Category Archives: Quantum mechanics

New Papers Dance

In spite of the “everything, etc.” that is life these days, I’ve managed to do a bit of science here and there, which has manifested as two papers. First, there’s the one about quantum physics, written with the QBism group at UMass Boston:

J. B. DeBrota, C. A. Fuchs and B. C. Stacey, “Symmetric Informationally Complete Measurements Identify the Essential Difference between Classical and Quantum” [arXiv:1805.08721].

We describe a general procedure for associating a minimal informationally-complete quantum measurement (or MIC) and a set of linearly independent post-measurement quantum states with a purely probabilistic representation of the Born Rule. Such representations are motivated by QBism, where the Born Rule is understood as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, the difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation — one just applies the Law of Total Probability (LTP) between the scenarios — while quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the essential difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born Rule obtained from a symmetric informationally-complete measurement (or SIC) minimizes this distinction in at least two senses—the first to do with unitarily invariant distance measures between the rules, and the second to do with available volume in a reference probability simplex (roughly speaking a new kind of uncertainty principle). Both of these arise from a significant majorization result. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions.

To get an overall picture of our results without diving into the theorem-proving, you can watch John DeBrota give a lecture about our work.

Second, there’s the more classical (in the physicist’s sense, if not the economist’s):

B. C. Stacey and Y. Bar-Yam, “The Stock Market Has Grown Unstable Since February 2018” [arXiv:1806.00529].

On the fifth of February, 2018, the Dow Jones Industrial Average dropped 1,175.21 points, the largest single-day fall in history in raw point terms. This followed a 666-point loss on the second, and another drop of over a thousand points occurred three days later. It is natural to ask whether these events indicate a transition to a new regime of market behavior, particularly given the dramatic fluctuations — both gains and losses — in the weeks since. To illuminate this matter, we can apply a model grounded in the science of complex systems, a model that demonstrated considerable success at unraveling the stock-market dynamics from the 1980s through the 2000s. By using large-scale comovement of stock prices as an early indicator of unhealthy market dynamics, this work found that abrupt drops in a certain parameter U provide an early warning of single-day panics and economic crises. Decreases in U indicate regimes of “high co-movement”, a market behavior that is not the same as volatility, though market volatility can be a component of co-movement. Applying the same analysis to stock-price data from the beginning of 2016 until now, we find that the U value for the period since 5 February is significantly lower than for the period before. This decrease entered the “danger zone” in the last week of May, 2018.

Recent Advances in Packing

The weekend before last, I overcame my reluctance to travel and went to a mathematics conference, the American Mathematical Society’s Spring Central Sectional Meeting. I gave a talk in the “Recent Advances in Packing” session, spreading the word about SICs. My talk followed those by Steve Flammia and Marcus Appleby, who spoke about the main family of known SIC solutions while I covered the rest (the sporadic SICs). The co-organizer of that session, Dustin Mixon, has posted an overall summary and the speakers’ slides over at his blog.

To Thems That Have

Occasionally, I think of burning my opportunities of advancing in the physics profession — or, more likely, just burning my bridges with Geek Culture(TM) — by writing a paper entitled, “Richard Feynman’s Greatest Mistake”.

I did start drafting an essay I call “To Thems That Have, Shall Be Given More”. There are a sizable number of examples where Feynman gets credit for an idea that somebody else discovered first. It’s the rich-get-richer of science.
Continue reading To Thems That Have

Multiscale Structure of More-than-Binary Variables

When I face a writing task, my two big failure modes are either not starting at all and dragging my feet indefinitely, or writing far too much and having to cut it down to size later. In the latter case, my problem isn’t just that I go off on tangents. I try to answer every conceivable objection, including those that only I would think of. As a result, I end up fighting a rhetorical battle that only I know about, and the prose that emerges is not just overlong, but arcane and obscure. Furthermore, if the existing literature on a subject is confusing to me, I write a lot in the course of figuring it out, and so I end up with great big expository globs that I feel obligated to include with my reporting on what I myself actually did. That’s why my PhD thesis set the length record for my department by a factor of about three.

