(He won’t; he’s too busy appropriating its respectability.)

Today, I learned that Boghossian’s institutional review board found that his actions in the “Grievance Studies” hoax constitute research misconduct.

**EDIT TO ADD:** Further commentary on the ethics of academic hoaxes.

- B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”

]]>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.

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.

]]>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.

So, if you’re wanting for some commentary on quantum mechanics, here goes:

Gleason’s theorem begins with a brief list of postulates, which are conditions for expressing “measurements” in terms of Hilbert spaces. To each physical system we associate a complex Hilbert space, and each measurement corresponds to a resolution of the identity operator — in Gleason’s original version, to an orthonormal basis. The crucial assumption is that the probability assigned to a measurement outcome (i.e., to a vector in a basis) does not depend upon which basis that vector is taken to be part of. The probability assignments are “noncontextual,” as they say. The conclusion of Gleason’s argument is that any mapping from measurements to probabilities that satisfies his assumptions must take the form of the Born rule applied to some density operator. In other words, the theorem gives the set of valid states *and* the rule for calculating probabilities given a state.

(It is significantly easier to prove the POVM version of Gleason’s theorem, in which a “measurement” is not necessarily an orthonormal basis, but rather any resolution of the identity into positive semidefinite operators, $\sum_i E_i = I$. In this case, the result is that any valid assignment of probabilities to measurement outcomes, or “effects,” takes the form $p(E) = {\rm tr}(\rho E)$ for some density operator $\rho$. The math is easier; the conceptual upshot is the same.)

I have a sneaky suspicion that a good many other attempted “derivations of the Born rule” really amount to little more than burying Gleason’s assumptions under a heap of dubious justifications. MGM don’t quite do that; what they present is more interesting.

They start with what they consider the “standard postulates” of quantum mechanics, which in their reckoning are five in number. Then they discard the last two and replace them with rules of a more qualitative character. Their central result is that the discarded postulates can be re-derived from those that were kept, plus the more qualitative-sounding conditions.

MGM say that the assumptions they keep are about state space, while the ones they discard are about measurements. But the equations in the three postulates that they keep could just as well be read as assumptions about measurements instead. Since they take measurement to be an operationally primitive notion — fine by me, anathama to many physicists! — this is arguably the better way to go. Then they add a postulate that has the character of noncontextuality: The probability of an event is independent of how that event is embedded into a measurement on a larger system. So, they work in the same setting as Gleason (Hilbert space), invoke postulates of the same nature, and arrive in the same place. The conclusion, if you take their postulates about complex vectors as referring to measurement outcomes, is that “preparations” are dual to outcomes, and outcomes occur with probabilities given by the Born rule, thereupon turning into new preparations.

Let’s treat this in a little more detail.

Here is the first postulate of what MGM take to be standard quantum mechanics:

To every physical system there corresponds a complex and separable Hilbert space $\mathbb{C}^d$, and the pure states of the system are the rays $\psi \in {\rm P}\mathbb{C}^d$.

We strike the words “pure states” and replace them with “sharp effects” — an equally undefined term at this point, which can only gain meaning in combination with other ideas later.

(I spend at least a little of every working day wondering why quantum mechanics makes use of complex numbers, so this already feels intensely arbitrary to me, but for now we’ll take it as read and press on.)

MGM define an “outcome probability function” as a mapping from rays in the Hilbert space $\mathbb{C}^d$ to the unit interval $[0,1]$. The abbreviation OPF is fine, but let’s read it instead as *operational preparation function.* The definition is the same: An OPF is a function ${\bf f}: {\rm P}\mathbb{C}^d \to [0,1]$. Now, though, it stands for the probability of obtaining the measurement outcome $\psi$, for each $\psi$ in the space ${\rm P}\mathbb{C}^d$ of sharp effects, given the preparation ${\bf f}$. All the properties of OPFs that they invoke can be justified equally well in this reading. If ${\bf f}(\psi) = 1$, then event $\psi$ has probability 1 of occurring given the preparation ${\bf f}$. For any two preparations ${\bf f}_1$ and ${\bf f}_2$, we can imagine performing ${\bf f}_1$ with probability $p$ and ${\bf f}_2$ with probability $1-p$, so the convex combination $p{\bf f}_1 + (1-p){\bf f}_2$ must be a valid preparation. And, given two systems, we can imagine that the preparation of one is ${\bf f}$ while the preparation of the other is ${\bf g}$, so the preparation of the joint system is some composition ${\bf f} \star {\bf g}$. And if measurement outcomes for separate systems compose according to the tensor product, and this $\star$ product denotes a joint preparation that introduces no correlations, then we can say that $({\bf f} \star {\bf g})(\psi \otimes \phi) = {\bf f}(\psi) {\bf g}(\phi)$. Furthermore, we can argue that the $\star$ product must be associative, ${\bf f} \star ({\bf g} \star {\bf h}) = ({\bf f} \star {\bf g}) \star {\bf h}$, and everything else that the composition of OPFs needs to satisfy in order to make the algebra go.

