We went out in the way a bad joke would have predicted. We lost against our own racism and sexism, our endemic illnesses whose symptoms were intensified by corrupt law enforcement and institutionally rotten mass media. Undone at the final hour by a bizarre codicil in a slaveowners’ constitution. Undone, pushed over the edge—but the edge was too close all along. When it really mattered, we proved ourselves incompetent: not able to handle our civil responsibilities, indeed, in a sense, not ready for adulthood. In the name of national glory, we have voted ourselves a government of the worst. And now a generation will grow up ignorant, poor and sick, if they get the chance to grow up at all. Many of the things we will lose will be things we can never regain, from international respect to endangered species to the lives of our loved ones.

Many good people will keep up the good fight and stir up, as John Lewis says, the good trouble.

The abyss has opened before us.

Whether the future we make for ourselves will have anything to commend it now depends upon our ability to stare into that abyss and make it blink.

]]>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:

Note that again, the publication appeared in *AIP Conference Proceedings.*

I’d like to know how far and wide the rot goes, but that’s the kind of question I’d need to crowdsource!

**EDIT TO ADD (later in the day):** The problem is not confined to articles *AIP Conference Proceedings.* It also hits articles published in *Physical Review A,* one of the flagship journals of physics. Here, for example, is Adán Cabello getting credit for a paper by Klimov and Muños:

He is also credited with a paper by Caball**ar**, Ocampo and Galapon, also in *Physical Review A.*

Of course, there may be multiple underlying software problems at work.

**EDIT TO ADD (slightly later still):** Paul Pukite writes in via Twitter to say,

yesterday I caught an entry in an SPIE conf misattributed to me. I had a cite in the same issue so thought it a 1 time mistake. But today it also removed cites to individual chapters in my book, which were redundant. Looks like crawling a new algorithm.

It looks like Google Scholar is also crediting Pukite with “Article#: INSPEC Accession Number,” by a certain “P. Options” (who may be a collaborator of a certain “P. Login,” active some years ago).

**EDIT TO ADD (21 January):** Marcus Appleby wrote in with an example where Google Scholar attributes a solo-author paper of his to seventy people:

I think that in this case, Google Scholar is taking the author list from *everyone who published in the first twenty-five papers* on the Quantum Physics arXiv during December 2004. No, really: take a look!

Let’s say you tell your students that arm folding is a genetic trait, with the allele for right forearm on top (R) being dominant to left forearm on top (L). Results from a large number of studies show that about 11 percent of your students will be R children of two L parents; if they understand the genetics lesson correctly, they will think that either they were secretly adopted, or Mom was fooling around and Dad isn’t their biological father. More of your students will reach this conclusion with each bogus genetic trait that you add to the lesson. I don’t think this is a good way to teach genetics.

Via PZ Myers, who is teaching genetics this semester and has an interest in getting it right.

]]>My answer: They don’t. But not for the reason that most physicists who bother with such speculations probably think.

As a university graduate *and* veteran of many science-blog comment threads, I’ve seen my share of arguments about the mind-body problem, the Hard Problem of Consciousness and all that. The older I get, the more I feel that the way these arguments set up the problems automatically forbids a solution, and then passes this failure off as profundity. One defines *Matter* as something entirely unlike *Mind,* and then one argues about how the latter can or cannot arise out of the former. After enough rounds of this, any elegant formulation of words that appears to provide a solution immediately becomes suspect: If it *appears* to make the problem more tractable, then it *must* be treating Matter and Mind *insufficiently differently.*

When I was younger, I embraced the idea that *reality*—you know, all the stuff out there and in here that is, like, real and all—is at bottom a *mathematical entity* of some kind. What can I say? I also listened to Foreigner and thought that racism in America was on the wane. Now that my day job is actually doing physics, as opposed to reading breathless books about it, I find that this attitude, the instinctive identification of the physical and the mathematical, is entirely optional. It says more about your own temperament than pretty much anything else, nothing in particular compels it, and you can cheerfully do research without it. Countervailing influences, some of which I also carried with me since childhood, eventually led me to find it a rather empty business!

My colleague Marcus Appleby wrote an essay about this general topic a while back, an essay called (straightforwardly enough) “Mind and Matter” (arXiv:1305.7381). It prompted me to formulate a slogan, which goes like this:

*Cognitive science led me to reject the Cartesian idea of “Mind.” Quantum physics makes it legitimate to reject the Cartesian idea of “Matter.” Confusion reigns when we discard one pole but not the other!*

Above and beyond that, I have grown to doubt that the imagery of “mindless mathematical laws” is sustainable, once we confront the daily practice and the philosophy of mathematics. To get a tiny taste of this, consider the following puzzle, posed by John Baez:

You might say the standard natural numbers are those of the form 1 + ··· + 1, where we add 1 to itself some finite number of times. But what does ‘finite number’ mean here? It means a standard natural number! So this is circular.

