A qubit is a thing that one *prepares* and that one *measures.* The mathematics of quantum theory tells us how to represent these actions algebraically. That is, it describes the set of all possible preparations, the set of all possible measurements, and how to compute the probability of getting a particular result from a chosen measurement given a particular preparation. To do something interesting, one would typically work with multiple qubits together, but we will start with a single one. And we will begin with the simplest kind of measurement, the *binary* ones. A binary test has two possible outcomes, which we can represent as 0 or 1, “plus” or “minus”, “ping” and “pong”, et cetera. In the lab, this could be sending an ion through a magnetic field and registering whether it swerved up or down; or, it could be sending a blip of light through a polarizing filter turned at a certain angle and registering whether there is or is not a flash. Or any of many other possibilities! The important thing is that there are two outcomes that we can clearly distinguish from each other.

For any physical implementation of a qubit, there are three binary measurements of special interest, which we can call the $X$ test, the $Y$ test and the $Z$ test. Let us denote the possible outcomes of each test by $+1$ and $-1$, which turns out to be a convenient choice. The *expected value* of the $X$ test is the average of these two possibilities, weighted by the probability of each. If we write $P(+1|X)$ for the probability of getting the $+1$ outcome given that we do the $X$ test, and likewise for $P(-1|X)$, then this expected value is $$ x = P(+1|X) \cdot (+1) + P(-1|X) \cdot (-1). $$ Because this is a weighted average of $+1$ and $-1$, it will always be somewhere in that interval. If for example we are completely confident that an $X$ test will return the outcome $+1$, then $x = 1$. If instead we lay even odds on the two possible outcomes, then $x = 0$. Likewise, $$ y = P(+1|Y) \cdot (+1) + P(-1|Y) \cdot (-1), $$ and $$ z = P(+1|Z) \cdot (+1) + P(-1|Z) \cdot (-1). $$

To specify the preparation of a single qubit, all we have to do is pick a value for $x$, a value for $y$ and a value for $z$. But not all combinations $(x,y,z)$ are physically allowed. The valid preparations are those for which the point $(x,y,z)$ lies on or inside the ball of radius 1 centered at the origin: $$ x^2 + y^2 + z^2 \leq 1. $$ We call this the *Bloch ball,* after the physicist Felix Bloch (1905–1983). The surface of the Bloch ball, at the distance exactly 1 from the origin, is the *Bloch sphere.* The points where the axes intersect the Bloch sphere — $(1,0,0)$, $(-1,0,0)$, $(0,1,0)$ and so forth — are the preparations where we are perfectly confident in the outcome of one of our three tests. Points in the interior of the ball, not on the surface, imply uncertainty about the outcomes of all three tests. But look what happens: If I am perfectly confident of what will happen should I choose to do an $X$ test, then my expected values $y$ and $z$ must both be zero, meaning that I am *completely uncertain* about what might happen should I choose to do either a $Y$ test or a $Z$ test. There is an inevitable tradeoff between levels of uncertainty, baked into the shape of the theory itself. One might even call that a matter… of principle.

We are now well-poised to improve upon the language in the news stories. The point that specifies the preparation of a qubit can be at the North Pole $(0,0,1)$, the South Pole $(0,0,-1)$, or anywhere in the ball between them. We have a whole continuum of ways to be intermediate between completely confident that the $Z$ test will yield $+1$ (all the way north) and completely confident that it will yield $-1$ (all the way south).

Now, there are other things one can do to a qubit. For starters, there are other binary measurements beyond just the $X$, $Y$ and $Z$ tests. Any pair of points exactly opposite each other on the Bloch sphere define a test, with each point standing for an outcome. The closer the preparation point is to an outcome point, the more probable that outcome. To be more specific, let’s write the preparation point as $(x,y,z)$ and the outcome point as $(x’,y’,z’)$. Then the probability of getting that outcome given that preparation is $$ P = \frac{1}{2}(1 + x x’ + y y’ + z z’). $$

An interesting conceptual thing has happened here. We have encoded the preparation of a qubit by a set of expected values, i.e., a set of probabilities. Consequently, all those late-night jazz-cigarette arguments over what probability means will spill over into the arguments about what quantum mechanics means. Moreover, and not unrelatedly, we can ask, “Why *three* probabilities? Why is it the Bloch sphere, instead of the Bloch disc or the Bloch hypersphere?” It would be perfectly legitimate, mathematically, to require probabilities for only two tests in order to specify a preparation point, or to require more than three. That would not be quantum mechanics; the fact that three coordinates are needed to nail down the preparation of the simplest possible system is a structural fact of quantum theory. But is there a deeper truth from which that can be deduced?

