# Quantum Bayesian Networks

## March 24, 2013

### Easy to Customize Application For Unit Conversion

Filed under: Uncategorized — rrtucci @ 4:31 am

As part of their job, experimental physicists must have a very good feeling for the size of the quantities that they are trying to measure when those quantities are expressed in the practical units of measurement commonly used in engineering. On the other hand, in order to make their equations clearer and simpler, theoretical physicists (especially high energy physicists) usually use natural units where Planck’s constant divided by 2 pi, the speed of light, and Boltzmann’s constant are all set to one. (i.e., $\hbar =c=k=1$). But when the time comes to compare a theory with lab measurements, even theorists are forced to deal with engineering units. Even if a theorist writes his theory in natural units, it’s still prudent for him to have a good idea of the typical size of the quantities involved in his theory, with those quantities expressed in engineering units.

Yesterday, while I was thinking about the sizes of certain things, it dawned upon me that JavaScript is an ideal language for writing a unit conversion application.

I wrote a little proof of concept unit conversion application last night. It inter-converts between

• temperature (in degrees Kelvin)
• energy (in eV)
• frequency (in Hz)
• wavelength (in meters)

assuming that $E=k T, E=hf, c=f\lambda$, where $T=$ temperature, $E=$ energy, $f=$ frequency, $\lambda =$ wavelength.)

WordPress.com does not allow the blogs it hosts for free to use JavaScript. It automatically edits JavaScript out. But if you have a WordPress brand blog that runs on your own website, then there is no such limitation for you, and you can add JavaScript code to your blog posts. (WordPress.com does allow its free blogs to display spreadsheets that are public and live in your google account. Cool!)

Since this blog of mine is a cheapie run by WordPress.com, I can’t insert JavaScript directly in this post. Instead I have downloaded the webpage to my website. Here it is. Below is a dead figure (jpg) of the page:

Though trivial, I find this little application very useful and highly mesmerizing. I keep plugging numbers into it and trying to remember physical situations I am familiar with in which those sizes arise.

Of course, there already exist lots of websites and computer widgets that do unit conversion for you. The advantage of my little application is that it is easy to customize and it runs on your computer’s web browser, without any need to download supplementary software libraries and without any need to be connected to the internet.

More detailed explanation of why I think JavaScript is an ideal language for this job

JavaScript is admirably suited for doing all the simple mathematics involved in unit conversion. Plus your computer does all the calculations involved without using any extra software other than a browser, which everyone has. If you use a script written in Perl or PHP or whatever, you either have to download much extra software into your computer, or you have to be connected to the internet and beg a remote server to do those very simple calculations for you.

Another nice thing about JavaScript for writing a unit conversion application is that the finished application is just a simple webpage. Any user can download the webpage into his computer. Then he can run the application simply by opening the webpage on his web browser. Alternatively, he can open the webpage with any text editor, whereupon he can view the source code and easily change/customize it to suit his personal taste and needs.

Besides using this application as is, you can use it as a template to write other unit converting applications that suit your taste and needs. Just change the details such as the number of rows, the text, the numerical constants and the formulas being used, and presto, you can change this to an application that, for example, expresses area in different units like acres, square feet, square miles, square meters, etc.

Even if you know very little about JavaScript, as long as you know a little html and one of the C family of languages (e.g., C, Java, C++. etc.), I think you will find it very easy to customize. Take me as an example: I knew next to nothing about JavaScript a day ago, but since I already know C, C++, Java and html, I found this application trivial to write. I just copied from lots of examples that I found on the internet. There are lots (way too many to mention individually) of VERY GENEROUS people who have posted numerous examples and tutorials and code snippets of html and JavaScript on the internet. Whenever I didn’t know how to do something in JavaScript, I just Googled with wild abandon until I found an example that did what I wanted to do.

JavaScript can also be an ideal language for writing applications that do things other than unit conversion. In general, if you want to make a webpage that interacts with the user and does a simple calculation for him based on his answer to a few button or menu choices or numerical inputs, then JavaScript is a nice way to do this.

