# Quantum Bayesian Networks

## April 23, 2012

### 2001 A Space Odyssey – The QC Dawn of Man

Filed under: Uncategorized — rrtucci @ 7:28 pm

A tribe of String Theorists is shown in the primeval African desert picking fleas off each other and scavenging for food, competing with pigs for wild seeds and roots. A leopard kills one member of the tribe, and later that day they are driven away from their watering hole by an enemy tribe of apes. The future looks very grim for this tribe of String Theorists. The LHC is very unlikely to shed much light on their theories, and it’s also very unlikely that a successor to the LHC, one capable of probing substantially higher energies, will be built by the world community in the next few decades.

Disgruntled and defeated, they sleep overnight in some crags surrounding a rocky crater. Upon awakening, they find that a black monolith, a Quantum Fog 9000 Quantum Computer, has appeared in the crater. They approach it shrieking and jumping, and eventually touch it cautiously.

Soon after encountering the Quantum Fog 9000 QC, one of the String Theorists sees QC simulations in a new light. He uses one to pound on a pile of bones, the skeletal remains of earlier theories. Shortly thereafter, the previously herbivorous tribe of String Theorists is shown eating meat. Presumably they’ve used their QC simulation to kill a pig.

Armed with QC simulations, the tribe of String Theorists attacks the enemy tribe at the watering hole and they kill its leader. As they beat him to death with their QC simulations, the other members of the enemy tribe are shown cowering back in fear.

Flush with victory, the leader of the tribe of String Theorists flings his QC simulation high up into the sky. In mid-flight, the QC simulation transmutes into a cylindrical spacecraft floating in space, orbiting a planet. It’s now millions of years after the dawn of Man, when String Theorists invented the first, primitive QC String Theory simulation tools. Now those tools have evolved into much more advanced QC simulations of quantum-gravity that are routinely used by advanced spacetime travelers.

References: (I may have borrowed some phrases from some of them)

## April 17, 2012

### Mirror, Mirror, on the Wall, What is the Most Symmetric State of Them All in Quantum SIT?

Filed under: Uncategorized — rrtucci @ 3:50 pm

SIT =Shannon Information Theory

Let $H({\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N})$ denote the classical entropy of a probability distribution of the random variables ${\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$.

Consider a Hilbert space ${\cal H}_{\underline{a}_1}\otimes{\cal H}_{\underline{a}_2}\otimes\ldots\otimes{\cal H}_{\underline{a}_N} = {\cal H}_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$. For any density matrix $\rho_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$ that acts on ${\cal H}_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$, let $S({\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N})$ be its von Neumann entropy.

Suppose $|\psi\rangle_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}\in {\cal H}_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$ is a ket (pure quantum state) with density matrix given by $\rho_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N} = [|\psi \rangle_{\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}][h.c.]$. Then it is well known (a consequence of the Schmidt decomposition or of the Araki-Lieb Inequality) that for this pure state

$S({\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N})=0$

$S(\underline{a}_J) = S(\underline{a}_{J^c})$

where $J$ and $J^c$ are disjoint sets whose union equals $\{1, 2, \ldots, N\}$, and where $\underline{a}_J = (\underline{a}_j)_{j\in J}$ and $\underline{a}_{J^c} = (\underline{a}_j)_{j\in J^c}$. For example, if $N=4$, then we have

$S(\underline{a}_1, \underline{a}_2, \underline{a}_3, \underline{a}_4)=0$
$S(\underline{a}_1) = S(\underline{a}_2, \underline{a}_3, \underline{a}_4)$ and permutations
$S(\underline{a}_1, \underline{a}_2) = S(\underline{a}_3, \underline{a}_4)$ and permutations

Pure quantum states thus have a very high degree of symmetry. So much so that we can make a model of their entropy as follows. I’ll call it the RUM (Roots of Unity Model) of pure states.

Define

$S(\underline{a}_J) = \left|\sum_{j\in J} \underline{a}_j \right|$

(note that the entropy contributions are summed coherently rather than incoherently. The latter is common for macroscopic classical systems)

where we now interpret ${\underline{a}_1,\underline{a}_2, \ldots, \underline{a}_N}$ as the $N$th roots of unity. In other words, we are now setting

$\underline{a}_j = {\rm exp}(i\frac{2\pi (j-1)}{N})$ for $j=1, 2, \ldots, N$.

The model works because

$\sum_{j=1}^N \underline{a}_j = 0$

so

$\sum_{j\in J} \underline{a}_j = - \sum_{j\in J^c} \underline{a}_j$

$S(\underline{a}_J) =|\sum_{j\in J} \underline{a}_j| = |\sum_{j\in J^c} \underline{a}_j| = S(\underline{a}_{J^c})$.