So, conceivably, the concept of ‘standard’ natural number, and the concept of ‘standard’ model of Peano arithmetic, are more subjective than most mathematicians think. Perhaps some of my ‘standard’ natural numbers are nonstandard for you!

One can remove the ambiguity in the informal definition of “natural number,” but not in a unique way. The resulting multiplicity is irrelevant for practical purposes—ah, but tonight, we’re concerned with *impractical purposes!* We can shove this befuddlement under the rug and try to move on to defining the real numbers, only to find that a similar difficulty arises there. There is a stupendous diversity of viable candidates for the position of “the real numbers.” Technically, they are known as *real closed fields.* They are equivalent to the extent that if you can prove a mathematical statement involving the numbers 0 and 1, the operations of addition and multiplication, and the less-than-or-equal comparison in one such arena, you can prove it in any other. But two different real closed fields do not have to be isomorphic: There might not be a one-to-one mapping between them that respects the structure of arithmetic.

Mathematics is that domain of life in which “two minds can know one thing,” as best as that is possible—but you have to work painfully hard to subtract the *minds* from that image, and the result might leave you with nothing at all.

What does this have to do with physics? Perhaps nothing, except for the mental entertainment and recuperative value obtained by taking a break *from* physics! The mathematics that finds application in physics is peculiarly robust against changes in the underpinnings—to how we choose to define the real numbers, for example. On the other hand, if we’re being honest, this might be an artifact of what kinds of problems the physicists have chosen to work on! And we have to admit the vexing hassles that even our best physical theories run into because we assume that spacetime is a continuum—i.e., because we habitually think that events happen along a line of real numbers.

Another thing I’ve lost my taste for with the passing years is the blog comment section. So, I don’t have a fantastically strong urge to engage in the FQXI essay contest officially. Instead, I’ll close with a pointer to a manifesto I posted recently, and in particular to the wild speculations in the latter sections.

]]>]]>A curious thing: in the four classes so far, the number of students attending has been, respectively, 19, 17, 15, 13. Assuming that the arithmetic progression continues, our final class will have $-1$ student. Some of Joachim’s colleagues have expressed an interest in coming along to see what $-1$ student looks like. This presents problems of a philosophical type.

\[ ax^2 + bx + c = 0 \] 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

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.

This round of depression isn’t just worse than my previous episodes: it’s different. My symptoms, the things that help, the things that make it worse — they’re different. I’ve spent the last four years learning how to manage depression, and now, at least to some extent, I need to start all over again.

It’s different because the world is genuinely terrible. That’s not the depression talking: that’s a reasonable, evidence-based assessment of reality. You know the joke, “just because you’re paranoid doesn’t mean they’re not out to get you?” Well, just because you’re depressed doesn’t mean the world’s not terrible. And just because you’re anxious doesn’t mean the world’s not terrifying.

Greta’s project of blogging once a weekday this month is one of the incentives that’s got me posting more often, though I’m not keeping up with her rate.

]]>- “Information and structure in complex systems,”
*Plektix*(24 October 2014).

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

- “An Information-Theoretic Formalism for Multiscale Structure in Complex Systems,” arXiv:1409.4708 [cond-mat.stat-mech].

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

*Multiscale Structure in Eco-Evolutionary Dynamics,*arXiv:1509.02958 [q-bio.PE].

In this post, I’ll carry the ideas laid out in these sources a little bit farther in a particular direction.

Imagine a system that is made up of a large number of pieces. A theme one encounters in many areas of science is that such a system is simpler to understand when the pieces are independent of one another, each one doing its own thing. In thermodynamics and statistical physics, for example, we spend a lot of time studying gases, and the easiest conceptual model of a gas to work with is an *ideal gas,* in which the atoms sail past each other without interacting. On the other hand, a system can also be simple to understand if all the pieces are so tightly correlated or so strongly bound together that they essentially move as a unit. Then, instead of having to understand lots of little things, we just have to understand one big thing, because knowing what any one part is doing tells us most all of what we need to know about the whole.

A similar high-level intuition is helpful in pure mathematics, as Terry Tao once explained. In mathematics, one finds *structured* collections, where the relations among the individual elements have a high degree of predictability, and *random or pseudo-random* collections, where the status of one element is not informative about the status of another. Moreover, one also encounters *hybrid* collections, which partake in features of both.