I have been experimenting with writing scientific pieces that are deliberately bite-sized to begin with. The first such experiment that I presented to the world, “Sporadic SICs and the Normed Division Algebras,” was exactly two pages long in its original form. The version that appeared in a peer-reviewed journal was slightly longer; I added a paragraph of context and a few references.

My latest attempt at a mini-paper (articlet?) is based on a blog post from a few months back. I polished it up, added some mathematical details, and worked in a comparison with other research that was published since I posted that blog item. The result is still fairly short:

New Paper Dance Macabre

C. A. Fuchs, M. C. Hoang and B. C. Stacey, “The SIC Question: History and State of Play,” arXiv:1703.07901 [quant-ph] (2017).

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 323. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

Also available via SciRate.

Aphorism

Last night I thought of a way to summarize why my current big research project appeals to me.

The SIC problem gives us the opportunity to travel all throughout mathematics, because, while the definition looks pretty small, the question is bigger on the inside.

For a taste of why this is so, try here:

Google Scholar Whisky-Tango-Foxtrottery

Google Scholar is seriously borked today. I heard about the problem when Christopher Fuchs emailed me to say that he had his Google Scholar profile open in a browser and happened to click the refresh button, whereupon his total citation count jumped by 700. After the refresh, his profile was full of things he hadn’t even written. Poking around, I found that a lot of publications in the American Institute of Physics’s AIP Conference Proceedings were being wildly misattributed, almost as if everyone who contributed to an issue was getting credit for everything in that issue.

For example, here’s Jan-Åke Larsson getting credit for work by Giacomo D’Ariano:

screenshot of Google Scholar

And here’s Chris picking up 38 bonus points for research on Mutually Unbiased Bases—a topic not far from my own heart!—research done, that is, by Ingemar Bengtsson:
Continue reading Google Scholar Whisky-Tango-Foxtrottery

More Google Scholar Irregularities

A few years ago, I noticed a glitch in a paper that colleagues of mine had published back in 2002. A less-than sign in an inequality should have been a less-than-or-equals. This might have been a transcription error during the typing-up of the work, or it could have entered during some other phase of the writing process. Happens to the best of us! Algebraically, it was equivalent to solving an equation
\[ ax^2 + bx + c = 0 \] with the quadratic formula,
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a},\] and neglecting the fact that if the expression under the square root sign equals zero, you still get a real solution.

This sort of glitch is usually not worth a lot of breath, though I do tend to write in when I notice them, to keep down the overall confusingness of the scientific literature. In this case, however, there’s a surprise bonus. The extra solutions you pick up turn out to have a very interesting structure to them, and they include mathematical objects that were already interesting for other reasons. So, I wrote a little note explaining this. In order to make it self-contained, I had to lay down a bit of background, and with one thing and another, the little note became more substantial. Too substantial, I learned: The journal that published the original paper wouldn’t take it as a Comment on that paper, because it said too many new things! Eventually, after a little more work, it found a home:

The number of citations that Google Scholar lists for this paper (one officially published in a journal, mind) fluctuates between 5 and 6. I think it wavers on whether to include a paper by Szymusiak and Słomczyński (Phys. Rev. A 94, 012122 = arXiv:1512.01735 [quant-ph]). Also, if you compare against the NASA ADS results, it turns out that Google Scholar is missing other citations, too, including a journal-published item by Bellomo et al. (Int. J. Quant. Info. 13, 2 (2015), 1550015 = arXiv:1504.02077 [quant-ph]).

As I said in 2014, this would be a rather petty thing to care about, if people didn’t rely on these metrics to make decisions! And, as it happens, all the problems I noted then are still true now.

17 Equations that Clogged My Social-Media Timeline

An image burbled up in my social-media feed the other day, purporting to be a list of “17 Equations that Changed the World.” It’s actually been circulating for a while (since early 2014), and purports to summarize the book by that name written by Ian Stewart. This list is typo-ridden, historically inaccurate and generally indicative of a lousy knowledge-distribution process that lets us down at every stage, from background research to fact-checking to copy-editing.
Continue reading 17 Equations that Clogged My Social-Media Timeline

3, 8, 24, 28, Eureka!

