybe at the end of this fun project, I’ll write a pedagogical paper with my ruminations.

I don’t expect that during the course of this project, I will discover anything fundamentally new that hasn’t been discovered and written about by others many times before. Although the subject of Maxwell demonology was viewed with much fear and bewilderment in the past, my impression is that since Maxwell’s time, significant inroads have been made into understanding the subject and it is nowadays considered fairly well understood and settled.

Truly, I will be standing on the shoulders of giants for this project. Maxwell demonology has a vast and illustrious history in Physics. Here are some of the more famous tracts on these dark arts

J. C. Maxwell, “Theory of Heat” (Appleton, London, 1871).
L. Szilard, Z. Phys. 53, 840 (1929).
L. Brillouin, J. Appl. Phys. 22, 334 (1951).
R. Landauer, IBM J. Res. Develop. 5, 183 (1961).
C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).
R. Landauer, Science 272, 1914 (1996).

In the past 5 years or so, a Japanese school led by Sagawa and Ueda has come out with some fine papers which are the culmination and synthesis of much previous work by others and some new one by them. These papers, to my mind, have exorcised Maxwell’s demon, at least for a long while. For example, here is one paper by them that I like very much:

“Minimal Energy Cost of Thermodynamic Information Processing: Measurement and Information Erasure”, by Takahiro Sagawa and Masahito Ueda
arxiv 0809.4098v3