## A Paradox of Toom's Rule?

Science is slow.  You can do things like continue a conversation with yourself (and a few commenters) that started in 2005.  Which is what I’m now going to do  The below is probably a trivial observation for one of the cardinals, but I find it kind of interesting.

Let’s begin by recalling the setup.  Toom’s rule is a cellular automata rule for a two dimensional cellular automata on a square grid.  Put +1 and -1′s on the vertices of a square grid, and then use the following update rule at each step: “Update the value with the majority vote of your own state, the state of your neighbor to the north, and the state of your neighbor to the east.”  A few steps of the rule are shown here with +1 as white and -1 as blue:

As you can see Toom’s rule “shrinks” islands of “different” states (taking away such different cells from the north and east sides of such an island.)  It is this property which gives Toom’s rule some cool properties in the presence of noise.

So now consider Toom’s rule, but with noise.  Replace Toom’s update rule with the rule followed by, for each and every cell a noise process.  For example this noise could be to put the cell into state +1 with p percent probability and -1 with q percent probability.  Suppose now you are trying to store information in the cellular automata.  You start out at time zero, say, in the all +1 state.  Then let Toom’s rule with noise run.  If p=q and these values are below a threshold, then if you start in the +1 state you will remain in a state with majority +1 with a probability that goes to one exponentially as a function of the system size.  Similarly if you start in -1.  The cool thing about Toom’s rule is that this works not just for p=q, but also for some values of p not equal to q (See here for a picture of the phase diagram.)  That is there are two stable states in this model, even for biased noise.

Contrast Toom’s rule with a two dimensional Ising model which is in the process of equilibriating to temperature T.  If this model has no external field applied, then like Toom’s rule there is a phase where the mostly +1 and the mostly -1 states are stable and coexist.  These are from zero temperature (no dynamics) to a threshold temperature T, the critical temperature of the Ising model. But, unlike in Toom’s rule, if you now add an external field, which corresponds to a dynamics where there is now a greater probability of flipping the cell values to a particular value (p not equal to q above), then the Ising model no longer has two stable phases.

In fact there is a general argument that if you look at a phase diagram as a function of a bunch of parameters (say temperature and applied magnetic field strength in this case), then the places where two stable regimes can coexist has to be a surface with one less dimension than your parameter space.  This is known as Gibbs’ phase rule.  Toom’s rule violates this.  It’s an example of a nonequilibrium system.

So here is what is puzzling me.  Consider a three dimensional cubic lattice with +1,-1 spins on its vertices. Define an energy function that is a sum over terms that act on the spins on locations (i,j,k), (i+1,j,k), (i,j+1,k), (i,j,k+1) such that E = 0 if the spin at (i,j,k+1) is in the correct state for Toom’s rule applied to spins (i,j,k), (i+1,j,k), and (i,j+1,k) and is J otherwise.  In other words the terms enforce that the ground state locally obey’s Toom’s rule, if we imagine rolling out Toom’s rule into the time dimension (here the z direction). At zero temperature, the ground state of this system will be two-fold degenerate corresponding to the all +1 and all -1 state.  At finite temperature this model well behave as a symmetric noise Toom’s rule model (see here for why.)  So even at finite temperature this will preserve information, like the d>2 Ising model and Toom’s CA rule.

But since this behaves like Toom’s rule, it seems to me that if you add an external field, then this system is in a bit of paradox.  On the one hand, we know from Gibb’s phase rule, that this should not be able to exhibit two stable phases over a range of external fields.  On the other hand, this thing is just Toom’s rule, laid out spatially.  So it would seem that one could apply the arguments about why Toom’s rule is robust at finite field.  But these contradict each other.  So which is it?

## 4 Pages

Walk up to a physicist at a party (we could add a conditional about the amount of beer consumed by the physicist at this point, but that would be redundant, it is a party after all), and say to him or her “4 pages.”  I’ll bet you that 99 percent of the time the physicist’s immediate response will be the three words “Physical Review Letters.”  PRL, a journal of the American Physical Society, is one of the top journals to publish in as a physicist, signaling to the mating masses whether you are OK and qualified to be hired as faculty at (insert your college name here).  I jest!  (As an aside, am I the only one who reads what APS stands for and wonders why I have to see the doctor to try out for high school tennis?)  In my past life, before I passed away as Pontiff, I was quite proud of the PRLs I’d been lucky enough to have helped with, including one that has some cool integrals, and another that welcomes my niece into the world.