One could go in multiple directions from here: What about tests with more than two outcomes? Systems composed of more than one qubit? Very quickly, the structures involved become more difficult to visualize, and familiarity with linear algebra — eigenvectors, eigenvalues and their friends — becomes a prerequisite. People have also tried a variety of approaches to understand what quantum theory might be derivable from. Any of those topics could justify something in between a blog post and a lifetime of study.

**SUGGESTED READINGS:**

- E. Rieffel and W. Polak,
*Quantum Computing: A Gentle Introduction*(MIT Press, 2011), chapter 2 - J. Rau,
*Quantum Theory: An Information Processing Approach*(Oxford University Press, 2021), section 3.3 - M. Weiss, “Python tools for the budding quantum bettabilitarian” (2022)

“Dead from the next pandemic? Dead from civil war? Dead from the combination pandemic-and-civil-war?”

]]>“Preprints accepted by the

The Supreme Court today is drooling with eagerness to kill Biden’s vaccine-or-test mandate, on the legal rationale of “we declare that we can, so we will”. So, first, congratulations to Omicron. Second, this makes it even more plain that they’ll throttle the EPA on the same “fuck any regulators that want to actually regulate” basis, in a few months.

The Democrats will probably lose at least the House in November (the map isn’t turning out as gerrymandered as a lot of folks expected, but it’s still bad enough). That’s the chance of court reform gone, with a reactionary majority free to uphold theocracy, sabotage the vote, treat LGBT people as subhuman, attack labor rights, fuck over press freedom (if *Roe* is gone, *NYT v Sullivan* can hardly be safe).

The Democrats will lose the Senate in 2024 (the map will be terrible unless 2022 goes amazingly for them). Oh, and two years of Republicans running the House means two years of Benghazi!-ing, a shutdown or two, doing everything possible to make Trump president again. Did we mention that one factor in that nominally not-so-bad map has been “incumbent protection”, i.e., baking in the MAGA?

… OK, maybe Trump will be dead by then, or too ill to be propped up on two feet. DeSantis seems the most likely heir at the moment. But whatever.

And supposing Biden wins in ’24? Not a lot he’ll be able to do with both the Senate and (probably) the House against him.

Point is, we’re on a three-year train to Fuckedville while the planet cooks around us.

I’ve seen people cast about for analogies for what’s in progress/likely to be coming. Turkey under Erdogan? Hungary under Orban? The Time of Troubles? There always seems to be some ingredient that makes the analogy not quite match, for me, but not a single option on the table looks good.

What was that old Adam Smith line about there being “a great deal of ruin in a nation”? Right now, we’re in the middle of measuring just how much ruin there is.

(Apropos, how much ruin is there in a health-care system?)

Autocracy is here. It just isn’t evenly distributed, yet.

]]>Nobody has answered “24x – 3”.

(Grandpa Stacey voice) I am disappointed.

]]>As the kids say, “Like, quantum and subscribe!”

]]>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”.

]]>**EDIT TO ADD (8 September):** To my surprise, I was able to edit those notes in the direction of being a paper. A few items came out after my post which lifted the burden of discussing certain topics and let a theme come together. Accordingly, see arXiv:2109.03186.

And I’m all, “So, instead of testing *one* effective field theory by putting matter into extreme conditions, you’re testing … multiple … effective field theories … by putting matter into extreme conditions.” I’m making my astonished face, can’t you tell?

Bartleby the Scrivener: I would prefer not to

and… scene

]]>- B. C. Stacey, “Canonical probabilities by directly quantizing thermodynamics” (PDF).

