Category Archives: Information Theory

Group Theory, Information Theory, Quantum mechanics, University education

New Textbook

Copies of a textbook surrounded by Oaxacan carved wooden animals

B. C. Stacey, A First Course in the Sporadic SICs. SpringerBriefs in Mathematical Physics volume 41 (2021).

This book focuses on the Symmetric Informationally Complete quantum measurements (SICs) in dimensions 2 and 3, along with one set of SICs in dimension 8. These objects stand out in ways that have earned them the moniker of “sporadic SICs”. By some standards, they are more approachable than the other known SICs, while by others they are simply atypical. The author forays into quantum information theory using them as examples, and the author explores their connections with other exceptional objects like the Leech lattice and integral octonions. The sporadic SICs take readers from the classification of finite simple groups to Bell’s theorem and the discovery that “hidden variables” cannot explain away quantum uncertainty.

While no one department teaches every subject to which the sporadic SICs pertain, the topic is approachable without too much background knowledge. The book includes exercises suitable for an elective at the graduate or advanced undergraduate level.

ERRATA:

In the preface, on p. v, there is a puzzling appearance of “in references [77–80]”. This is due to an error in the process of splitting the book into chapters available for separate downloads. These references are arXiv:1301.3274, arXiv:1311.5253, arXiv:1612.07308 and arXiv:1705.03483.

Page 6: “5799” should be “5779” (76 squared plus 3), and M. Harrison should be added to the list of co-credited discoverers. The most current list of known solutions, exact and numerical, is to my knowledge this presentation by Grassl.

Page 58: “Then there are 56 octavians” should be “Then there are 112 octavians”.

Information Theory, Quantum mechanics

On Reconstructing the Quantum

It’s manifesto time! “Quantum Theory as Symmetry Broken by Vitality” [arXiv:1907.02432].

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using a reference measurement. This program reduces the physical question of why the quantum formalism is empirically successful to the mathematical question of why complete sets of equiangular lines appear to exist in complex vector spaces when they do not exist in real ones. My primary goal is to clarify motivations, rather than to present a closed book of numbered theorems, and consequently the discussion is more in the manner of a colloquium than a PRL.

Also available via SciRate.

Information Theory, Quantum mechanics

New Paper Dance

Another solo-author outing by me: “Invariant Off-Diagonality: SICs as Equicoherent Quantum States” [arXiv:1906.05637].

Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric structures in state space. For the case of a qubit, we find an easy construction of qubit SICs (Symmetric Informationally Complete POVMs). SICs in dimension 3 and 8 are also shown to be equicoherent.

Also available via SciRate.

Group Theory, Information Theory, Quantum mechanics

Sporadic SICs and exceptional Lie algebras

A while back, I had a bit of a sprawling conversation about certain geometrical oddities over multiple threads at the n-Category Café. I finally got organized enough to gather these notes together, incorporating edits for clarity and recording one construction I haven’t found written in the literature anywhere.

Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy.

UPDATE (30 March 2019): Thanks to a kind offer by John Baez, we’re going through this material step-by-step over at a blog with a community, the n-Category Café:

  • Part 1: Definitions and preliminaries
  • Part 2: Qutrits and E6
  • Part 3: The Hoggar lines, E7 and E8
Information Theory, Quantum mechanics

Triply Positive Matrices

One more paper to round out the year!

J. B. DeBrota, C. A. Fuchs and B. C. Stacey, “Triply Positive Matrices and Quantum Measurements Motivated by QBism” [arXiv:1812.08762].

We study a class of quantum measurements that furnish probabilistic representations of finite-dimensional quantum theory. The Gram matrices associated with these Minimal Informationally Complete quantum measurements (MICs) exhibit a rich structure. They are “positive” matrices in three different senses, and conditions expressed in terms of them have shown that the Symmetric Informationally Complete measurements (SICs) are in some ways optimal among MICs. Here, we explore MICs more widely than before, comparing and contrasting SICs with other classes of MICs, and using Gram matrices to begin the process of mapping the territory of all MICs. Moreover, the Gram matrices of MICs turn out to be key tools for relating the probabilistic representations of quantum theory furnished by MICs to quasi-probabilistic representations, like Wigner functions, which have proven relevant for quantum computation. Finally, we pose a number of conjectures, leaving them open for future work.

This is a sequel to our paper from May, and it contains one minor erratum for an article from 2013.

Information Theory, Quantum mechanics, Science history

QBism and the Ithaca Desiderata

Time again for the New Paper Dance!