Henceforth we will abbreviate $\sum_{j\in J} \underline{a}_j = \sum \underline{a}_J$

I bet a lot of people have discovered RUM before, but I myself discovered it on my own yesterday. I find RUM very helpful because (besides being good for numbing pain and inducing forgetfulness 🙂 ) it helps me visualize better the entropy of a pure quantum state. For example, it “explains” to me why quantum conditional entropies can be negative. Indeed, suppose $J$ and $K$ are two disjoint subsets of $\{1,2,\ldots,N\}$. Then

$S(\underline{a}_J|\underline{a}_K) = S(\underline{a}_{J\cup K} ) - S(\underline{a}_K) =|\sum \underline{a}_{J \cup K}| - |\sum \underline{a}_K|$

Also, the Araki-Lieb and sub additivity inequalities

$|S(\underline{a}_J) - S(\underline{a}_K)| \leq S(\underline{a}_J ,\underline{a}_K)\leq S(\underline{a}_J) + S(\underline{a}_K)$

become the well known triangle inequalities

$\left|\;\;|\sum\underline{a}_J| - |\sum\underline{a}_K|\;\;\right| \leq |\sum\underline{a}_{J \cup K}|\leq |\sum \underline{a}_J| + |\sum \underline{a}_K|$

## April 9, 2012

### Quantum Bayesian Networks, The Ultimate Mixology Guide

Filed under: Uncategorized — rrtucci @ 7:30 pm

Check out my new paper

An Introduction to Quantum Bayesian Networks for Mixed States
by Robert R. Tucci (arxiv abstract)

The paper is a fairly elementary pedagogical introduction. It’s sort of a prelude to a more researchy paper that I am trying to write next.

I’m very pleased that I was able to draw all the pictures of the paper in LaTex “native” fashion, using only LaTex macros (xypic and cancel). The LaTex Gods have been generous to me once again.

This was the original cover of the paper, but my editor nixed it:

Picture Credits:

The drawing of a man making snow angels on the sand of a sandbox is of course The Vitruvian Man by Leonardo da Vinci. So why did Leonardo write backwards? Historians don’t know for certain but many believe it was because Leonardo was left-handed and he wanted to write quickly with a quill pen without smudging the ink with his hand.

The drawing of a letter U with an eyeball resting on the fat (classical) leg of the U could have been a sketch done by an abstract modernist painter like Dali or Picasso, or it could have been a depiction of a creature from another planet, the u-tube cyclops?, but it’s really a drawing made by John Archibald Wheeler (professor at Princeton Univ. for almost 40 years, PhD advisor to Richard Feynman, Hugh Everett and many, many others). The drawing symbolizes the feedback between the observer and the observed in quantum mechanics.

## April 1, 2012

### China’s Muppet Communiqué

Filed under: Uncategorized — rrtucci @ 12:01 am

It seems to me that cyberspying between the US and China is at an all time high right now, and destined to keep growing. As evidence of this, consider the following news articles. Note also the huge number of responses that these articles have elicited throughout the web.

In light of all this spying activity, it didn’t surprise me one bit when I read the following news item today.

WASHINGTON DC — April 1, 2012 — This secret communique, emanating from the highest ranks of the Chinese Communist Party, was recently intercepted by Julian Assange’s WikiLeaks organization:

 To all venerable members of Chinese Communist Party: Dear comrades, disturbing news is: (1) Quantum Computer (量子计算机) has been called “Hydrogen Bomb of Cyberspace” and “Doomsday Machine” by American Imperialists puppet masters (these quotes obtained by venerable cyberspy number 1337). (2) IBM already has blueprint for QC and thinks can build QC in the next 5-10 years. (this data obtained by venerable cyberspy number 1709) (3) NSA is building top-secret cyberspying facility in Utah(cost of 2 billion USD = 12 billion yuan = 4 billion bowls of rice). Completion date: 2013, year of the SNAKE- not good. (this data obtained by venerable cyberspy number 3059) Said General Secretary of the Communist party Xiao Xiao Shūjì: USA is weak, heavily indebted country ruled by rich, 1% of people. China will counter American QC imperialist threat with 5 year and 12 year plans to build DRAGON QC. Cost of 200 billion USD = 1200 billion yuan = 400 billion bowls of rice (each bowl ALSO INCLUDES an egg roll on the side). Completion date: 2014, year of the HORSE- good luck year. Our selling bullet points: (for PowerPoint presentation to members of Chinese Communist Party) Will build DRAGON QC in top-secret Foxconn-Lenovo facility, using smart, productive child labor, and world-class Taiwanese technology. Top Secret Facility will be called Lucky Golden QC Facility Number 1. Will hire, at twice their current salary, most diabolical executives from Goldmann Sachs to do business planning best scientists from American and European universities to do scientific planning phone hacking human hyenas from News Corps to demoralize competition Will use DRAGON QC for codebreaking, development of biological weapons, monitoring of our citizens, and spying on other countries. Will share QC technology with Iran, North Korea, Syria and Nigeria. Will deploy DRAGON QC near US mainland, at our new Bejucal Base in Cuba. Venerable Chinese super spy number 007 has been sleeper mole in USA since 1997. He or she graduated from Harvard. He or she worked at Goldmann Sachs under tutelage of Darth Vader (Lawrence Summers), but left Goldmann Sachs in 2012 because too toxic. Now he or she is highly paid lobbyist like Newt Gingrich. Venerable secret agent 007 advises: “America is China’s muppet. Let’s rip out their eyeballs”. America is slow turtle. China poised to win QC race and dominate cyberspace. “We’ve caught them napping.” (Custer’s last words). (Hee, hee, this is old American joke, according to venerable cyberspy number 1200. Americans pretty funny) Xinhua News Agency, official press agency of the government of the People’s Republic of China, has come up with the following inspirational posters to inform our citizenry of the true news: Five minutes to midnight (Doomsday is when America builds QC) Inside View of American Spy Agency NSAThe Heart of Darkness (in America)

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