## March 14, 2013

### Be a QC MOOC-her

Filed under: Uncategorized — rrtucci @ 6:41 pm

In English, “to mooch” means to get something without paying for it. The word “moocher” has been forever seared into American culture and consciousness by the superb jazz song “Minnie the Moocher”, sung by the highly charismatic and entertaining black performer Cab Calloway (Wikipedia entry for song/ lyrics/ youTube of Calloway singing it). “Minnie the Moocher” was sung by Calloway himself in the hilarious movie “The Blues Brothers” (Wikipedia entry for movie).

In a previous post, I spoke about how MOOCs (Massive Online Open Courses) are revolutionising higher education. You don’t have to pay anything to take them, at least for now, so taking one amounts to mooching.

But can one mooch specifically on the subject of QCs? Yes! There are at least two current MOOC offerings that, in my opinion, are highly relevant to quantum computing:

• (Coursera) Probabilistic Graphical Models, taught by Stanford Prof. Daphne Koller

Next Session: Apr 8, 2013 (11 weeks long)

This course is based on the masterful, 1200 page book on Bayesian and Markov Networks by Daphne Koller and Nir Friedman. (I’ve spoken previously in this blog about their book, here and here). Daphne Koller and Andrew Ng, both Stanford Profs., are the founders of Coursera.

• (EdX) Quantum Mechanics and Quantum Computation, taught by Berkeley Prof. Umesh V. Vazirani

Classes for current term already started on: Feb 11, 2013

Very plain and elementary exposition of quantum computing covering only two algorithms (Shor and Grover). This course makes a very strong case for the claim that there are only two known algorithms for quantum computing.

Daphne’s course never mentions quantum mechanics, but I think that bayesian networks are FUNDAMENTAL to quantum computers. So I enthusiastically recommend her course to all QC aficionados.

I recommend Daphne’s course more strongly than Umesh’s. For one thing, Umesh’s course is very standard and Daphne’s is one of a kind. Besides, Daphne belongs to the Computer Science tribe that believes in applications (i.e., the programmers, the shakers and movers of the internet, the Big Shoulders, the brawny Code Butchers for the World), whereas Umesh belongs to that other CS tribe that uses the word application maybe once every five years (i.e., the complexity theorists). (Quantum complexity theorists are also notable for how slowly they move. Sometimes I suspect that molasses or Heinz ketchup is coursing through their veins. I think the last time a quantum complexity theorist invented a new, mildly useful QC algorithm was before you were born, sonny.)

Another bit of news: Recently, a “symposium” was held to celebrate John Preskill’s 60th birthday. The conference also celebrated the instituti which John founded, the IQIM. (I believe IQIM stands for I-CK-Y M-ashuganas. I believe the IQIMs are very active as a group, and that they are even considering putting out a calendar for selling power tools like electric drills and such, with scantily clad women posing with the tools.)

## March 4, 2013

### In Love With My E-Reader

Filed under: Uncategorized — rrtucci @ 8:00 pm

I’m currently hard at work trying to adapt Judea Pearl’s causality theory to quantum mechanics. I believe I’m making lots of progress, maybe enough to come out eventually (soon?) with a paper with my findings. But until then, I have to talk about something, right? So let me mention a device that has delighted me very much of late.