The idea is that Boltzmann’s rule $p(E_n) \propto e^{-E_n / k_B T}$ pops up really naturally when you ask for a rule that plays nicely with the *composing-together* of uncorrelated systems. This, in turn, gives a convenient expression to the idea that classical physics is what you get when you handle quantum systems sloppily.

It’s common to present “the Copenhagen interpretation” as a kind of dynamical collapse model, in which wavefunctions are ontic entities (like a sophomore’s picture of the electromagnetic field) that evolve according to the Schrödinger equation, except in moments of “measurement” that take place in unspecified conditions. This portrayal is typically intended to make “the Copenhagen interpretation” sound like a mutant form of Newtonian mechanics where $F = ma$ almost always, except at peculiar instants when $F$ suddenly becomes $ma/2$ and then switches back again. Of course, this is abhorrent and pathological.

When I was a child, my parents bought me a magnet from a museum gift shop. It had a long handle, likely made deliberately to resemble a magic wand, and as educational toys go, it served its function, since I went around poking all sorts of things to see if the magnet would grab them. I suspect this is a common enough type of learning experience. One discovers, for example, that it will pick up paperclips but not pennies. Having calibrated one’s understanding of the magnet, one can then use it as a tool — say, by telling which of two matchboxes is filled with paperclips, or that something is different about a wire coil connected to a battery versus one that is not.

What concerned Bohr himself was that this transition — between the calibration phase, when an object is under scrutiny, and its later use as a laboratory instrument — is conceptually nontrivial. First a lens is a strangely curved block of glass we must work to comprehend, and then it is a means to overthrow Aristotle. There are not two different dynamical laws, but two different *languages*.

Here’s how John Wheeler put it:

“Bohr stresses […] that the stick we hold can itself be an object of investigation, as when we run our fingers over its surface. The same stick, when grasped firmly and used to explore something else, becomes an extension of the observer or—when we depersonalize—a part of the measuring equipment. As we withdraw the stick from the one role, and recast it in the other role, we transpose the line of demarcation from one end of it to the other. The distinction between the probed and the probe, so evident at this scale of the everyday, is the without-which-nothing of every elementary phenomenon, of every closed quantum process.”

[From “Law Without Law”, in the Wheeler–Zurek collection, p. 206]

The commonalities and contrasts with QBism should be evident enough. Extension of the observer, yes; depersonalize to mere dead “equipment”, no, for it is the latter move that gets one into trouble with Wigner’s Friend. And, on a perhaps more practical level where the choice of research problems is concerned, Bohr takes the quantum formalism pretty much as given and leaves “the quantum principle” not explicitly defined.

It may also be illustrative to consider how Rovelli’s “Relational Quantum Mechanics” treats this point. I tentatively infer that Rovelli thinks giving a special role to an agent means imposing two different dynamical laws, one for systems of agent-type and another for all nonagent physical entities. Even if he doesn’t spell it out, that seems to be the mindset he operates with, and the background he relies upon. Of course, he balks at that dichotomy. I would, too!

]]>Nearly two decades ago now, there was a dust-up when a couple French TV personalities got a clutch of physics and mathematics papers published, and even received PhD’s, and their “work” turned out to be nonsense. (The Wikipedia article on *l’affaire Bogdanov* is currently not terrible, and it contains more pointers to details than almost anyone could honestly desire.) The news stories about the incident really played up the “even the physicists can’t tell if the papers are nonsense or not” angle. That rather oversells the case, though. I read the Bogdanovs’ “Topological field theory of the initial singularity of spacetime” when I was a first-year grad student, and I could see through it. If you know what a Lagrangian is, and the fog doesn’t intimidate you, then you can tell something is wrong. If you don’t know what a Lagrangian is, you’re probably not reading theoretical physics papers yet.

So, what went wrong?

Nobody knows for sure, but we can hazard a plausible guess. Peer-reviewing is an obligation, and not a task that is often relished. It’s like jury duty: We understand the importance of what the system is trying to uphold, but on any given day, we’d rather be doing something else.

