Category Archives: Quantum mechanics


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).


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.)

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

We all knew it

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.

Google Scholar Irregularities

Google Scholar is definitely missing citations to my papers.

The cited-by results for “Some Negative Remarks on Operational Approaches to Quantum Theory” [arXiv:1401.7254] on Google Scholar and on INSPIRE are completely nonoverlapping. Google Scholar can tell that “An Information-Theoretic Formalism for Multiscale Structure in Complex Systems” [arXiv:1409.4708] cites “Eco-Evolutionary Feedback in Host–Pathogen Spatial Dynamics” [arXiv:1110.3845] but not that it cites My Struggles with the Block Universe [arXiv:1405.2390]. Meanwhile, the SAO/NASA Astrophysics Data System catches both.

This would be a really petty thing to complain about, if people didn’t seemingly rely on such metrics.

EDIT TO ADD (17 November 2014): Google Scholar also misses that David Mermin cites MSwtBU in his “Why QBism is not the Copenhagen interpretation and what John Bell might have thought of it” [arXiv:1409.2454]. This maybe has something to do with being worse at detecting citations in footnotes than in endnotes.


One day, I’ll be able to explain the story behind how I got into this, but looking back on all the oddities of it, I’m not sure that a medium other than manga could do it justice.

My Struggles with the Block Universe [arXiv:1405.2390]

Christopher A. Fuchs, Maximilian Schlosshauer (foreword), Blake C. Stacey (editor)

This document is the second installment of three in the Cerro Grande Fire Series. Like its predecessor arXiv:quant-ph/0105039, “Notes on a Paulian Idea,” it is a collection of letters written to various friends and colleagues, most of whom regularly circuit this archive. The unifying theme of all the letters is that each has something to do with the quantum. Particularly, the collection chronicles the emergence of Quantum Bayesianism as a robust view of quantum theory, eventually evolving into the still-more-radical “QBism” (with the B standing for no particular designation anymore), as it took its most distinctive turn away from various Copenhagen Interpretations. Included are many anecdotes from the history of quantum information theory: for instance, the story of the origin of the terms “qubit” and “quantum information” from their originator’s own mouth, a copy of a rejection letter written by E. T. Jaynes for one of Rolf Landauer’s original erasure-cost principle papers, and much more. Specialized indices are devoted to historical, technical, and philosophical matters. More roundly, the document is an attempt to provide an essential ingredient, unavailable anywhere else, for turning QBism into a live option within the vast spectrum of quantum foundational thought.

As the comment field says, “CAUTION, do not unthinkingly print from a printer: 2,348 pages, 4 indices, 6 figures, with extensive hyperlinking.”

MSwtBU was originally submitted to the arXiv on 10 May 2014, the anniversary of the predecessor volume and before that of the Cerro Grande Fire, which started the whole business. To my knowledge, it is the longest item currently on the arXiv.

omg 2000+ pages. There goes my free time.
— Dave Bacon, via Twitter

New Paper Dance

B. C. Stacey, “SIC-POVMs and Compatibility among Quantum States” [arXiv:1404.3774]:

An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack’s “Conditions for compatibility of quantum state assignments” [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. Along the way, I correct two mathematical errors in Caves, Fuchs and Schack’s paper. One error is a minor nit to pick, while the other is a missed opportunity.