The news has been so unrelentingly bad these past few weeks that I’m taking momentary refuge in good old numerology. I happened to re-read this blog post by John Baez about the free modular lattice on 3 generators. This is a nice bit of pure math that features rather prominently the numbers 3, 8, 24 and 28. The numerological part is that I noticed the same numbers popping up in a problem that I had studied for other reasons, so I figured it would be fun to write about, even if my 28 isn’t exactly equal to Baez’s 28, so to speak.
Continue reading 3, 8, 24, 28, Eureka!

New Paper Dance

M. Appleby, C. A. Fuchs, B. C. Stacey and H. Zhu, “Introducing the Qplex: A Novel Arena for Quantum Theory,” arXiv:1612.03234 [quant-ph] (2016).

Abstract:

We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of modern quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, mutually exclusive experiments mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. Along the way, we derive a condition for the existence of a $d$-dimensional SIC.

Also available through SciRate, where I have a whole profile.

New Paper Dance

B. C. Stacey, “Geometric and Information-Theoretic Properties of the Hoggar Lines” (2016), arXiv:1609.03075 [quant-ph].

We take a tour of a set of equiangular lines in eight-dimensional Hilbert space. This structure defines an informationally complete measurement, that is, a way to represent all quantum states of three-qubit systems as probability distributions. Investigating the shape of this representation of state space yields a pattern of connections among a remarkable spread of mathematical constructions. In particular, studying the Shannon entropy of probabilistic representations of quantum states leads to an intriguing link between the questions of real and of complex equiangular lines. Furthermore, we will find relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.

All things told, SIC-POVMs are just about the damnedest things I’ve ever studied in mathematics.

(Also listed on SciRate.)

A Frabjous, Albeit Delayed, Day

David Mermin thanked me for finding a glitch in one of his papers. I can retire now, right?

The matter concerns “Hidden variables and the two theorems of John Bell” [Reviews of Modern Physics 65, 3 (1993), pp. 803–15]. Specifically, we turn our attention to Figure 4, the famous “Mermin pentagram,” reproduced below for convenience.

mermin-hiddenvariables-fig4

The caption to this figure reads as follows:

Ten observables leading to a very economical proof of the Bell–KS theorem in a state space of eight or more dimensions. The observables are arranged in five groups of four, lying along the legs of a five-pointed star. Each observable is associated with two such groups. The observables within each of the five groups are mutually commuting, and the product of the three observables in each of the six groups is $+1$ except for the group of four along the horizontal line of the star, where the product is $-1$.

In that last sentence, “three observables in each of the six groups” should instead read “four observables in each of the five groups” (in order to agree with the diagram, and to make sense).

Glitches and goofs can happen to anybody. I’m embarrassingly prone to them myself. I also have the pesky kind of personality that is inclined to write in when I find them. This has led to a journal-article erratum once before, and now that I think about it, it provided the seeds for two papers of my own. As they say about Wolverine, being per-SNIKT-ety pays off!

(Incidentally, it took two months for this latest erratum to appear. A sensible system could have done it in as many days, but that’s scientific publishing for you.)

My Year in Publications

This is, apparently, a time for reflection. What have I been up to?

And so this is Korrasmas
Things have been Done
Kuvira is fallen
A new ‘ship just begun

Kor-ra-sa-mi
We all knew it
Kor-ra-sa-mi
now-ow-ow-owwwwwww

Well, other than watching cartoons?

At the very beginning of 2014, I posted a substantial revision of “Eco-Evolutionary Feedback in Host–Pathogen Spatial Dynamics,” which we first put online in 2011 (late in the lonesome October of my most immemorial year, etc.).

In January, Chris Fuchs and I finished up an edited lecture transcript, “Some Negative Remarks on Operational Approaches to Quantum Theory.” My next posting was a solo effort, “SIC-POVMs and Compatibility among Quantum States,” which made for a pretty good follow-on, and picked up a pleasantly decent number of scites.

Then, we stress-tested the arXiv.

By mid-September, Ben Allen, Yaneer Bar-Yam and I had completed “An Information-Theoretic Formalism for Multiscale Structure in Complex Systems,” a work very long in the cooking.

Finally, I rang in December with “Von Neumann was Not a Quantum Bayesian,” which demonstrates conclusively that I can write 24 pages with 107 references in response to one sentence on Wikipedia.