Quantum Bayesian Networks

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.

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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!
dominoes

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

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"

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.

January 9, 2013

The Devil and Reverend Bayes

Filed under: Uncategorized — rrtucci @ 2:44 pm

While Daniel Webster (or Reverend Bayes) argues, the devil whispers in the judge's ear (jpeg from Wikipedia)

While Daniel Webster (or Reverend Bayes) argues, the devil whispers in the judge’s ear (jpeg from Wikipedia)

The short story “The Devil and Daniel Webster” (1936), by Stephen Vincent Benét, appears in many anthologies of classic American short stories. It is commonly read in American high school English classes. It was even made into an academy award-winning movie in 1941.

In the short story, a New Hampshire farmer named Jabez Stone is down on his luck so he sells his soul to the devil in return for 7 years of prosperity. When the devil, called Mr. Scratch, finally comes to collect on his contract, fellow New Hampshireman and legendary orator Daniel Webster takes the devil to court, and acts as Jabez’s lawyer. The judge and jury are to be hand-picked by Mr. Scratch on condition that they all be Americans. Mr. Scratch of course picks some of the most despicable figures in American history. I won’t tell you who wins the court battle. I hope you’ll read the short story to find out.

The vivid imagination and skilful pen of Stephen Vincent Benét are amply in display in this short story. Benét’s prose has a strong musical and rhythmical quality to it. This is not surprising, for he also wrote a lot of poetry. He even won a Pulitzer prize for his wonderful, long narrative poem “John Brown’s Body”. I’ve personally read a fair amount of Benét’s work, because I like his style so much. Unfortunately, he died at the early age of 44 of a heart attack. However, he did write quite a lot of stuff, considering his early demise.

A contest between a man and the devil: Such contests have been described since time immemorial in numerous works of art. I’m hard pressed to think of a more universally appealing or riveting type of contest.

In my version of the contest, a beautiful mistress called Miss Thermica (short for Miss Thermodynamics, or Miss Hottie) has been flirting with the devil. Mr. Scratch alleges that because she was given so much beauty, he now owns her soul, which is called The Second Law. Revered Bayes, a legendary orator, and one of the few men in existence with enough courage to confront Old Scratch, takes Ole pointy ears to court to prevent him from taking possession of the Second Law. Want to read a transcript of the court proceedings for this case? Riveting stuff! If so, then check out my new papers

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