If you want to find the problems in a *really* bad paper, don’t ask an ordinary peer-reviewer. You’re better off with the scientists who congregate online to talk their subject because they have passion for it. And that’s what happened with the Bogdanov papers. The formal reviewers flubbed them, but sci.physics.research and the physics blogs delivered.

The old hands in the online forums will also be the ones most seasoned by claims of perpetual-motion machines and disproofs of relativity. They’ll have a more practiced critical eye and be less likely to get snowed by obfuscation. That long stretch of background exposition which sounds correct? It’s no guarantee that the *new* idea in the paper is — it could have been copied out of any standard text — and it can even be a warning sign. (We don’t need the 1001th introduction to why quantum gravity is a puzzle, or what the P versus NP problem is. Why do you, Mr. Author, think we do?)

The Bogdanovs had a pretty good success rate, getting *six* articles published. They also posted an un-refereed preprint that doesn’t appear to have been formally submitted anywhere (and whose mathematics was also wrong).

So, going by the numbers, physics — hardest of the hard sciences — must be six times as defective as postmodern philosophy, or Science, Technology and Society studies, or whatever it was that Alan Sokal demolished by getting his hoax paper into *Social Text*, right?

This is also a better success rate than Bohannon’s sting on open-access journals (157/304) or the “grievance studies” hoax (7/20, counting charitably). Of course, nobody who found the “grievance studies” hoax impressive thinks the Bogdanov affair exposes physics as rotten postmodern tripe!

(We might also note the irony that theme Sokal and Bricmont’s book keeps hitting is that the “fashionable nonsense” is gender-essentialist, tying femininity with menstruation for example. A good helping of queer and trans theory would harmonize perfectly with the stronger parts of their critique! Yet it is the people who appoint themselves their biggest fans who wish to erase queer and trans theory from the academy as just so much “grievance studies”. Likewise, I’d bet cash money that many Jordan Peterson fanboys regard the Sokal hoax as checkmate for postmodernism. But I’m having a hard time telling Lacan’s “This torus really exists and it is exactly the structure of the neurotic” apart from Peterson’s “I really believe that’s a representation of DNA.”)

I, too, once found Sokal’s prank basically the funniest thing ever. Taking on board what Sokal himself said shortly thereafter — “From the mere fact of publication of my parody I think that not much can be deduced” — I have to speculate, though, that the same point could have been made with more light and less heat even if his paper had been published with the journal’s contrivance, say as an April Fool’s item. The real point was not that the editors of one journal could be asleep at the wheel regarding one individual paper, but that many highly-regarded people had (apparently) said things about physics that were dubious at best, and without getting pushback from their academic communities. In retrospect, many years and two books from Sokal later, I cannot say that the prank aspect did a particularly good job of clarifying that. If anything, it seems to have convinced people of the merits of a glamorous but uninformative methodology.

]]>We’d all be so much better off, had the lessons of fourth grade only stuck.

(Also, the voice actor for the engineer/architect type character in a lot of those David Macaulay adaptations was Brian Blessed, which is pretty nice.)

I met David Macaulay once when he gave a talk at MIT and I happened to hear about it in time. I’m glad I had the chance to interact even briefly with the fellow who wrote and drew the books I grew up on, you know?

Some years prior to that, I saw James Burke of *Connections* fame in person, but I didn’t have the chance to ask him a question after his talk.

That’s part of a good story, actually. It was at a defense-industry trade conference, which (I presume as part of some PR package) hosted “Astro Bowl”, a trivia-team competition for high schools around the state. It was pretty much like academic team, which I also did, except that all the questions were about astronomy and space exploration. This was Huntsville, Alabama, and sponsoring that kind of thing was very much what the local industry would do. Did I mention that the event was at a convention center named for Wernher von Braun? Anyway, we were answering trivia questions in one room, and in the next banquet hall over, James Burke was going to deliver a keynote. So, during a gap in our schedule, I ducked in, missing the beginning of his speech and having to leave before the Q&A.