I recently bought an e-reader—Amazon’s Kindle “Paperwhite” $119 model. This model just came out on Oct 1, 2012. Its screen is 6 inches, 2-point touch sensitive, black/white e-ink and illuminated. I absolutely love it. I bought it mainly for reading research papers in pdf form, downloaded from arXiv or elsewhere. I find I no longer have to make paper copies of research papers I want to read. I just download their pdfs from my computer onto the e-reader and read them there. I find that reading them on the e-reader is more comfortable than reading them on a computer screen, and almost as comfortable as reading a paper copy. But the e-reader beats paper copies in that you can carry a huge number of papers in one lightweight, sleek, compact device, and you can do computer searches on them. My device has a 2 GB storage (some of that is used up by the operating system). That’s enough memory to store more than a thousand papers. ArXiv papers that were written originally in LaTex and turned into pdf, are quite legible on my e-reader, including all the mathematical symbols, and you can change the viewing size. Of course, I also use my e-reader to read novels. I’ve done quite a lot of that too. I find my e-reader to be a MUCH MORE convenient and pleasant way of reading novels than paper-backs or hard-backs. I believe this for many reasons that I won’t go into here, since this is a blog devoted to quantum computing. Talking about quantum computing, let me mention that Jeff Bezos, the visionary founder of Amazon.com, is a believer in the future of quantum computers; so much so that in 2012, he invested in D-Wave. So I am proud to use this blog to throw in a plug for his e-reader, although I would recommend it even if Jeff Bezos had no interest in quantum computers. Even if Jeff were a computational complexity theorist, I would still recommend this e-reader. ## February 27, 2013 ### My Prediction For 2013: “Professor, I’ll have some fries with that exam” Filed under: Uncategorized — rrtucci @ 4:34 pm On January of each year, it has become my habit to post in this blog a yearly prediction. I forgot to make one this year, so here it is. This prediction is not directly related to quantum computers, but it might be related to them indirectly. The prediction has to do with the future role of universities and their professors in scientific research. Although currently most quantum computing research is done by university professors, this may not be true for much longer. The role of university professors is currently undergoing rapid, drastic changes. In the future, university professors may be so busy teaching or performing clerical duties that they won’t be able to devote sufficient time to their scientific research activities. If you are young and want to do research in quantum computing, maybe you should consider starting a company rather than becoming a university professor. University level online education has been around for more than a decade, some of it offered by universities themselves. However, most universities, afraid that such modern teaching techniques could spell the end of their highly lucrative racket, have portrayed online education as being only a supplementary educational tool, in no sense to be thought of as a serious alternative to a campus education. But the times, they are a-changing. The Khan Academy, started in 2006, put universities to shame, showing how just one person can teach Calculus and other engineering subjects FOR FREE to tens of millions of people, in dozens of languages. Meanwhile, famous American universities have continued to increase their yearly tuitions to astronomical levels (MIT yearly tuition for 2012-2013:$42,000).

The newest trend in online education, and one which is catching on like wildfire (Coursera only started 2 years ago and already has had millions of enrollees), is MOOCs, massive open online courses, so called because these courses have “massive” enrolments, sometimes in the tens of thousands of students. Enrollment is free, at least for now. Tests are only of the multiple choice kind and machine graded (like SATs), but you get to ask your questions to thousands of attentive live ears.

Two of the biggest MOOC players are Coursera and EdX. Coursera was started by Stanford profs, and EdX by MIT/Harvard profs. I like to call Coursera the robber baron capitalist team and EdX the Marxist/Stallman-ist communist one. The capitalists have venture capital funding and tout themselves as a startup, whereas the communists were started as an effort to stymie the capitalists, and they have no VC funding (they are funded by charitable grants). The capitalists and the communists are in fierce competition. Currently, the capitalists are trouncing the communists. Here is the score board, as of Feb 2013:

Capitalist Team (Coursera)
62 Universities (https://www.coursera.org/universities).
323 courses.

http://en.wikipedia.org/wiki/Coursera

https://www.coursera.org/

Communist Team (EdX)
12 Universities but only 3 (MIT, Harvard, Berkeley) currently offering courses.
25 courses.

http://en.wikipedia.org/wiki/EdX

https://www.edx.org/

I think that the EdX people are destined to lose the competition and fade away into oblivion. To begin with, their name really sucks. Furthermore, my impression is that their heart isn’t really in what they are doing. Their goal seems to be merely to perpetuate the old thinking that online education is a nice supplement but not an alternative to campus education, and that universities are a not-for-profit institution (that might be true technically, but is it true spiritually?). Nobody believes those fibs anymore.

My prediction: (I’m not saying that this prediction is necessarily what I want the future to be. I’m just saying that I believe that the inexorable market forces will produce this result in the end.)