The general approach of the references I bullet-pointed above is that we can use *information theory* to make these heuristic discussions quantitative. One conclusion is that we should not aim to quantify a system’s “complicatedness” by a single number. Instead, it is more illuminating to devise *curves* that indicate *how much* structure is present *at different scales.*

We start by saying that a system $\mathcal{S}$ is composed of pieces, or components. We’ll denote the set of components of the system $\mathcal{S}$ by $S$. It is important to distinguish the two, because we could take the same components and arrange them in a different way to create a different system. We express the arrangement or the patterning of the components by defining an **information function.** For each subset $T \subset S$, the information function $H$ assigns a nonnegative number $H(T)$, which expresses how much information is necessary to say what all the pieces in $T$ are doing. One can actually prove quite a lot from the starting point that if $H$ is to be an information function, it had better satisfy a few basic axioms.

First, as we already said, $H(T) \geq 0$ for any $T \subset S$.

Second, if $T \subset V \subset S$, then $H(T) \leq H(V)$. We can call this property **monotonicity**.

Third, if we have two subsets $T \subset S$ and $V \subset S$, not necessarily nested in one another or anything like that, the total information assigned to their union, $H(T \cup V)$, must be limited. Information pertinent to the components in $T$ can be pertinent to the components in $V$. For one reason, $T$ and $V$ might have some components in common. And even those components that are not shared between the two sets might be correlated in some way such that the amount of information necessary to describe the whole lot is reduced. So, we require that

\[

H(T \cup V) \leq H(T) + H(V) – H(T \cap V),\] which we call **strong subadditivity**. Given two components $s$ and $t$ within $S$, the **shared information** expresses the difference between what is required to describe them separately versus describing them together:

\[

I(s;t) = H(\{s\}) + H(\{t\}) – H(\{s,t\}).\] It follows from strong subadditivity that the shared information is always nonnegative.

A **descriptor** of a system is an entity outside that system which tells us about it in some way. Mathematically speaking, we take the system $\mathcal{S}$ and augment it with a new component that we can call $d$, to form a new system whose component set is $S \cup \{d\}$. The information function of the augmented system reduces to that of the original system $\mathcal{S}$ when applied to subsets of $S$. The values of the augmented system’s information function on subsets that include the descriptor $d$ tell us how $d$ shares information with the original components of $\mathcal{S}$.

The **utility** of a descriptor $d$ is

\[ U(d) = \sum_{s \in S} I(d;s). \] Given the basic axioms of information functions that we listed above, we can define an **optimal descriptor** as the one which has the largest possible utility, given its own amount of information. That is, if we invest an amount of information $y$ in describing the system $\mathcal{S}$, then an optimal descriptor has $H(d) = y$, and it relates to $\mathcal{S}$ in such a way that $U(d)$ is as large as the basic axioms of information functions allow. This defines a linear programming problem whose solution is the **optimal utility**, and so the theory of linear programming lets us prove helpful results about how the optimal utility varies as a function of $y$. Taking the derivative of the optimal utility gives the **marginal utility of information**, or MUI.

We prove a fair bit about the MUI in the paper where we introduced it. For systems of the ideal-gas type, where information applies to one component at a time, the MUI is a low and flat function: Investing one bit of description buys one unit of utility, until the whole system is described. On the other hand, if all the components are bound together and “move as one,” then investing a small amount of information buys us utility on a large scale, because that small amount applies across the board. In this case, the MUI starts high and falls sharply.

When the construction of a system is specified in detail, it is sometimes possible to make a finer degree of analysis, which reveals features that a first application of a structure index can overlook. For example, take the two systems defined by James and Crutchfield (2016). These are three-component systems composed of random variables whose joint states are chosen by picking a row at random from a table, with uniform probability. When systems are defined as collections of random variables with some joint probability distribution, we can use the **Shannon information** (a.k.a., Shannon entropy, Shannon index) as our information function $H$. The **dyadic** and **triadic** systems are defined respectively by the tables

\[

D = \left(\begin{array}{ccc}

0 & 0 & 0 \\

0 & 2 & 1 \\

1 & 0 & 2 \\

1 & 2 & 3 \\

2 & 1 & 0 \\

2 & 3 & 1 \\

3 & 1 & 2 \\

3 & 3 & 3

\end{array}\right);

\ T =

\left(\begin{array}{ccc}

0 & 0 & 0 \\

1 & 1 & 1 \\

0 & 2 & 2 \\

1 & 3 & 3 \\

2 & 0 & 2 \\

3 & 1 & 3 \\

2 & 2 & 0 \\

3 & 3 & 1

\end{array}\right).