Wait, wht?!?  Yes, in “Coherence-Preserving Quantum Bits” the acknowledgement include a reference to my brother’s newborn daughter.  Certainly I know of no other paper where such acknowledgements to a beloved family member is given.  The other interesting bit about that paper is that we (okay probably you can mostly blame me) originally entitled it “Supercoherent Quantum Bits.”  PRL, however, has a policy about new words coined by authors, and, while we almost made it to the end without the referee or editor noticing, they made us change the title because “Supercoherent Quantum Bits” would be a new word.  Who would have thought that being a PRL editor meant you had to be a defender of the lexicon?  (Good thing Ben didn’t include qubits in his title.)

Which brings me to the subject of this post.  This is a cool paper.  It shows that a very nice quantum error correcting code due to Bravyi and Haah admits a transversal (all at once now, comrades!) controlled-controlled-phase gate, and that this, combined with another transversal gate (everyone’s fav the Hadamard) and fault-tolerant quantum error correction is universal for quantum computation.  This shows a way to not have to use state distillation for quantum error correction to perform fault-tolerant quantum computing, which is exciting for those of us who hope to push the quantum computing threshold through the roof with resources available to even a third world quantum computing company.

What does this have to do with PRL?  Well this paper has four pages.  I don’t know if it is going to be submitted or has already been accepted at PRL, but it has that marker that sets off my PRL radar, bing bing bing!  And now here is an interesting thing I found in this paper.  The awesome amazing very cool code in this paper  is defined via its stabilizer

I I I I I I IXXXXXXXX; I I I I I I I ZZZZZZZZ,
I I IXXXXI I I IXXXX; I I I ZZZZ I I I I ZZZZ,
IXXI IXXI IXXI IXX; I ZZ I I ZZ I I ZZ I I ZZ,
XIXIXIXIXIXIXIX; Z I Z I Z I Z I Z I Z I Z I Z,

This takes up a whopping 4 lines of the article.  Whereas the disclaimer, in the acknowledgements reads

The U.S. Government is authorized to
reproduce and distribute reprints for Governmental pur-
poses notwithstanding any copyright annotation thereon.
Disclaimer: The views and conclusions contained herein
are those of the authors and should not be interpreted
as necessarily representing the official policies or endorse-
ments, either expressed or implied, of IARPA, DoI/NBC,
or the U.S. Government.

Now I’m not some come-of-age tea party enthusiast who yells at the government like a coyote howls at the moon (I went to Berkeley damnit, as did my parents before me.)  But really, have we come to a point where the god-damn disclaimer on an important paper is longer than the actual definition of the code that makes the paper so amazing?

Before I became a ghost pontiff, I had to raise money from many different three, four, and five letter agencies.  I’ve got nothing but respect for the people who worked the jobs that help supply funding for large research areas like quantum computing.  In fact I personally think we probably need even more people to execute on the civic duty of getting funding to the most interesting and most trans-form-ative long and short term research projects. But really?  A disclaimer longer than the code which the paper is about?  Disclaiming, what exactly?  Erghhh.

## Non-chaotic irregularity

In principle, barring the intervention of chance, identical causes lead to identical effects.  And except in chaotic systems, similar causes lead to similar effects.  Borges’ story “Pierre Menard” exemplifies an extreme version of this idea: an early 20′th century writer studies Cervantes’ life and times so thoroughly that he is able to recreate several chapters of “Don Quixote” without mistakes and without consulting the original.

Meanwhile, back at the ShopRite parking lot in Croton on Hudson, NY,  they’d installed half a dozen identical red and white parking signs, presumably all from the same print run, and all posted in similar environments, except for two in a sunnier location.

The irregular patterns of cracks that formed as the signs weathered were so similar that at first I thought the cracks had also been printed, but then I noticed small differences. The sharp corners on letters like S and E,  apparently points of high stress, usually triggered near-identical cracks in each sign, but not always, and in the sunnier signs many additional fine cracks formed.

Another example of reproducibly irregular dynamics was provided over 30 years ago by Ahlers and Walden’s experiments on convective turbulence, where a container of normal liquid helium, heated from below, exhibited nearly the same sequence of temperature fluctuations in several runs of the experiment.

## El Naschie works on entanglement now

El Naschie (top), shown photoshopped in with three Nobel laureates.