B. C. Stacey, “QBism and the Ithaca Desiderata” [arXiv:1812.05549].

In 1996, N. David Mermin proposed a set of desiderata for an understanding of quantum mechanics, the “Ithaca Interpretation”. In 2012, Mermin became a public advocate of QBism, an interpretation due to Christopher Fuchs and Ruediger Schack. Here, we evaluate QBism with respect to the Ithaca Interpretation’s six desiderata, in the process also evaluating those desiderata themselves. This analysis reveals a genuine distinction between QBism and the IIQM, but also a natural progression from one to the other.

Information Theory, Quantum mechanics

The State Space of Quantum Mechanics is Redundant

There was some water-cooler talk around the office this past week about a paper by Masanes, Galley and Müller that hit the arXiv, and I decided to write up my thoughts about it for ease of future reference. In short, I have no reason yet to think that the math is wrong, but what they present as a condition on states seems more naturally to me like a condition on measurement outcomes. Upon making this substitution, the Masanes, Galley and Müller result comes much closer to resembling Gleason’s theorem than they say it does.

So, if you’re wanting for some commentary on quantum mechanics, here goes:
Continue reading The State Space of Quantum Mechanics is Redundant →

Information Theory, Quantum mechanics, Relativity, Statistical mechanics

What I Do

At the moment, I’m taking a quick break from reading some rather dense mathematical prose, and I spent yesterday plugging away at a draft of my research group’s next technical publication. This led me to reflect on a lesson that I think a lot of science education leaves out: Even in a technical article, you have to have a story to carry the progression through. “These are all the boffo weird roadside attractions we found while proving the theorems in our last paper” is honest, but not adequate.

Our research project is the reconstruction of the mathematical formalism of quantum theory from physical principles. We tease apart the theory, identify what is robustly strange about it — for many more quantum phenomena can be emulated with classical stochasticity than are often appreciated — and try to build a new representation that brings the most remarkable features of the physics to the forefront. In special relativity, we have Einstein’s postulates, and the dramatic tension between them: Inertial observers can come to agree upon the laws of physics, but they cannot agree upon a standard of rest. In thermodynamics, we have the Four Laws, which come with their own dramatic tension, in that energy is conserved while entropy is nondecreasing. Both of these theories are expressed in terms of what agents can and cannot do, yet they are more than “mere” engineering, because they apply to all agents. Or, to say it another way, it is to the benefit of any agent to pick up the theory and use it as a guide.

What, then, is the analogue for quantum theory? If the textbook presentation of quantum physics is like the formulae for the Lorentz transform, with all those square roots and whatnot, or the Maxwell relations in thermo, with all those intermingling partial derivatives that we invent hacks about determinants to remember, what is quantum theory’s version of Einstein’s postulates or the Four Laws?

That’s the grandiose version, anyway. The reason I got invited to speak at an American Mathematical Society meeting is that the geometric structures that arise in this work are vexingly fascinating. You want about Galois fields and Hilbert’s 12th problem? We’ve got ’em! How about sphere packing and unexpected octonions? We’ve got those, too! And the structure that leads down the latter path turns out, on top of that, to yield a new way of thinking about Mermin’s 3-qubit Bell inequality. It is all lovely, and it is all strange.

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.

Information Theory, Quantum mechanics

New Paper Dance, Encore

This time, it’s another solo-author outing.

B. C. Stacey, “Is the SIC Outcome There When Nobody Looks?” [arXiv:1807.07194].

Informationally complete measurements are a dramatic discovery of quantum information science, and the symmetric IC measurements, known as SICs, are in many ways optimal among them. Close study of three of the “sporadic SICs” reveals an illuminating relation between different ways of quantifying the extent to which quantum theory deviates from classical expectations.

Information Theory, Plectics, 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.

Group Theory, Information Theory, Quantum mechanics, Seminar Summaries

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.

Information Theory, Popularization, Quantum mechanics, Relativity, Science history

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 →

Bibliophilia, Information Theory, Plectics, Quantum mechanics

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:

Information Theory, Quantum mechanics, Software

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.

Information Theory, Plectics, Statistical mechanics

Multiscale Structure, Information Theory, Explosions

I’d like to talk a bit about using information theory to quantify the intuition that a complex system exhibits structure at multiple scales of organization. My friend and colleague Ben Allen wrote an introduction to this a while ago:

Ben’s blog post is a capsule introduction to this article that he and I wrote with Yaneer Bar-Yam:

I also cover this topic, as well as a fair bit of background on how to relate probability and information, in my PhD thesis:

In this post, I’ll carry the ideas laid out in these sources a little bit farther in a particular direction.
Continue reading Multiscale Structure, Information Theory, Explosions →