\]

If we compute the MUI for these two systems in the manner described above, we find that the MUI for the dyadic system is the same as for the triadic. In fact, the information functions for the two systems, computed according to the Shannon scheme, agree for all subsets $U$. However, we can detect a difference between their structures by *augmenting* them. Let the three components of the dyadic system be written $d_1$, $d_2$ and $d_3$, and introduce a new variable $\Delta_{12}$ which takes the value 1 when the state of $d_1$ and $d_2$ are the same, and is 0 otherwise. This new **ancilla** variable is determined completely by the original system, and is sensitive to the particular values taken by the original system components $d_1$, $d_2$ and $d_3$. We can define two other ancillae in the same way, $\Delta_{13}$ and $\Delta_{23}$. Carrying out this construction for the dyadic example system, we find that the three ancillae form a completely correlated block system (that is biased towards the joint state 000).

In contrast, for the triadic example, defining three ancillae in the same way, we find that they form a **parity-bit system** (with odd parity). The ancillary system has four possible joint states, all of which are equally probable, and so it has two bits of information overall. For each possible joint state of the ancillary system, the original system can be in one of two joint states, with equal probability. Therefore, we see that the three bits of information necessary to specify the state of the original system break down into a pair of bits for describing the ancillae, plus one more bit of additional detail.

The logical extreme of this approach is to define an ancilla for each possible value of each component of a system. I like to call this “exploding” the original system. Applying the our theory to an exploded system can reveal new details of organization, at the price of increasing the number of components one must consider. For example, consider a two-component system defined by picking two successive characters at random out of a large corpus of English text. The shared information between the two components quantifies how much knowing the value of the first character helps us predict the value of the second. However, knowing that the first character is a $Q$ is a stronger constraint than knowing that it is, say, a $T$, because fewer characters can follow a $Q$. We can express this in our theory by exploding the two-component system, defining new components that represent the events of the first character being a $Q$ and the first character being a $T$. This is the same basic idea as that of the *partial information decomposition*; however, our axioms do not force the user to introduce a distinction between “input” and “output” variables as the PID framework does.

This post will be a gentle fantasy, because sometimes, in the Situation, we need that, or because that’s all I can do today.

Last month, Evelyn Lamb asked, how should we revamp the Breakthrough Prize for mathematics? This is an award with $3 million attached, supported by tech billionaires. A common sentiment about such awards, a feeling that I happen to share, is that they go to people who have indeed accomplished good things, but on the whole it isn’t a good way to spend money. Picking one person out of a pool of roughly comparable candidates and elevating them above their peers doesn’t really advance the cause of mathematics, particularly when the winner already has a stable position. Lamb comments,

$\$3$ million a year could generously fund 30 postdoc years (or provide 10 3-year postdocs). I still think that wouldn’t be a terrible idea, especially as jobs in math are hard to come by for fresh PhD graduates. But […] more postdoc funding could just postpone the inevitable. Tenure track jobs are hard to come by in mathematics, and without more of them, the job crunch will still exist. Helping to create permanent tenured or tenure-track positions in math would ease up on the job crisis in math and, ideally, make more space for the many deserving people who want to do math in academia. […] from going to the websites of a few major public universities, it looks like it’s around $2.5 million to permanently endow a chair at that kind of institution.

I like the sound of this, but let’s not forget: If we have $3 million *per year,* then we don’t have to do the same thing every year! My own first thought was that if you can fund 10 postdocs for three years apiece, you can easily pay for 10 new open-source math textbooks. In rough figures, let us say that it takes about a year to write a textbook on material you know well. Then, the book has to be field-tested for at least a semester. To find errors in technical prose, you need to find people who *don’t* already know what it’s *supposed* to say, and have them work through the whole thing.

If we look at, say, what MIT expects of undergrad math majors, we can work up a list of courses:

- Calculus (2 semesters)
- Differential equations
- Linear algebra
- Algorithms and computation
- Discrete mathematics
- Abstract algebra
- Complex analysis
- Probability and statistics

That’s roughly nine semesters of material, all of which is stable enough that a good book could be useful for *decades.*

Really, we should have solved the textbook problem long ago, roughly at the time when the Web became a thing. Recently, the American Mathematical Society started hosting Open Math Notes, “course notes, textbooks, and research expositions in progress.” This is a good thing, but it’s only a step in the right direction. What we *ought* to have is a repository, officially stamped with AMS approval (or that of some suchlike organization), so that anybody who has to teach a class on a core math subject can go to one place and find everything they need. Not many *students* might have the gumption (or the *leisure time*) to sit down and work through the whole curriculum on their own, but we really ought to support the *teachers*—and by doing that, we benefit the autodidacts too.

This would be something like OpenStax, but depth-first: We focus on one major that we know really darn well, and we do it right. And then, God willing, we build a website that isn’t so bloated with JavaScript that my laptop gives up before the site can load. (Seriously, you bastards: Text. Pictures. Click to download file. We had this sorted twenty goddamn years ago. In the rapidly approaching post-Net-Neutrality age, we really ought to be cutting our bandwidth requirements wherever we can.) And we provide the content in as flexible a manner as possible. Want an ebook to plop onto your reader? Click to download the whole thing! Want a page on a specific topic, something like a Wikipedia article that’s actually edited and curated? Here’s a URL right to it!