My prediction is that in future (in the next 10-20 years?), most research will no longer be done by universities but rather by companies or public labs, like Bell Labs, Google, IBM, NIST, CERN, etc. The most prestigious universities will become for profit businesses. Each of them will have dozens or even thousands of frachisees throughout the world (The McDonald’s model). Franchisees will be entrusted with running miniscule (by current university campus standards) outlets where students can take supervised tests. Also, those outlets will be used to give the occasional lab course and to hold social meetups. Students will do most of their learning online. They will no longer have to travel to another country to go to McMIT or McHarvard or McStanford. Most university professors, with the exception of the very few superstars that actually “act” in the training videos, will perform menial roles similar to those of McDonald’s minimum wage employees, and have to ask questions like, would you like some fries with that exam? Most present day university professors are royal peacocks enamored with themselves. This will be quite a blow to their egos.

## February 19, 2013

### Quantum Noyce, We Need You!

Filed under: Uncategorized — rrtucci @ 5:38 pm

American Experience” (AE) is a superb series of documentaries about quirky and transformational events of American History. It’s made by American public TV (more specifically, by the PBS station WGBH in Boston). AE has produced about 250 episodes in its 25 years of existence. Wikipedia has a complete list of all the episodes that AE has produced throughout the years.

I find most AE programs fascinating. Not only do they present the dry facts of history, but also its gossipy details, making it come alive as a vivid portrayal of human beings, with all their strengths and weaknesses. The AE and Ken Burns’ documentaries would make a wonderful complement to any American History course.

I would recommend any of the AE episodes. They are all great. However, one has to begin somewhere, and this being a blog about quantum computing, let me emphasize one recent episode entitled “Silicon Valley” about the history of microprocessors (the precursors and lesser siblings of quantum computers).

Here is the teaser, taken from the PBS website, for the “Silicon Valley” episode:

In 1957, decades before Steve Jobs dreamed up Apple or Mark Zuckerberg created Facebook, a group of eight brilliant young men defected from the Shockley Semiconductor Company in order to start their own transistor business. Their leader was 29-year-old Robert Noyce, a physicist with a brilliant mind and the affability of a born salesman who would co-invent the microchip — an essential component of nearly all modern electronics today, including computers, motor vehicles, cell phones and household appliances.

The entire film (82 mins.) and its transcript are available at this PBS webpage (Information magically available to anyone. What a wonderful age we live in, a golden age for education and research, reminiscent of 1453, when Constantinople fell to the Turks and the Renaissance Period officially began).

Check out also

http://en.wikipedia.org/wiki/Robert_Noyce

The Wikipedia article on Noyce includes links to his milestone patents. Noyce, together with Gordon Moore and Andy Grove, went on to found the Intel company. Intel built the first microprocessor and is today the world’s largest semiconductor maker. A company that was once insanely great but which has lost its bearings and its once highly innovative spirit. I say this because Intel has done next to nothing to help develop the next generation of microprocessors, the quantum computer ones. (Related blog posts: Intel-The Reluctant Natural, and Intel Waffles)

Call me a zealot, or a fanatic, but there is no doubt in my mind that quantum computers will be built some day soon. As Lubos Motl pointed out recently and quite eloquently (see comment 47 here), there appears to be nothing in fundamental physics that forbids us from building QCs. Furthermore, over the last ten years, experimentalists have made smooth and substantial progress in taming the dragon of decoherence.

Now all we need is for our Quantum Noyce to show up. Surely, our Messiah has already been born and is walking among us.

Other AE episodes of scientific & technological interest (this is a subset of the Wikipedia list of all AE episodes):

INNOVATIONS
Edison’s Miracle of Light
Wright Stuff, The (on the Wright brothers)
Telephone, The
Big Dream, Small Screen (on television)
Race for the Super Bomb
Wizard of Photography, The
Streamliners: America’s Lost Trains
Great Transatlantic Cable, The
Living Weapon, The (on biological weapons)
Panama Canal
Grand Coulee Dam

AVIATION/SPACE
Great Air Race of 1924, The
Satellite Sky, The
Race to the Moon

Silicon Valley

BIOGRAPHIES
Henry Ford
Richest Man in the World, The: Andrew Carnegie
Rockefellers, The (1)
Rockefellers, The (2)

Of course, all the Nova episodes are also of scientific & technological interest.

## February 8, 2013

### Lubos Motl on the Road to Damascus

Filed under: Uncategorized — rrtucci @ 9:42 pm

Tales of religious conversion are of widespread interest.