Ultimately, the same math has to work out, after we swap the words around, because the final structure is self-dual: The same set of rays ${\rm P}\mathbb{C}^d$ provides the extremal elements both of the state space and of the set of effects. So, if we take the dual of the starting point, we have to arrive in the same place by the end.

But is either choice of starting point more *natural*?

I find that focusing on the measurement outcomes is preferable when trying to connect with the Bell and Kochen–Specker theorems. In the former, the “preparation” is fixed, while in the latter, it can be arbitrary, but either way, we don’t say much *interesting* about it. The action lies in the choice of measurements, and how the rays that represent one measurement can interlock with those for another. So, from that perspective, putting the emphasis on the measurements and then deriving the state space is the more “natural” move. It puts the mathematics in conceptual and historical context.

That said, on a deeper level, I don’t find either choice all that compelling. To appreciate why, we need only look again at that arcane symbol, ${\rm P}\mathbb{C}^d$. That is the setting for the whole argument, and it is completely opaque. Why the complex numbers? Why throw away an overall phase? What is the meaning of “dimension,” and why does it scale multiplicatively when we compose systems? (A typical justification for this last point would be that if we have $n$ completely distinct options for the state of one system, and we have $m$ completely distinct options for the state of a second system, then we can pick one from each set for a total of $nm$ possibilities. But what are these options “completely distinct” with respect to, if we have not yet introduced the concept of measurement? Why should dimension be the quantity that scales in such a nice way, if we have no reason to care about vectors being orthogonal?) All of this cries out for a deeper understanding.

]]>When I heard about *J. Con. Id.,* I couldn’t help thinking that I have myself supported some unpopular scientific opinions. A few times, that’s where my best professional judgment led me. When my colleagues and I have found ourselves in that position, we set forth our views by publishing … in *Nature.*

- M. J. Wade et al., “Multilevel and kin selection in a connected world,”
*Nature***463**(2010), E8–E9. - N. David Mermin, “Physics: QBism puts the scientist back into science,”
*Nature***507**(2014), 421–23.

(I have to admit that the 2010 comment is not as strong as it could have been. It was a bit of a written-by-committee job, with all that that implies. I recommend that every young scientist go through that process … once. Better papers in the genre came later. And for my own part, I think I did a better job distinguishing all the confusing variants of terminology when I had more room to stretch, in Chapter 9 of arXiv:1509.02958.)

]]>Over on his blog, Peter Woit quotes a scene from the imagination of John Horgan, whose *The End of Science* (1996) visualized physics falling into a twilight:

A few diehards dedicated to truth rather than practicality will practice physics in a nonempirical, ironic mode, plumbing the magical realm of superstrings and other esoterica and fretting about the meaning of quantum mechanics. The conferences of these ironic physicists, whose disputes cannot be experimentally resolved, will become more and more like those of that bastion of literary criticism, the Modern Language Association.

OK (*cracks knuckles*), a few points. First, “fretting about the meaning of quantum mechanics” has, historically, been damn important. A lot of quantum information theory came out of people doing exactly that, just with equations. The *productive* way of “fretting” involves plumbing the meaning of quantum mechanics by *finding what new capabilities quantum mechanics can give you.* Let’s take one of the least blue-sky applications of quantum information science: securing communications with quantum key distribution. Why trust the security of quantum key distribution? There’s a whole theory behind the idea, one which depends upon the quantum de Finetti theorem. Why is there a quantum de Finetti theorem in a form that physicists could understand and care about? Because Caves, Fuchs and Schack wanted to prove that the phrase “unknown quantum state” has a well-defined meaning for personalist Bayesians.

This example could be augmented with many others. (I selfishly picked one where I could cite my own collaborator.)

It’s illuminating to quote the passage from Horgan’s book just before the one that Woit did:

This is the fate of physics. The vast majority of physicists, those employed in industry and even academia, will continue to apply the knowledge they already have in hand—inventing more versatile lasers and superconductors and computing devices—without worrying about any underlying philosophical issues.