We *could* have done this long ago, but it was never in anybody’s short-term interest to get enough folks organized to *do* it. And when people do get organized, they run off to “innovative” schemes that create niche products, benefiting those who needed the least help in the first place. But with $3 million, I bet we could make it happen.

Another thing that $\$3$ million could probably pay for? An indie film. Aronofsky and friends made $\pi$ for under $70,000 and brought in over $3 million at the box office, for crying out loud. Wouldn’t it be neat to make, say, a dramedy about the intersecting lives of young mathematicians—less schmaltzy than *A Beautiful Mind,* perhaps tapping into the two-body problem for conflict.

Anyone can be unlucky in love, even if they stay in the same city their entire lives. But academic shuffling is particularly hostile to romance. The short-term contracts mean that when you arrive in a new country, if you’re interested in finding a long-term partner, you have something like two years to identify and convince a person you’ve just met to agree to follow you wherever you might end up in the world, and you won’t be able to tell them where that will be. If you happen to have different citizenships (which is likely), you have to take into account immigration issues as well—your partner may not be able to follow you without a spousal visa, which can mean a rather hasty life-long commitment, or, depending on the marriage laws of the country in question, a total impossibility.

Don’t tell me there’s not a screenplay in that!

]]>My country is now an onrushing catastrophe.

One remarkable thing about the disaster unfolding around us is that it has something of a fractal character. Zoom in on a small part of it, and you find the themes of the whole: Endemic sexism and racism; mass media so institutionally rotten they whiff anything important; contempt for science, expertise and basic adulthood maturity… Systemic failures playing out on the grand scale, but also leaving their signatures in the little moment-to-moment moves. Little eddies amid the maelstrom.

A crisis on all scales demands responses at all scales. Here is one action to support, in the small-to-medium range: Don’t let science go down the memory hole!

The safety of US government climate data is at risk. Trump plans to have climate change deniers running every agency concerned with climate change. So, scientists are rushing to back up the many climate databases held by US government agencies before he takes office.

We hope he won’t be rash enough to delete these precious records. But: better safe than sorry!

The

Azimuth Climate Data Backup Projectis part of this effort. So far our volunteers have backed up nearly 1 terabyte of climate data from NASA and other agencies. We’ll do a lot more! We just need some funds to pay for storage space and a server until larger institutions take over this task.

The project has already met its first funding goal, but more can’t hurt, and it’s open for contributions until 31 January. With more cash on hand, they can “back up more data, create a better interface for getting it, and put more work into making sure it’s error-free and authenticated.”

Just a few weeks ago, we saw a state government try to cover up the science of climate change, and there’s no reason to think that our new federal government will do anything less.

After the 31st, to support helping at larger scales, there’s the American Civil Liberties Union and Planned Parenthood.

]]>This paper represents an elaboration of the lectures delivered by one of us (CAF) during “Course 197 — Foundations of Quantum Physics” at the International School of Physics “Enrico Fermi” in Varenna, Italy, July 2016. Much of the material for it is drawn from arXiv:1003.5209, arXiv:1401.7254, and arXiv:1405.2390. However there are substantial additions of original material in Sections 4, 7, 8 and 9, along with clarifications and expansions of the older content throughout. Topics include the meaning of subjective probability; no-cloning, teleportation, and quantum tomography from the subjectivist Bayesian perspective; the message QBism receives from Bell inequality violations (namely, that nature is creative); the import of symmetric informationally complete (SIC) quantum measurements for the technical side of QBism; quantum cosmology QBist-style; and a potential meaning for the holographic principle within QBism.

**2.** M. Appleby, C. A. Fuchs, BCS and H. Zhu, “Introducing the Qplex: A Novel Arena for Quantum Theory.”

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.

**3.** BCS, “Geometric and Information-Theoretic Properties of the Hoggar Lines.”

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.

**4.** BCS, “Sporadic SICs and the Normed Division Algebras.”

Recently, Zhu classified all the SIC-POVMs whose symmetry groups act doubly transitively. Lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.

Also, some things I’d written earlier found official homes in 2016. This paper—which I worked on because I’d never taken material all the way from a lecture recording to written form, and it sounded like a fun thing to try—became a book chapter.

