“OK,” you might say, while wondering what the big deal is.

“In fact, I am going to measure *all speeds* as the time it takes to travel a standard unit of distance.”

“Uh, hold on.”

“And this means that, contrary to what you learned in Big University, zero is not a speed! Because the right way to think of speed is the time it takes to travel 1 standard distance unit, and an object that never moves never travels.”

Now, you might try to argue with me. You could try to point out all the things that my screwy definition would break. (For starters, I am throwing out everything science has learned about inertia.) You could try showing examples where scientists I have praised, like Feynman or whoever, speak of “a speed equal to zero”. When all that goes nowhere and I dig in further with every reply, you might justifiably conclude that I am high on my own supply, in love with my own status as an iconoclast. Because that is my *real* motivation, neither equations nor expertise will sway me.

Yes, it’s time for another installment in my occasional series, *Friends Don’t Let Friends Learn Topic X from Eliezer Yudkowsky.* For those who don’t know, Yudkowsky is an autodidact and fanfiction writer who, like E. L. James, portrays insufferable characters as admirable and thereby gives the whole medium a bad name. Unlike James, he also fills his work with bad science. Because he scratches an emotional itch for people enamored of the idea that they are above emotion, he has become influential in circles you would rather avoid.

Among many other things that Yudkowsky has famously attempted to explain is the concept of probability. The bit I want to zoom in upon today is the time that he argued that 0 and 1 are not probabilities. He grounds this headscratcher in the statement that you can’t turn a probability of 1 into a ratio by the function $f(p) = p/(1-p)$, because you’d be dividing by 0. This and everything that followed is just getting high off his own supply. One could try showing how he presumes his own conclusion. One could try showing how he breaks the basic idea that probabilities by their nature add up to 100% (given an event *E*, what can Yudkowsky say is the probability of the event *E*-or-not-*E*?). One could even observe that the same E. T. Jaynes he praises in that blog post uses 1 as a probability, for example in Chapter 2 of *Probability Theory: The Logic of Science* (Cambridge University Press, 2003). If you really want to cite someone he admires, you could note that Eliezer Yudkowsky uses 1 as a probability when trying (and failing) to explain quantum mechanics, because he writes probability amplitudes of absolute value 1.

As an academic, I have to hold myself back from developing all those themes and more. But the additional wrongness that comes in when he turns to quantum mechanics is worth pausing to comment upon.

Yudkowsky loves to go on about how *the map is not the territory,* to the extent that his fandom thinks he coined the phrase, but he is remarkably terrible at understanding which is which. Or, to be a little more precise, he is actively uninterested in appreciating that the question of what to file under “map” versus “territory” is one of *the* big questions that separate the different interpretations of quantum mechanics. He has his desired answer, and he argues for it by assertion.

He’s also just ignorant about the math. Stepping back from the details of what he gets wrong, there are bigger-picture problems. For example, he points to a complex number and says that it can’t be a probability because it’s complex. True, but so what? The Fourier transform of a sequence of real numbers will generally have complex values. Just because one way of expressing information uses complex numbers doesn’t mean that every perspective on the problem has to. And, in fact, what he tries to do with two complex numbers — one amplitude for each path in an interferometer — you can actually do with three real numbers. They can even be probabilities, say, the probability of getting the “yes” outcome in each of three yes/no measurements. The quantumness comes in when you consider how the probabilities assigned to the outcomes of different experiments all fit together. If probabilities are, as Yudkowsky wants, always part of the “map”, and a wavefunction is mathematically equivalent to a set of probabilities satisfying some constraint, then a wavefunction belongs in the “map”, too. You can of course argue that *some* probabilities are “territory”; that’s an argument which smart people have been having back and forth for decades. But that’s not what Yudkowsky does. Instead, through a flavor swirl of malice and incompetence, he ends up being too much a hypocrite to “steelman” the many other narratives about quantum mechanics.

The one thing in Thompson’s presentation that I didn’t particularly like is how he introduces derivatives of trig functions. It presumes that the reader has a lot of trig identities in their back pocket, and it makes a simplification that is hard to justify without going into limits, a topic that Thompson doesn’t explicitly teach. I’ve tried my hand at a replacement that appeals to the way he *does* teach.