But there just isn’t a clean dividing line between “underlying philosophical issues” and “more versatile computing devices”! In fact, the foundational question of what the nature of “quantum states” really are overlaps with the question of which quantum computations can be emulated on a classical computer, and how some preparations are better *resources* for quantum computers than others. Flagrantly disregarding attempts to draw a boundary line between “foundations” and “applications” is my day job now, but quantum information was already getting going in earnest during the mid-1990s, so this isn’t a matter of hindsight. (Feynman wasn’t the first to talk about quantum computing, but he was certainly influential, and the motivations he spelled out were pretty explicitly foundational. Benioff, who preceded Feynman, was also interested in foundational matters, and even said as much while building quantum Hamiltonians for Turing machines.) And since Woit’s post was about judging whether a prediction held up or not, I feel pretty OK applying a present-day standard anyway.

In short: Meaning matters.

But then, Horgan’s book gets the Einstein–Podolsky—Rosen thought-experiment completely wrong, and I should know better than to engage with what any book like that on the subject of what quantum mechanics might mean.

To be honest, Horgan is unfair to the Modern Language Association. Their convention program for January 2019 indicates a community that is actively engaged in the world, with sessions about the changing role of journalism, how the Internet has enabled a new kind of “public intellectuals”, how to bring African-American literature into summer reading, the dynamics of organized fandoms, etc. In addition, they plainly advertise sessions as open to the public, which I can only barely imagine a physics conference doing more than a nominal jab at. Their public sessions include a film screening of a documentary about the South African writer and activist Peter Abrahams, as well as workshops on practical skills like how to cite sources. That’s not just valuable training, but also a topic that is actively evolving: How do you cite a tweet, or an archived version of a Wikipedia page, or a post on a decentralized social network like Mastodon?

Dragging the sciences for supposedly resembling the humanities has not grown more endearing since 1996.

All this came up in the context of physics being done by artificial intelligence. If anything, the idea of “machines replacing physicists” is *less* plausible to me now than it was two decades ago, because back then, there was at least a *chance* that AI would have had something to do with understanding how human minds work, rather than just throwing a bunch of GPUs at a problem and calling the result “machine learning”. This perspective is informed in part by long talks with a friend whose research area *is* machine learning, and who is quite dissatisfied with the common approach to it. Specifically, they work in computer vision, where the top-notch algorithms still identify the Queen’s crown as a shower cap and can be fooled into calling a panda a vulture. People have problems, but not *those* problems.

What research is it that has prompted the specter of the Machine solving the Theory of Everything? Honestly, the “machine learning in the string landscape” language sounds to me like a new coat of paint over a general approach that physicists have been using for as long as we’ve had computers. Pose a problem, get your grad students to feed it into the computer, obtain numerical results, see what conjectures those results suggest, and if you’re lucky, prove those conjectures. In this particular case, the conjectures eventually proven might not ultimately connect to experiment, but that’s the old problem of *quantum gravity being hard to study,* not a new problem about the way physics is being done. And in order to put a question to a computer, you have to (get your grad student to) phrase it very carefully. You can’t ask the computer for a heuristic argument based on conjectural features that a nonperturbative theory of quantum gravity might have, were we to know of one. You have to talk about structures you can define (structures that you might *guess* are relevant to a nonperturbative theory of quantum gravity). For example, you might define a Calabi–Yau space in terms of an affine cone over a toric variety, which you in turn define in terms of a convex lattice polygon, etc., eventually converting the problem into one that you can code up. There’s still the big gap between your work and experiment, and there’s still the lack of a well-defined over-arching theory, but you’ve made your little corner of “experimental mathematics” less vague.

You might even be moving science in a healthy direction, by taking some of the ideas that have grown under the “string theory” umbrella and making them less a matter for physicists, and more a concern for the people who like to map extraordinarily complex mathematical structures for their own sake — the people who, for example, stay up late thinking about the centralizers of the Monster group. That shift could be a mutually beneficial development.

]]>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.

]]>We answer several questions that have been Frequently Asked about QBism. These remarks (many of them lighthearted) should be considered supplements to more systematic treatments by the authors and others.

Notwithstanding its great influence in modern physics, the EPR thought-experiment has been explained incorrectly a surprising number of times.

22 pages; an unknown number of bridges burned.

]]>Very Serious People: “I am aghast that Sarah Jeong said that we should #CancelWhitePeople!”

Me: “Ghasp! As an intensely white person myself, I—” [looks at demographics of who voted for Roy Moore time and time again] “… I can’t really object to that.”

Very Serious People: “If you replace ‘white’ with any other group, then what she said is incredibly racist.”

Me: “You have a good—” [holds finger to earpiece] “Hold on, I am being informed that words have meaning, and changing words can … change meaning?”

Generally, I continue to be impressed by how intensely the chattering class can clutch their pearls whenever a woman says something impolite in a setting where impoliteness is both ethically justified and culturally the norm, while the President^{*} repeatedly demonizes the very idea of a free press before audiences that salivate for fascism.

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.

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.

]]>