**5.** C. A. Fuchs and BCS, “Some Negative Remarks on Operational Approaches to Quantum Theory.”

Over the last 10 years there has been an explosion of “operational reconstructions” of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind. There is no doubt that this is invaluable work, particularly for our understanding of the intricate connections between so many quantum information protocols. But to me, it seems to miss the mark for an ultimate understanding of quantum theory; I am left hungry. I still want to know what strange property of matter forces this formalism upon our information accounting. To play on something Einstein once wrote to Max Born, “The quantum reconstructions are certainly imposing. But an inner voice tells me that they are not yet the real thing. The reconstructions say a lot, but do not really bring us any closer to the secret of the `old one’.” In this talk, I hope to expand on these points and convey some sense of why I am fascinated with the problem of the symmetric informationally complete POVMs to an extent greater than axiomatic reconstructions.

This is now a chapter in *Quantum Theory: Informational Foundations and Foils,* G. Chiribella and R. W. Spekkens, eds. (Springer, 2016.)

**6.** BCS, “SIC-POVMs and Compatibility among Quantum States.”

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. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.

This was published as *Mathematics* **4,** 2 (2016): 36.

**7.** BCS, “Von Neumann Was Not a Quantum Bayesian.”

Wikipedia has claimed for over three years now that John von Neumann was the “first quantum Bayesian.” In context, this reads as stating that von Neumann inaugurated QBism, the approach to quantum theory promoted by Fuchs, Mermin and Schack. This essay explores how such a claim is, historically speaking, unsupported.

This began with me getting irritated by one sentence on Wikipedia and writing 25 pages in response. (I fully endorse spite as a motivator for science!) It was published as *Phil. Trans. Roy. Soc. A,* **374,** 2068 (2016), 20150235.

A taste will suffice:

The Second Law of Thermodynamics is acknowledged in everyday life, in sayings such as “Ashes to ashes,” “Things fall apart,” “Rust never sleeps,” “Shit happens,” You can’t unscramble an egg,” “What can go wrong will go wrong,” and (from the Texas lawmaker Sam Rayburn), “Any jackass can kick down a barn, but it takes a carpenter to build one.”

By scrapping all the actual *thermodynamics* from his discussion of the Second Law (“Hello? Do you plan to have dinosaurs on your dinosaur tour?”), Pinker degrades it to the level of folk wisdom. And thus he gets his name up in lights for the *revolutionary* statement that life in the state of Nature is nasty, brutish and short. And *then,* after reducing science to a matter of earthy quips, he has the gall to turn around and claim that “the inherent tendency toward disorder” is *underappreciated.*

Another taste:

Poverty, too, needs no explanation. In a world governed by entropy and evolution, it is the default state of humankind. Matter does not just arrange itself into shelter or clothing, and living things do everything they can not to become our food. What needs to be explained is wealth. Yet most discussions of poverty consist of arguments about whom to blame for it.

The problem is that poverty, in practice, is not *simple.* Poverty has a particular character that folksy wisdom about barns and jackasses does not explain. It is not the same everywhere, and part of understanding those differences *is* a question of blame. (Why, for example, is economic mobility more of a sick joke in the United States than in other developed nations?) “What needs to be explained is wealth,” Pinker writes—but part and parcel of that is explaining the *why, specifically* of poverty.

Quoting somebody who is not Steven Pinker—William A. Darity, Jr., writing for *The Atlantic*:

Estimates generated from the 2013 round of the Federal Reserve’s Survey of Consumer Finances indicate that black households have

one-thirteenthof the wealth of white households at the median. We have concluded that the average black household would have to save 100 percent of its income for three consecutive years to close the wealth gap. The key source of the black-white wealth gap is the intergenerational effects of transfers of resources. White parents have far greater resources to give to their children via gifts and inheritances, so that the typical white young adult starts their working lives with a much greater initial net worth than the typical black young adult. These intergenerational effects are blatantly non-meritocratic.Indeed, the history of black wealth deprivation, from the failure to provide ex-slaves with 40 acres and a mule to the violent destruction of black property in white riots to the seizure and expropriation of black-owned land to the impact of racially restrictive covenants on homeownership to the discriminatory application of policies like the GI Bill and the FHA, created the foundation for a perpetual racial wealth gap.

Blacks working full time have lower levels of wealth than whites who are unemployed. Blacks in the third quintile of the income distribution have less wealth (or a lower net worth) than whites in the lowest quintile. Even more damning for any presumption that America is free of racism is our finding that black households whose heads have college degrees have $10,000 less in net worth than white households whose heads never finished high school.

Reached for comment, Ta-Nehisi Coates tweeted, “Seriously. Did not think it was possible for me to undersell white supremacy. Was wrong.”

But “shit happens,” right? No point in pointing any fingers, is there, Blamey McBlamington?

(Pinker’s *Edge* essay found via Dan Graur and PZ Myers.)

The following comments are meant to be representative, not exhaustive.