Further modifications may come as people apprise me of all the things I missed. I do wish to keep it short and sweet, rather than adding multiple new chapters.

]]>As I have written before, it is very difficult to provide substantive criticism of a “theory” that has no substance. I could point to individual things that make no sense, but the people who care don’t need my help, and the people who don’t won’t be convinced by anything I say. (“I asked ChatGPT to summarize the paper, and I found the results quite inspirational!”) I could try to provide a little media literacy, like *feel free to ignore any science “news” that’s just a press release from the guy who made it up.* But again, if you’re thirsty for something else, that will hardly satisfy. (“Reality is all on the blockchain, buy GameStop!”)

I promise this is going somewhere.

A certain bakery has a special deal on muffins. They sell mystery boxes for those who like to live dangerously: mix-and-match sets of three muffins apiece. Each day, Alice, Bob and Charlie buy a mystery box together, and each day, Alice, Bob and Charlie take one muffin apiece back to their respective laboratories for analysis. They each have two testing devices — say, a device that can test whether a muffin is positive for dairy, and another device that can test whether it is positive for tree nuts. We’ll call these $X$ tests and $Y$ tests for short. Each day, Alice chooses either to do an $X$ test or a $Y$ test. Bob likewise chooses, independently of Alice, and so does Charlie. Importantly, each muffin can only be tested *once.* Maybe the test destroys the muffin, or maybe it takes so long to do one that they eat their muffins immediately afterward. Whatever the rationale, one test per muffin — that’s a rule of the parable.

We can write what they choose to do in a compact way. For example, if all three of them choose to do the $X$ test on their respective muffins, we’ll write $A_X B_X C_X$. If Bob and Charlie choose to do the $Y$ test but Alice goes instead with the $X$ test, we’ll write $A_X B_Y C_Y$. And so on. We can also write the results compactly, using $+1$ to stand for a positive result and $-1$ to stand for a negative one. (We could also record the outcomes with zeros and ones, or with trues and falses, greens and blues, etc. Using $+1$ and $-1$ is just a notation that will turn out to be helpful in a moment.) So, for example, if Alice chooses $Y$, Bob chooses $X$ and Charlie goes with $Y$, the results might be $(+1, -1, -1)$. Or they might be $(+1, +1, +1)$, or perhaps $(-1, +1, -1)$.

Over many days of muffin investigation, comparing their notes, they find a dependable pattern. Whenever *two of them* choose to do the $Y$ test, then the *product of their results* is always $+1$. The specific outcome varies randomly from day to day, but there’s never only one $-1$, and they never get all three results being $-1$. From this pattern, they can draw a couple conclusions. First, once two of them obtain their results, the result of the third is predictable. Let’s say their choices are $A_Y B_X C_Y$, as in the previous example, and both Alice and Bob get the result $+1$. Then we can predict that Charlie will get $+1$, because that’s the only way the product of the three numbers can be $+1$. Or, suppose their choices are $A_Y B_Y C_X$, and both Bob and Charlie get a $-1$. The two of them report their results and wait for Alice. Knowing Bob and Charlie’s results, we can predict that Alice will report a $+1$ outcome, because that’s the only way the product of the three outcomes is $+1$. A minus times a minus makes a plus, and so a third minus would spoil the plus.

If we had used a different notation for the outcomes, like “green” and “blue” instead of $+1$ and $-1$, then we could express this pattern by saying that whenever two of them choose to do the $Y$ test, an even number of the results will be blue.

Now, we deduce something else from the pattern. We can make a prediction about what happens under very different conditions. What about the days when all three choose to measure $X$?

