Quantum Bayesian Networks

August 20, 2012

Up Yours John Pierce (Said With Echo On)

Filed under: Uncategorized — rrtucci @ 9:41 pm

Check out my new paper

“Capacity Region for Quantum Wiretap Coding”, by Robert R. Tucci, arXiv:1208.3193 (abstract here)

This paper is my answer to John Pierce’s piercing comments. I thank him for goading me from the grave to work on this problem. The paper is very likely to contain numerous flaws, but it’s the best I can do (at least until I learn more).

Let me place it in context. It’s really just the final in a series of 4 papers

  1. “An Introduction to Quantum Bayesian Networks for Mixed States”, by Robert R. Tucci, arXiv:1204.1550 (abstract here)

  2. “Some Quantum Information Inequalities from a Quantum Bayesian Networks Perspective”, by Robert R. Tucci, arXiv:1208.1503 (abstract here)

  3. “Shannon Information Theory Without Shedding Tears Over Delta \& Epsilon Proofs or Typical Sequences”, by Robert R. Tucci, arXiv:1208.2737 (abstract here)

  4. “Capacity Region for Quantum Wiretap Coding”, by Robert R. Tucci, arXiv:1208.3193 (abstract here)

All along, my main goal was merely to write paper 4, but I soon realized that I would have to write papers 1,2,3 first, if people were to follow my crazy notation, which seems to be unlike what everyone else uses.

You may have noticed that

first paper ~ 20 pages
second paper ~ 30 pages
third paper ~ 40 pages
fourth paper ~ 50 pages.

It looks like my verbosity is increasing with time, but no need to worry. I will not attempt to imitate B. Russell’s 10,000 page Principia Mathematica, a book that took 10 years to write and one that nobody has ever read. No chance of that for I burnt out all my fuses while writing paper 4. Now I’m totally exhausted and totally depleted of any new ideas.

How I Swam Across the River Shannon

Filed under: Uncategorized — rrtucci @ 9:18 pm

Check out my new paper

“Shannon Information Theory Without Shedding Tears Over Delta \& Epsilon Proofs or Typical Sequences”, by Robert R. Tucci, arXiv:1208.2737 (abstract here)

The River Shannon is Ireland’s longest river, spanning 224 miles. It starts near the center of Ireland, flowing first south, then west, finally emptying into the Atlantic Ocean. The city of Limerick is situated near the end of the river, on its estuary (where the fresh water and sea water meet). One can swim across the Shannon at many places if one so desires.

My own desire was to swim across the mighty River of Shannon information theory. I had tried to do so many times before. But every time I had dipped my toes into the frigid waters of a standard textbook like the one by Cover and Thomas, I had nearly drowned. And then, I watched the London 2012 Olympics on the telly. And my life was changed forever.  After observing in action that most brilliant of all British Olympic athletes, Sir Rowan Atkinson, I realized that I too had the makings of an Olympian. I immediately saw myself decisively beating Michael Phelps and Lord Byron on a race across the River Shannon, in the doggie paddle swimming style (a style also known as the “breast stroke” in XXX rated blogs). Doing this with the theme song for “Chariots of Fire” playing in the background.

But first I had to learn how to swim more than 3 yards without drowning.

From the movie “Johnny English Reborn”. Johnny kicking thug

I decided that Cover and Thomas was too mathematically rigorous, too fancy pantsy for an athletically challenged body like mine to master. I would have to find a less athletic, more zen-like method to defeat my opponents. I was undaunted. Years of carefully study of British scientific works of the highest caliber (e.g., Harry Potter, Q’s inventions for James Bond, Wallace and Gromit’s Inventions, 10,000 pages of B. Russell’s Principia Mathematica, the 20 cylinder Rolls Royce car engine) had taught me that no scientific problem is unsolvable by the clever British mind. I wondered. What fiendishly high tech MI7 device would Johnny English use if he were caught in my predicament? And then it came to me. Floaties, I needed training floaties. 

I soon conceived of the technical equivalent of floaties for information theory. The steepest descent method. This zen-like method is much easier on the body of an athlete than the method of typical sequences usually favored by Computer Science jocks.

Success came quickly after that. At this moment I stand in the rarefied air of the highest of 3 podia, ready to receive my gold medal… 

Wake up Bob. Wake up!!! It’s 6 AM and you’ve got to take the girls to school before you go to work. 

As the River Shannon ends with a Limerick, I end my story with one too:

A spider that spun webs quite odd
thought Revered Bayes was quite mod
and Claude Shannon too
and quantum bits too
So what did she do?
She wove with her goo
q nets for the Rev and for Claude

References

  1. See Comments Section for comments from Prof. Neri Merhav
  2. YouTube Video “Mr. Bean at the Opening Ceremony of The Olympic Games of 2012 in London”

The Secret Romance Between Information Theory Inequalities and Networks

Filed under: Uncategorized — rrtucci @ 8:30 pm

Check out my new paper:

“Some Quantum Information Inequalities from a Quantum Bayesian Networks Perspective”, by Robert R. Tucci, arXiv:1208.1503 (abstract here)

There are

the inequalities of SIT (Shannon Information Theory) and of Thermodynamics (e.g. that minor inequality \delta S \geq 0)

and then there are

networks,

and never the twain shall meet?

Of course not.

In classical SIT, a well known inequality is the so called Data Processing Inequality which says that any Markov chain

\underline{c}\leftarrow\underline{b}\leftarrow\underline{a}

satisfies

H(\underline{c}:\underline{a})\leq H(\underline{b}:\underline{a})

In other words, the correlation between \underline{a} and \underline{b} is larger than the correlation between \underline{a} and node \underline{c}, which is more distant from \underline{a} than node \underline{b}.

But a Markov Chain is just a super simple network. Why stop there? Why not consider more complicated networks? And why limit yourself to classical SIT. Why not quantum SIT too? See my new paper for a pedagogical introduction to the secret romance between SIT and Bayesian networks, a romance which goes on in both classical and quantum physics.

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