A famous one is the one of Saint Paul the Apostle (5 AD-67 AD), from Tarsus, Turkey. During the first half of his life, Saint Paul was an orthodox Jew who ruthlessly and zealously persecuted those Jews who were Christians, i.e., followers of the teachings of Jesus Christ who believed Jesus to be the Messiah promised to Israel in the Old Testament. (Jesus was already dead and supposedly resurrected by then.) During a trip from Jerusalem to Damascus, Paul fell from his horse and hit his head, whereupon he had a vision of Jesus Christ, enveloped by a great light (of millions of lumens). The light of the vision was so powerful that it left him blind for 3 days, at the end of which he regained his sight thanks to a miracle performed on him by a saint called Ananias of Damascus. After the vision, Saint Paul dedicated the rest of his life to spreading the gospel of Jesus Christ. He went on to found many chapters of the Christian Fraternity throughout the Mediterranean basin. The 13 epistles of the New Testament are due to him. (An epistle is a book from the New Testament of the Bible in the form of a letter from an Apostle: eg. Saint Paul’s epistle to the Romans.) Saint Paul appears to have been on friendly terms with the first Christian pope, Saint Peter, although there is at least one documented incident when they had a spat (Incident at Antioch)

More recently, Lubos Motl, an orthodox String Theorist committed to persecuting the Smolin/Woit crackpots of the world, fell from his bicycle and hit his head during a trip from Jerusalem Street to Damascus Street in his home town of Pissner, Czechia. The fall blinded Lubos for 3 days, but then a Czech porn star and amateur quantum computerist named Ana restored his sight. In gratitude, Lubos has now decided to spend the rest of his life spreading the sacred word of how quantum computers ain’t all that dumb, since they are just a trivial consequence of string theory. Lubos intends to write at least 130 modern epistles, aka blog posts, about quantum computing and to spread the QC faith throughout the String Theory basin. He has even uttered, though reluctantly, some mildly flattering words about the current QC pope, Scott Aaronson. Lubos and il Papa are in complete agreement in their beliefs that there is nothing in physics that will prevent us from building quantum computers. Despite this agreement, since these are two dogs that love to bark, they still find reasons to have friendly barking matches with each other. Such barking matches can be quite amusing to witness since these are undoubtedly two very smart dogs. Who knows what the QC future holds: maybe The Lubos Quantum Computer, built by Rumanian slave child laborers at the Lubos Plant in Czechia?

The Conversion of St. Paul, by Nicolas-Bernard Lepicie, 1767. (Lubos falls off from his bicycle, which resembles a horse, by artistic license)

Ananias restoring the sight of Saint Paul, by
Pietro da Cortona, 1631 (St. Ananias can be seen patting Lubos on the head for being a good String Theorist. His daughter Ana can be seen right behind him.)

## February 1, 2013

### Quantum Discord Brains

Filed under: Uncategorized — rrtucci @ 5:59 pm

Of course, my blog on quantum computing would not be complete without a post on quantum discord. So here it goes:

They number in the hundreds and make a lot of noise.

They hate quantum entanglement because they say it is too taxing to their brains. “Too hard. Doesn’t taste good”, they say.

They appear to be involved in deep conversation with each other about an important scientific subject. “Quantum discord brains taste better than quantum entanglement brains”, they cry.

I for one can’t understand what they are saying. Their conversation sounds to me like simply a series of grunts and moans, punctuated by an occassional wistful cry of either:

“Let’s eat more quantum discord brains”

“Quantum discord brains is a unique quantum resource”

“We don’t need to eat no stinking quantum computing algorithms. Quantum discord brains is all the nutrition we need”

“Let’s meet at Vlatko Vedral’s home. He good zombie leader. He set up 2012 quantum discord zombie jamboree for us”.

I left my heart in San Francisco“. “I left my brain in Singapore CQT” is a popular song among zombies.

References

The Quantum Discord Zombie Hordes are upon us. This would make a good poster for a quantum computing algorithms or a quantum entanglement conference. (I’m not the painter of this awesome drawing. I’m not sure of its provenance)

## January 30, 2013

### Blogging can be Hard!

Filed under: Uncategorized — rrtucci @ 7:27 pm

Just deleted a blog post. Realized after posting it that it was dull and dumb. Some days I have a very hard time producing a blog post that I can feel mildly proud of.