The Journal of Quantum Information Science will not be getting any of my papers starting today, because today is when I learned that they recently published the following gemA Resolution of Cosmic Dark Energy via a Quantum Entanglement Relativity Theory, by M. El Naschie.

Upon closer inspection, it isn’t hard to see why they published this paper. It’s because  ”El Naschie is very highly regarded in the community” and is “always spoken of as a possible Nobel prize candidate”. And as the great man himself has said, ”Senior people are above this childish, vain practice of peer review”, so there was no need for that.

Oh, but despite the apparent lack of peer review, they do have a \$600 article processing charge for open access. I wonder what costs these charges are meant to offset if the “submit” button just puts the article straight into the publication? Hmmm, I hope that the journal didn’t simply accept money in exchange for publishing the paper under the pretense of “open access”! Golly, that would be unethical.

The name of this post was shamelessly stolen from Gil Kalai’s popular series Test Your Intuition. But today’s post will be testing our physics intuition, rather than our mathematical intuition. Although this is a quantum blog, we’ll look at the behavior of a classical fluid.

The question is: what happens when you soak a washcloth with water and then ring it out… in zero gravity?

Think about it for a few minutes before watching the result of the actual experiment below.

Posted in Physics, Puzzle, Science | 1 Comment

## Sydney Postdoctoral Fellowships

The University of Sydney is now accepting applications for the Sydney Postdoctoral Fellowships. These are very competitive and prestigious university-wide fellowships, with terrific salary and benefits: a 3 year appointment with a A\$99,000 annual salary and a A\$25,000 discretionary research fund for travel, visitors, or equipment.

Because they are so competitive, you have to arrange for a faculty sponsor before applying. If you are interested in applying and joining the quantum physics group, then read the Sydney Postdoctoral Fellowship guidelines here:http://bit.ly/ZZ2r26. If you qualify, send me an email (sflammia at physics.usyd.edu.au) with a cover letter that briefly describes your qualifications and a short CV. In particular, please specify how you see yourself fitting in and complementing work within the rest of the quantum physics group at Sydney.

There are a few relevant deadlines: the deadline to secure a faculty sponsor is Friday 3 May, but the actual application deadline is 31 May, so you will have plenty of time to finish the application. However, because of the initial deadline, I will only consider applications before Tuesday the 30th of April.

## Q-Tube

This is a rare gem: Four lectures on quantum mechanics by Paul Dirac… on YouTube! Here’s the first one.

Also, the Q+ online lecture series continues to go strong, bringing in a steady stream of high-quality speakers. This month constitutes the “Nobel lecture”, and will be given by Dietrich Leibfried of NIST Boulder, in lieu of Dave Wineland, on April 23rd at 5pm UK time. The title is “Towards scalable quantum information processing and quantum simulation with trapped ions”, and it’s sure to be a great talk. Though the number of live video streams will be limited, you can go to the Q+ website to reserve a spot, or wait until after the lecture and watch a recording.

Posted in Quantum | 5 Comments

## Reflections on the Discord Bubble

The following is a guest post by Marco Piani.

A couple of months ago Steve wrote a post on “the discord bubble“. Let me try to provide a summary of his post and of his thoughts.

There have been too many works, too often of insufficient quality either technically or conceptually (“pointless” works in Steve’s words) about a property of quantum systems and correlations—the so-called quantum discord—that has not been proven yet to be key in our understanding of nature or of the `inner workings’ of quantum information processing. That is, such flurry of activity does not appear to be justified and is rather due to “hype”; most importantly, the product of such activity is way too often of questionable quality. Steve goes on to suggest that the bubble needs to be put under control, so that we would go down to a reasonable rate of publications on the subject, hopefully of higher average quality. In order to assess such quality, Steve proposes some rules of thumb—check them out in Steve’s post—to be applied hopefully by the authors themselves. In his words, Steve’s intention was

“not to trash the subject as intrinsically uninteresting; rather, [he wanted] to highlight the epidemic of pointless papers that constitute the discord bubble. [He hoped] that thinning the herd will increase the quality of the results in the field and decrease the hype surrounding it, because it has really gotten completely out of control.”

While I found Steve’s post essentially to the point and I furthermore highly appreciated his invitation to highlight in the comments good research on quantum discord—because there is good research on the topic—I feel that there is room for more reflection and discussion, in particular within—but of course not limited to—the community working on discord-related problems. To start it, I would like to express some of my own opinions.

