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.

# Category Archives: Quantum mechanics

# 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:

- B. C. Stacey, “Multiscale Structure of More-than-Binary Variables,” arXiv:1705.03927 [cond-mat.stat-mech] (2017).

# 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:

- M. Appleby, S. Flammia, G. McConnell and J. Yard, “SICs and Algebraic Number Theory,” arXiv:1701.05200 [quant-ph] (2017).

# 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:

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:

- B. C. Stacey, “SIC-POVMs and Compatibility among Quantum States,”
*Mathematics***4,**2 (2016): 36 arXiv:1404.3774 [quant-ph].

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.

# My 2016 in Official Physics Writing

It feels like a good time to enumerate the science things I’ve written or co-written over this past year. In reverse chronological order:

Continue reading My 2016 in Official Physics Writing

# 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.

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.)

# New(-ish) Publications

I’ve had a few scholarly items come out in the past several weeks—new stuff, and updated versions of old stuff. Here are their coordinates:

Continue reading New(-ish) Publications

# 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 begunKor-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.

# Epistricted Trits

In quantum mechanics, we are always calculating probabilities. We get results like, “There is a 50% chance this radioactive nucleus will decay in the next hour.” Or, “We can be 30% confident that the detector at position *X* will register a photon.” But the nature and origin of quantum probabilities remains obscure. Could it be that there are some kind of “gears in the nucleus,” and if we knew their alignment, we could predict what would happen with certainty? Fifty years of theorem-proving have made this a hard position to maintain: quantum probabilities are more exotic than that.

But what we *can* do is reconstruct a *part* of quantum theory in terms of “internal gears.” We start with a mundane theory of particles in motion or switches having different positions, and we *impose a restriction on what we can know* about the mundane goings-on. The theory which results, the theory of the knowledge we can have about the thing we’re studying, exhibits many of the same phenomena as quantum physics. It is clearly not the whole deal: For example, quantum physics offers the hope of making faster and more powerful computers, and the “toy theory” we’ve cooked up does not. But the “toy theory” can include many of the features of quantum mechanics deemed “mysterious.” In this way, we can draw a line between “surprising” and “truly enigmatic,” or to say it in a more dignified manner, between *weakly nonclassical* and *strongly nonclassical.*

The ancient Greek for “knowledge” is *episteme* (επιστημη) and so a restriction on our knowledge is an *epistemic restriction,* or *epistriction* for short.

A *trit* system is one where every degree of freedom has three possible values. Looking at Figure 4 of Spekkens’ “Quasi-quantization: classical statistical theories with an epistemic restriction,” we see that the valid states of an epistricted trit follow the same pattern as the SIC-allied Mutually Unbiased Bases of a quantum trit. But that is a story for another day.