It’s not known whether Pythagoras proved the theorem we named for him—or if any of the stories about him are more than legends, really. When you go back that far, the history of mathematics and science becomes semi-legendary. The best one can typically do for “evidence” is a fragment of a lost book quoted in another book that happened to survive, and all of it dating to decades or centuries after the events ostensibly being chronicled. Did Pythagoras actually *prove* the theorem we named after him, or did he merely *observe* that it held true in a few special cases, like the 3-4-5 right triangle? Tough to say, but the latter would have been easier, and it would seem to appeal to a number mystic, for whom it’s all about the ~~Benjamins~~ successive whole numbers. Pythagoras himself probably wrote nothing, and nothing in his own words survives. It’s not clear whether his contemporaries viewed him as a mathematician or primarily as a propounder of an ethical code. (Even only 150 years after the time he purportedly lived, the ancient authorities disagreed about whether Pythagoras was a vegetarian, with Aristoxenus saying no and Eudoxus yes.) If Pythagoras had never lived, and a cult had attributed their work to that name in ritual self-denial; if the stories of his visiting Egypt and being the son of a Tyrian corn merchant began as parables and were later taken as biography—it would be hard to tell the result from what we have today. (And, in fact, groups of mathematicians *do* sometimes publish under a collective pseudonym: witness the Bourbaki collective.)

Typical, really: Indian and Chinese people do the actual work, and the white guy who likely *didn’t* gets all the credit.

I’ll outsource the criticism of the “logarithms” part:

Once again [the] simple attribution to John Napier is exactly that, simplistic and historically misleading. We can find the principle on which logarithms are based in the work of several earlier mathematicians. We can find forms of proto-logarithms in both Babylonian and Indian mathematics and also in the system that Archimedes invented to describe very large numbers. In the fifteenth century

Triparty, of the French mathematician Nicolas Chuquet we find the comparison between the arithmetical and geometrical progressions that underlay the concept of logarithms but if Chuquet ever took the next step is not clear. In the sixteenth century the German mathematician Michael Stifel studied the same comparison of progressions in hisArithmetica integraand did take the next step outlining the principle of logarithms but doesn’t seem to have developed the idea further.It was in fact John Napier who took the final step and published the first set of logarithmic tables in his book

Mirifici Logarithmorum Canonis Descriptioin 1614. However the Swiss clockmaker and mathematician, Jost Bürgi developed logarithms independently of Napier during the same period although his book of tables,Arithmetische und Geometrische Progress Tabulen, was first published in 1620.

The “calculus” line is a mess. For starters, in at least one version circulating online, it’s got an extra “=” thrown in, which makes the whole thing gibberish. The $df$ over $dt$ notation is due to Leibniz, but the list attributes it to Newton, his bitter enemy (and a pretty bitter guy overall, by many accounts). Pierre de Fermat understood quite a bit of the subject before Newton worked on it, getting as far as computing the maxima and minima of curves by finding where their tangent lines are horizontal. And the philosophy of setting up the subject of calculus using limits is really a nineteenth-century approach to its foundations.

Inverse-square gravity was considered before Newton, and imaginary numbers before Euler.

Credit for the normal distribution should also go to de Moivre (earlier than Gauss) and Laplace (contemporaneous).

Maxwell never wrote his Equations in that manner; that came later, with Heaviside, Gibbs, Hertz and vector calculus. The simplification provided by the vector calculus is really nothing short of astonishing.

The idea of entropy came via Clausius, who found inspiration in the work of Carnot. The statement that entropy either increases or stays the same, which we could write as $dS \geq 0$, predates Boltzmann. What Boltzmann provided was an understanding of how entropy arises in *statistical* physics, the study of systems with zillions of pieces whose behavior we can’t study individually, but only in the aggregate. If you want to attribute an equation to Boltzmann in recognition of his accomplishments, it’d be better to use the one that is actually carved on his tombstone,

$$S = k \log W.$$

I am not sure that $E = mc^2$ is the proper way to encapsulate the essence of relativity theory. It is a consequence, not a postulate or a premise. The Lorentz transformation equations would do a better job at cutting to the heart of the subject. Note that these formulae are named after Lorentz, not Einstein; to put the history very, very briefly, Lorentz wrote the equations down first, but Einstein understood what they meant. (And the prehistory of $E = mc^2$ is pretty fascinating, too.)

Plucking out the Schrödinger equation (the list omits the umlaut because careless) does a disservice to the history of quantum mechanics. There are ways of doing quantum physics without invoking the Schrödinger equation: Heisenberg’s matrix mechanics, the Dirac–Feynman path integral, and the one it’s my day job to work on. In fact, not only did Heisenberg’s formulation come first, but we didn’t know what Schrödinger’s work *meant* until Max Born clarified that the square of the size of Schrödinger’s complex number $\Psi$ is a *probability*.