## Is there a bubble? What is it at its origin?

Yes, there is a bubble. Yes, there is too much hype. Yes, there are too many papers on the subject that are of questionable quality.

On one hand, the subfield of the `general quantumness of correlations’—as I like to think of discord and related concepts—still lacks a killer application to justify the present level of activity. The explosion of interest in the subfield was in a good part due to hints that such quantumness could help explain the quantum advantage in some noisy models of (limited) quantum computation. Such hints, to my knowledge, remain just hints; that is, too little to justify so many papers. We are still searching for a task/protocol that convincingly highlights discord or similar related properties as `resources’  or `the key’ in a general enough scenario—see this related post by Valerio Scarani. On the other hand, as Steve pointed out, many papers on discord are characterized by a low scientific merit, technically and/or conceptually.

I believe that the “discord bubble” is due to a combination of several factors, which have all come together to create the `perfect bubble’. Some are:

1) we live in a “publish or perish” (academic) world: if hype starts to be generated regarding a topic, many researchers will flock to that topic in the hope to score well in the number and kind of publications—high-impact journals—and the corresponding citations they are bound to get;

2) when there is a `new’ topic (like `discord’ with respect to the good ol’ `entanglement’) the technical and conceptual threshold to contribute new results is lower; on the other hand, quantum information is by now a mature field and new exciting results in `traditional’ theoretical topics—quantum Shannon theory, quantum error correction, quantum computation, … —require a high level of expertise and skills;

3) some form of `confirmation bias’: positive statements on the value and significance of new and old results on the topic are easily accepted and perpetuated, especially in light of 1) and 2).

Let me stress that these factors are mostly independent of the topic itself—discord. I am pretty sure that other fields of science have their own bubbles.

## Is it worth working on discord and related issues?

Quantum information processing aims at exploiting quantum features to provide us with new, powerful means to manipulate information. Reaching this goal requires the best possible understanding of such features.

More specifically for our case, there are quantum features that 1) are proper of bi- and multi-partite systems (i.e., they do not have a real correspondent for single systems)  and 2) do not reduce to entanglement, e.g, they can be present also in the absence of entanglement.  For example, one of these properties is related to the celebrated no-cloning theorem; another one is the unavoidable disturbance introduced by local measurements. In this sense there is a `general quantumness of correlations’, which, for the above reasons, I believe is worth investigating.

Of course, an argument like “[PROPERTY] could be useful, so we should study it” is not  enough, if not substantiated. There should be some concrete evidence and some convincing perspective of such usefulness in order to motivate the related investment of resources. I think that what we know about the general quantumness of correlations, although not enough to justify the number of papers dealing with it, satisfies these requirements, at least partially. More clearly:

We have concrete evidence that the general quantumness of correlations is a useful concept to consider.

Such usefulness goes from foundations—e.g., addressing the measurement problem and the emergence of classicality—to the alternative take on the no-cloning theorem mentioned above, to the study of quantum effects in the `locking’ of classical correlations in quantum information theory.

On the other hand:

We have not yet found a way to think about and exploit the general quantumness of correlations that makes such quantumness worthy of the central stage of quantum information processing, or of the title of `resource’.

As mentioned above, the protocols so far designed that pinpoint discord as the relevant `resource’ at play can be considered contrived. More generally, I doubt that discord will ever achieve a status of resource similar to entanglement. I rather believe that, for example, discord is—with the risk of sounding like I am trying to be witty—something that makes it possible to make something else impossible. For example, discord makes it impossible to access `in a classical way’  part of the information content of correlations.

All in all, I do believe that the general quantumness of correlations will further prove its usefulness, both as a conceptual tool and as concrete property present in distributed systems. What we have to remember is that such usefulness is not well established and that most of the work on the subject should still go in the direction of clarifying the value and applicability of the concept rather than, e.g., calculating discord in all sorts of physical systems.

Let me add that I consider the study of the general quantumness of correlations also as an attempt to think about the quantum from a different/larger perspective—different/larger with respect to what was previously done. In my case, thinking in terms of the `general quantumness of correlations’ has helped me, for example, to better understand entanglement itself.

## What should people who work or consider working on discord do? What about the rest of the quantum information community?