The prize for winning Astro Bowl was a trip to a space shuttle launch, so we were pretty motivated. Our chemistry teacher had gathered a few of us to form a team — this was the first year the competition was being held, and we didn’t really know how it would work. Neither, it seems, did the organizers. They sent us a packet of material to study, and we figured that material indicated the general subject matter that would be fair game, so we brushed up on those topics. Turns out, they asked questions *directly* from the worksheets and glossaries and such that they had mailed out, including some real esoterica like being able to identify shuttle missions by their STS numbers.

Well, we did a pretty mediocre job on the first day, barely squeaking into eligibility for the second. So, we gathered at my house and spent the night memorizing the exact mathematical minimum of each item in the info packet that uniquely identified it.

Come the next day.

“Question: What is the red—”

(bzzp)

“Yes, Grissom High?”

“‘Photosphere’, Steve!”

And that’s the story of how we won a free trip to see the space shuttle go into space.

Winning trivia contests got me two other trips to Florida, because I also did Academic Team, and the school that won state each year got to go to the nationals at Disney World, jointly sponsored by Disney, Pepsi and Panasonic. And it was every bit as surreal a hellscape cross-fertilization of capitalist faux-largesse and conniving college-placement gamesmanship as that would suggest.

Academic Team was mostly a way to hang out with people I liked and to get out of school during the day, so it was fine, though. For the first chunk of the year, there were competitions at the different high schools around the city (which was an education in relative economic advantage, for sure). If you won the city, you went to district, then regionals and then state, as tournaments typically go.

There were also various other competitions floating around the schedule. I have a fond memory of the one at Vanderbilt, which was special because college kids wrote the questions and so you could get real oddballs thrown in, like how to kill Ganon in the original Zelda or the ways to order hash browns at Waffle House. Fuck if we didn’t *own* on that. Where else was there to hang out in Huntsville, AL, except for Waffle House?

My own best moment, though, was a “Who am I?” question, where you had to guess the person or character’s name based on a purported self-description. “Who am I? In my youth, I read alchemists like Paracelsus and Albertus Magnus—”

(bzzp)

“Doctor Victor Frankenstein!”

The guy does a legit double take — like, who the hell could get that from *that* — and then recovers. “Please, it’s pronounced Fraahnk-en-shteen.”

Speaking of relative economic advantage: We were definitely the rich kids within Huntsville, and pretty clearly not so in a broader context, like at state or the Vanderbilt tournament, which drew from schools across the southeast. The Vanderbilt meet, actually, let a school send as many teams as they wanted to pay the registration fee for.

And shutting down the teams from the private schools who could field half a dozen of them? Was pretty fuckin’ sweet.

If you want to make a movie out of it, you can call this part of the story “the power of teamwork”. A lot of the schools who did well at regionals or state and such actually only had one good member on their team, who would be a glory hog and try to carry it all themselves. They’d jump in and answer early, sometimes earning big, but failing often enough to be weak against an adversary who could gather up the pieces they missed. That would work against a lot of teams, but what do you know, we learned how to respect each others’ capabilities. I could lean back and let Daniel take a question or vice versa. So, we could divide the field, and when matched against an egotist, we won — well, pretty much every time.

I did Academic Team in 11th and 12th grades, and both years, we won state and got to fly down to Orlando for nationals. We stayed at the Contemporary, drank excessive quantities of free soda, and rode the monorail often enough that “!Por favor manténgase alejado de las puertas!” became a running joke.

The first time, the summer after my 11th grade, we came in sixth in the nation. That is, we made it to the final round, where we placed sixth out of six. The year after that, we cleaned up on the first day, which by the way the matchups were arranged meant that we could skip directly to the semifinals on the last day. Then, we kind of gave up during the semifinals and barely brought our game, so we washed out and didn’t proceed to the finals at all. It was the summer after we graduated high school. Who the fuck cared?

]]>And about saying in January 2017, “I’m really not feeling that good about our ability to handle the next epidemic that comes our way.”

Now, turning out right when you’d much rather have been wrong is of course a complicated feeling. But I realized something. If you ever see a physicist getting out of his lane and opining about a subject that is not physics, you can direct him to me. I will then instruct him to bow before me, because I am his fucking god.

]]>