The number of names in that last paragraph—and I wasn’t even trying—is a clue that factoids and bullet points are not a good way of learning physics.

Yes, Robert May did write about the logistic map,

$$x_{t+1} = k x_t(1-x_t),$$

but he was hardly the first to poke at it. In his influential paper “Simple mathematical models with very complicated dynamics,” there’s a moment which expresses pretty well how science happens sometimes:

How are these various cycles arranged along the interval of relevant parameter values? This question has to my knowledge been answered independently by at least 6 groups of people, who have seen the problem in the context of combinatorial theory, numerical analysis, population biology, and dynamical systems theory (broadly defined).

Also, d’Alembert was not named “d’Almbert.”

]]>First, the 3. Consider the Pauli matrices:

$$\sigma_x = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right),\qquad \sigma_y = \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right),\qquad \sigma_z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right).$$

Note that of these three matrices, only $\sigma_y$ is antisymmetric, and also note that we have

$$\sigma_z \sigma_x = -\sigma_x\sigma_z = i\sigma_y.$$

This much is familiar, though that minus sign gets around. For example, it is the fuel that makes the GHZ thought-experiment go, because it means that

$$\sigma_x \otimes \sigma_x \otimes \sigma_x = -(\sigma_x \otimes \sigma_z \otimes \sigma_z)(\sigma_z \otimes \sigma_x \otimes \sigma_z)(\sigma_z \otimes \sigma_z \otimes \sigma_x).$$

And this leads us to where the 8 comes into the story. Let’s consider the finite-dimensional Hilbert space made by composing three qubits. This state space is eight-dimensional, and we build the *three-qubit Pauli group* by taking tensor products of the Pauli matrices, considering the $2 \times 2$ identity matrix to be the zeroth Pauli operator. There are 64 matrices in the three-qubit Pauli group, and we can label them by six bits. The notation

$$\left(\begin{array}{ccc} m_1 & m_3 & m_5 \\ m_2 & m_4 & m_6\end{array}\right)$$

means to take the tensor product

$$(-i)^{m_1m_2} \sigma_x^{m_1} \sigma_z^{m_2} \otimes (-i)^{m_3m_4} \sigma_x^{m_3} \sigma_z^{m_4} \otimes (-i)^{m_5m_6} \sigma_x^{m_5} \sigma_z^{m_6}.$$

Now, we ask: Of these 64 matrices, how many are symmetric and how many are antisymmetric? We can only get antisymmetry from $\sigma_y$, and (speaking heuristically) if we include too much antisymmetry, it will cancel out. More carefully put: We need an odd number of factors of $\sigma_y$ in the tensor product to have the result be an antisymmetric matrix. Otherwise, it will come out symmetric. Consider the case where the first factor in the triple tensor product is $\sigma_y$. Then we have $(4-1)^2 = 9$ possibilities for the other two slots. The same holds true if we put the $\sigma_y$ in the second or the third position. Finally, $\sigma_y \otimes \sigma_y \otimes \sigma_y$ is antisymmetric, meaning that we have

$$9 \cdot 3 + 1 = 28$$

antisymmetric matrices in the three-qubit Pauli group. In the notation established above, they are the elements for which

$$m_1 m_2 + m_3 m_4 + m_5 m_6 = 1 \mod 2.$$

**Puzzle:** This has a secret geometrical meaning in terms of the Fano plane. What is it?

**Hint:** $28 = 7 \cdot 4 = 7 \cdot (7 – 3)$.

We have a 3 (the nontrivial elements of the single-qubit Pauli group), an 8 (the dimension of the three-qubit Hilbert space), and a 28 (the number of antisymmetric three-qubit Pauli operators). Does this story have a 24 as well? Sneakily, yes—because the same combinatorial notation that we used to enumerate the 28 antisymmetric Pauli operators also enumerates the 28 bitangents to a quartic curve. This is a lovely piece of nineteenth-century geometry, in the genre of the 27 lines on a cubic surface. It connects to Galois theory, as this paper explains:

- D. Plaumann, B. Sturmfels and C. Vinzant, “Quartic curves and their bitangents,”
*Journal of Symbolic Computation***46**(2011), 712–33.

I didn’t think that 24 entered into this numerology until I read a little more deeply about quartic curves, and I learned that a generic plane quartic has 24 flex points.

The story of the free modular lattice on 3 generators is a story about how 3 things together build up an interesting collection of 28 things that live in an 8-dimensional space. I find it rather cute that an 8-dimensional space also yields an interesting collection of 28 things built up from 3 things in this other way.

[Cross-posted from the *n*-Category Café.]