PAM Dirac was notoriously taciturn. (See this “Interview With Dirac“) I should be more like him and stop blogging and incriminating myself, exposing my appalling ignorance and narrow-mindedness. When you aren’t too bright, like me, it’s a good strategy to be taciturn. That way others will think that you are silent because you are busy having deep thoughts like a Dirac. They won’t realize that the reason you are so quiet is that you are still trying to understand the first slide of the Powerpoint presentation.

## January 22, 2013

### Causality and Quantum Mechanics

Filed under: Uncategorized — rrtucci @ 4:39 pm

This blog is a travelogue of sorts for a globe trekker of sorts (me). It’s a Captain’s log for my tiny one man boat. Here is my newest entry.

These are exciting times for this traveller. In the past few weeks, I’ve been gearing up for my next trip, in which I plan to visit the country of Causality. It’s a huge country, so one could easily get lost in it. To prevent that from happening to me, I’ll be doing a journey that has been done before with much success. I’ll be retracing the steps of an earlier, very famous and intrepid, Bayesian networks explorer, Captain Judea Pearl, who has done for Causality what Captain Cook did for the Pacific.

One can often reverse some of the arrows of a Bayesian net without changing the full probability distribution that the net represents. This leads some people to say that Bayesian networks are not causal. I like to say instead that the basic theory of B nets, all by itself, is causally incomplete. Judea Pearl has taught us that one can build a super-structure on top of that basic foundation, to address causality issues more fully. The object of the superstructure is to compare at least two B Nets, the original one and another one obtained from the first by doing an “intervention” or “operation, or “surgery” on it. Pearl has even devised a “calculus” for that superstructure.

Here are some references recounting captain Pearl’s Pacific/Causality voyages:

Wikipedia has two entries (here and here) on Causality that are pretty good.

The importance of Causality, or the big C, has been recognized by mankind for thousands of years in fields as diverse as: religion, philosophy, logic, history, jurisprudence, psychology, medicine, pharmacology, epidemiology, mathematical probability and statistics, and physics.

C is a type of bond that connects, in varying degrees, two events across time, or two statements in an if-then clause in logic, or two events in probability theory.

In physics, which is my gig, C is everywhere you look. It could be called Nature’s Hammer.

C is virtually synonymous, or at least joined at the hip, with what we call a force. In Newtonian mechanics, a force causes an acceleration. The force is the cause and the acceleration is the effect. A force by any other name is still a cause. The force of a punch to the gut or a hammer crushing glass. The force of a giant maelstrom pulling a sea vessel, like a twig, inexorably towards its center. All manifestations of C. Since forces are a major concern in physics, C is too. But its relationship to force is not the only reason why physicists are enamored with C.

C often relates two events close in time and space. So any temporal phenomenon in physics, which is just about everything in physics, is deeply related to C. For example, in special relativity, which tells us how the clocks in different inertial frames are related, C is there. It’s lurking beneath the surface, in all that talk about simultaneity of events, light cones, spacelike and timelike intervals and frame dependance of observations. Other examples of phenomena related to C because of their temporal nature: the Second Law of thermodynamics, and processes with feedback. As in the case of special relativity, C is good at imposing restrictions on what a system can do.

And then, as is its wont, quantum mechanics throws a spanner into the C works. Quantum mechanics supercedes our classical ideas of what C is and how it should behave. Purely classical physics is based on differential equations with initial and/or boundary conditions. These seem to imply that nature and C are deterministic. That C is merely the hand of destiny. But then quantum mechanics tells us that no, nature and C have an inescapable, intrinsic probabilistic aspect to them. But it gets worse. Quantum mechanics tells us that nature and C are not just gambling with an ordinary probability theory, but with a probability theory based on complex amplitudes, and that makes it rife with quantum entanglement and coherence weirdness (and richness). And God only knows how C will manifest itself in our final theory of quantum gravity, but it’s certain to be in an interesting way, full of flair and panache.