The effort worth investing—as individual researchers, as a scientific (in particular, quantum information) community, and as a society (e.g., in terms of funding)—in the study of the general quantumness of correlations is not easy to determine. This actually holds in general, for essentially any research subject. What is worrisome about a bubble, is that it can reduce the efficiency of the procedures in place—in particular the peer-review process—to shape the activity of the community, to reallocate its resources, and to induce researchers to adopt a good practice. [We actually know that the peer-review process, in particular in the standard form associated to publication in journals, does not work perfectly in general.]  As personal experience goes, I have often rejected low-quality papers on discord only to see them published in another journal. So it is important to recognize the presence of a bubble and bring the problem to the attention of the community, so that extra care can be taken in assessing the value of papers, both in terms of correctness and relevance. While I endorse Steve’s rules of thumb to assess the quality of papers related to discord, I would like to address the problem of the discord bubble with a list of suggestions mostly comprising general good practices.

So, to the people working/interested in working on discord:

a) re-evaluate why you are working on the topic. A useful exercise it that of imagining explaining to someone who is NOT already working on discord why he should be interested in investigating the subject. If whatever reason you provide is based on previous results in the field, make sure that you have checked the source at a sufficient level of detail to be sure that those results—and in particular any related strong claim—would be convincing for him, and, most importantly, that they are convincing for you;

b) be aware of the large body of work that already exists in quantum information processing: what you think is new and exciting may well be already known (either published, or easy to see and part of the `folklore’);

c) analyze critically your work and that of the others, both as author and as referee: cite only literature that is significant and relevant to your paper and reject papers that do not provide substantial advance in neither understanding nor applications;

d) focus on providing more—and possibly conclusive—evidence that the study of the quantumness of correlations is justified: we need `killer applications’ and `killer concepts’;

e) entanglement theory has been a very fruitful field of study because entanglement is a fundamental concept and it can be understood/analyzed as a resource in a reasonable, operationally justified framework—that of distant labs, where the quantum operations allowed are only local, at most coordinated by classical communication. Trying to mimic successful stories might be a good idea, but there are dangers involved. Let us avoid creating the Bizzarro version of entanglement.

To the rest of the quantum information community:

Be critical and open-minded at the same time.

While complaining about the existence of a discord bubble is more than reasonable and, from my point of view, quite welcome, attacking the study of discord per se is unjustified. Please challenge whoever makes claims that are too strong, question observations and calculations that you judge irrelevant, push people who work on discord to meet the highest standards in international research, reject papers when they do not meet such standards. But please do not dismiss a talk or a paper just because it deals with discord; evaluate it only on the basis of its specific scientific merit.  Furthermore, if you feel like it, you can ponder for some minutes on questions like “Is there anything quantum about a distributed quantum state that is not solely due to entanglement?”, and “Can we make use of it?”.

## On the fractal nature of bubbles

The best results of recognizing that there is a `discord bubble’ and of taking corresponding action—maybe on the lines suggested above—would be, on one hand, to improve the research activity on discord, and, on the other hand, to avoid the risk that the subject and the people working on it acquire a bad reputation.

Let me mention that there are scientists who think that research on quantum computation is itself a bubble—see this post by Scott Aaronson. Is this further proof that bubbles can have a fractal structure? Of course the `accusation’ of being bubbles is at different levels for discord and quantum information, and not just in the sense of the `fractal level’. The criticism towards quantum computing is mostly about its realizability, while discord, although a useful conceptual tool, has not convincingly been proven to be a `resource’, even in theory.  Nonetheless, there is something in common about the two accusations of being a bubble: hype—again. Indeed, Scott writes about the accusations of the quantum computing skeptic M. I. Dyakonov:

“Dyakonov fumes about how popular articles, funding agency reports, and so forth have overhyped progress in quantum computing, leaving the conditions out of theorems and presenting incremental advances as breakthroughs.  Here I sadly agree.”

So, let me add a point to the above list of suggestions:

f) try to reduce—or at least do not contribute to—the hype.

This will have the effect of making the topic of discord less attractive for people whose work will not actually improve the standing of the topic, and improve its reputation within the quantum information community, potentially attracting good researchers.

The above-mentioned post by Scott is also notable for having started a high-quality discussion in the comments section on whether quantum computing deserves the accusation of being a bubble. It would be great to have a similar discussion on discord in the comments below. I expressed already my personal opinion: we are in front of a discord bubble, but there is merit in studying the general quantumness of correlations. I would be particularly happy to have a discussion on whether—and why—my opinion is too harsh or too mild, and to receive, as Steve already asked in his post, motivated suggestions about works that should survive a `pop’ of the discord bubble.