C and quantum mechanics are both ubiquitous topics in Physics, so these two topics are likely to overlap in many ways, some already known, others awaiting to be discovered.

So far, Captain Pearl has not done much exploring of C in quantum mechanics. He has not sailed his uniquely outfitted man of war, the HMS Bayesian Networks, too close to those shores yet. This is a great opportunity for quantum information theory and quantum computing explorers.

Causality a work!

Bullet ripping through Jack of Hearts. High speed photography by MIT’s Harold (Doc) Edgerton.

Artist Harry Clarke’s 1919 illustration for “A Descent into the Maelström”

Wikipedia entry for Edgar Allan Poe’s short story “Descent into the Maelstrom” here

## January 12, 2013

### Conditional Ageing Inequality or Me Raising CAIN

Filed under: Uncategorized — rrtucci @ 7:52 pm

As mentioned in my previous post, the one entitled “The devil and Reverend Bayes“, this week I published two papers on arXiv. Below, you’ll find an excerpt taken from the introduction section of one of those papers. This excerpt sums up nicely what I’ve been doing in the last 3 months. Thanks to Henning Dekant for suggesting that I look at the work of Sagawa and Ueda.

This paper originated as an attempt to understand a series of papers (Refs.[5] to [11]) by Sagawa, Ueda and coworkers (S-U) in which they claim that the standard Second Law of thermodynamics does not apply to non-equilibrium processes with feedback (i.e., Maxwell demon type processes). They give a generalization of the Second Law that they claim does apply to such processes. Although I agree in spirit with much of what S-U are trying to do, and I profited immensely from reading their papers, I disagree with some of the details of their theory. I discuss my disagreements with the S-U theory in a separate paper, Ref.[12]. The goal of this paper is to report on my own theory for generalizing the Second Law so that it applies to processes with feedback. My theory agrees in spirit with the S-U theory, but differs from it in some important details.

Let me explain the rationale behind my theory.

Suppose we want to consider a system in thermal contact but not necessarily in equilibrium with a bath at temperature $T$. Let $\underline{X}$ denote all non-thermal variables (fast changing, not in thermal equilibrium) and let $\underline{\Theta}$ denote all thermal variables (slow changing, in thermal equilibrium) describing both the system and bath. Let $\tau$ denote time. For any operator $\Omega_\tau$, define $\Omega_\tau|_{\tau=\tau_1}^{\tau_2}= \Omega_{\tau_2}-\Omega_{\tau_1}$. My slight generalization of the Second Law is

$S_\tau(\underline{\Theta}_\tau|\underline{X}_\tau)|_{\tau=0}^{\tau}\geq 0,$       (1)

where $S(\underline{a}|\underline{b})$ is the conditional entropy (i.e., conditional spread) of $\underline{a}$ given $\underline{b}$. I call Eq.(1) the conditional ageing inequality (CAIN). The standard Second Law corresponds to the special case when there are no $\underline{X}_\tau$ variables, in which case Eq.(1) reduces to

$S_\tau(\underline{\Theta}_\tau)|_{\tau=0}^{\tau}\geq 0 .$          (2)

The standard Second Law could be described as unconditional ageing, or simply as ageing.

Now, what is the justification for the CAIN? The justification for the Second Law Eq.(2) is that the superoperator that evolves the overall probability distribution in the classical case (or the overall density matrix in the quantum case), from time 0 to $\tau$, increases entropy because it can be shown to be doubly stochastic in the classical case (or unital\footnote{A superoperator is unital if it maps the identity matrix to itself.} in the quantum case). The justification for the CAIN is the same, except that the evolution superoperator is doubly stochastic (or unital) only if the non-thermal variables are held fixed during the evolution. The CAIN is not true for all evolution superoperators. Our hope is that it applies to systems of interest that commonly occur in nature.

The goal of this paper is to study the consequences of the CAIN. In particular, we apply the CAIN to four cases of the Szilard engine: for a classical or a quantum system with either one or two correlated particles.

Besides proposing this new inequality that we call the CAIN, another novel feature of this paper is that we use quantum Bayesian networks for our analysis of Maxwell demon type processes.

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