Posted in Bubbles, Quantum, Quantum Computing | 4 Comments

## Simple circuit "factors" arbitrarily large numbers

Last Thursday, at the QIP rump session in Beijing, John Smolin described recent work with Graeme Smith and Alex Vargo [SSV] showing that arbitrarily large numbers $N$ can be factored by using this constant-sized quantum circuit

to implement a compiled version of Shor’s algorithm.  The key to SSV’s breathtaking improvement is to choose a base for exponentiation, $a$, such that the function $a^x bmod N$ is periodic with period 2.  (This kind of  simplification—using a base with a short period such as 2, 3, or 4—has in fact been used in all experimental demonstrations of Shor’s algorithm that we know of).  SSV  go on to show that an $a$ with period 2 exists for every product of distinct primes $N=pq$, and therefore that the circuit above can be used to factor any such number, however large.  The problem, of course, is that in order to find a 2-periodic base $a$, one needs to know the factorization of $N$. After pointing this out, and pedantically complaining that any process requiring the answer to be known in advance ought not to be called compilation, the authors forge boldly on and note that their circuit can be simplified even further to a classical fair coin toss, giving a successful factorization whenever it is iterated sufficiently many times to obtain both a Head and a Tail among the outcomes (like having enough kids to have both a girl and a boy).   Using a penny and two different US quarters, they successfully factor 15, RSA-768, and a 20,000-bit number of their own invention by this method, and announce plans for implementing the full circuit above on state-of-the art superconducting hardware.  When I asked the authors of SSV what led them into this line of research, they said they noticed that the number of qubits used to do Shor demonstrations has been decreasing over time, even as the number being factored increased from 15 to 21, and they wanted to understand why.  Alas news travels faster than understanding—there have already been inquiries as to whether SSV might provide a practical way of factoring without the difficulty and expense of building a large-scale quantum computer.

## Viacheslav Belavkin

I am sad to report that Viacheslav Belavkin recently passed away. One of the rarified few who made important contributions to quantum information in the 1970′s, Belavkin’s work was and is very significant in both physics and mathematics.

Belavkin won the Main State Prize of Russia (formerly the Lenin prize) in 1996, jointly with Stratanovich, for his contributions to stochastic calculus and the theory of quantum measurement.

I only met Belavkin once, while he was visiting the Perimeter Institute. He was dining alone, and had just finished dinner and was about to pay, so he told the server put the bill on the tab for Belavkin. Overhearing this, I turned to him and said “Oh, you’re Belavkin!” He was clearly pleased that I knew who he was, and over a beer he shared with me some of the very interesting history of the early days of quantum information theory. You can read for yourself Belavkin’s perspective on the early days of the field through his potted autobiography. (The link is to a web cache, since sadly the University of Nottingham has taken down his personal webpage.)

It was clear from my conversation with him that he was the quintessential jaded ex-Soviet scientist who had seen everything done 10 years ahead of its rediscovery in the West. He wasn’t very modest about his own discoveries, either. I distinctly remember him saying the following, “My first paper on quantum information theory was in 1972. And Stratanovich had some in the 60′s.” To his great credit, though, he was essentially correct! Much of his work was rediscovered in the 90′s, often in less generality.

Chris Fuchs told me a classic story about Belavkin, which I’ll recall as best I can. Sometime in the 90′s, Chris was at a conference along with many smart people working on continuous measurement and feedback control of quantum systems. When it was Belavkin’s turn to talk, he calmly took the chalk and began to recap the talks from the morning session where people had been presenting their recent work.

“This morning we heard a talk by Prof. Smith in which he proved the following theorem.”  Belavkin calmly scrawled the statement of the theorem on the board, in a formal style,

Theorem 1 [Smith, 1995]. For all $x$, there exists a $y$ such that

He continued, “Then we heard a talk by Prof. Jones, where he proved the following.”  Once more, he carefully wrote the statement of the theorem on the board, just below the first one.  ”And finally we heard from Prof. Brown, who demonstrated this theorem.”  Again, he patiently wrote the formal statement of the theorem on the board, with the name and date for attribution.

Belavkin paused for dramatic effect, then began writing new dates to the right of the theorems.  ”In 1972 I proved Theorem 1.  In 1976 I proved Theorem 2.  And in 1985 I proved Theorem 3.  Now we will hear about some new results.”

Here is the notice of his passing from the University of Nottingham. I hope that they will make his old webpage available again.

Posted in Announcement, Obituary | 7 Comments