We’ve used capital letters to write their choices. Let’s use lowercase letters to denote the *properties* of the muffins being measured. An $A_X$ measurement — that is, Alice doing the $X$ test — uncovers the value of $a_X$. Likewise, Bob doing the $Y$ test reveals the $b_Y$ muffin property, and so on. The pattern so far is that

$$ a_X b_Y c_Y = +1, $$

and

$$ a_Y b_X c_Y = +1, $$

and

$$ a_Y b_Y c_X = +1. $$

Here comes the neat trick. We multiply all three of these facts together.

$$ (a_X b_Y c_Y)(a_Y b_X c_Y)(a_Y b_Y c_X) = +1. $$

We rearrange the variables, putting like with like:

$$ a_X b_X c_X a_Y^2 b_Y^2 c_Y^2 = +1.$$

On any given day, we don’t know the value of $a_Y$ or $b_Y$ or $c_Y$ before the test, and if Alice chooses $X$ instead of $Y$, then we’ll never learn $a_Y$ for that day, and likewise for Bob and Charlie. But the values do have to be waiting there, don’t they? Each has to be either $+1$ or $-1$, even if we never learn which: That’s just a fact determined at the bakery.

The square of $-1$ is the same as the square of $+1$, so on every day,

$$ a_Y^2 = b_Y^2 = c_Y^2 = +1. $$

Therefore, we can drop all the $Y$ factors from our previous equation!

$$ a_X b_X c_X = +1. $$

And now we have a prediction: on those days when all three independently choose to do the $X$ test, the product of their answers will be

$$ A_X B_X C_X = +1. $$

This follows from the dependability of the pattern found on the two-$Y$-test days and the basic assumption that the tests are testing properties intrinsic to the muffins, properties waiting to be found. The tests don’t have to be for dairy and tree nuts — any properties will do. The objects being tested don’t have to be muffins, either. Anything that can come in sets of three is suitable. From the dependable pattern in the two-$Y$-test events, we can draw a conclusion about the triple-$X$-test events. The product of the $X$ results is going to be $+1$.

How general this result is! From the one pattern, we deduce the other, whether we find that first pattern in muffins, cookies, apples, sand grains…

Electrons? Photons?

Ay, there’s the rub.

It is possible to make triplets, not of baked goods but of subatomic particles, that fit the two-$Y$-test pattern. After doing the $Y$ test on the first two particles, for example, the result of an $X$ test is predictable, using the same rule we described above.

*But the prediction about what happens when each particle gets the $X$ test does not hold.*

The rules of quantum mechanics imply that there is a way an experimenter can prepare triplets of particles such that they should predict that the product of an $X$ result and two $Y$ results will always be $+1$, while the product of three $X$ results will always be $-1$.

By rights, this ought to be impossible. But the imagination of nature is more subtle than our own!

Doing the calculation to get the quantum-mechanical answer is not all that hard. It’s a spot of matrix algebra that’s doable by hand and doesn’t require knowing much more than what an “eigenvector” is. (The readings below have more details.) The much more difficult part is knowing what to make of it!

The predictions of quantum mechanics — which have been checked in the lab, directly and indirectly, and found to work superbly — clash with the basic and seemingly bulletproof calculation we have done here. How can that calculation fail to apply? Where is the gap that means one pattern doesn’t have to imply the other? If an argument whose fundamental premise is that *measurements record a property of the thing being measured* is inconsistent with quantum physics, our spectacularly successful guide to living in reality, then what does that say about reality?

Well, ask two rabbis and you’ll get three opinions. It’s not even clear what making progress on that kind of “what does it all mean?!” question even looks like. If people disagree on the meaning but all use the same math to make the same predictions, on what grounds can we even prefer one proposal about the meaning over another? Gut reaction? Preserving intuitions from classical physics? Remember, classical physics is what turned out to be wrong, so we had to invent quantum mechanics to fix it!

Perhaps one way to keep the question from going stale and vacuous is to find a new way of expressing quantum theory itself. After all, there’s no grand guarantee that the formulation of the theory which is good for finding the spectrum of atomic hydrogen (and all the other topics we traditionally assign to undergraduates each semester) is equally good for all questions. And one feature of the standard presentation is that the really weird stuff, such as what we’ve confronted today, tends to be buried several chapters in. When the books lay out “axioms” or “postulates” for quantum mechanics, blatant defiance of intuition about intrinsic properties isn’t among them. Instead, it is deduced as a consequence, deep into the algebra. But perhaps that is a historical accident, due to the way we got here and not to nature itself!

**READINGS**

The parable of the muffins is my attempt at rephrasing a thought-experiment by N. David Mermin, who learned of a *four*-particle scenario devised by Greenberger, Horne and Zeilinger and found that it could be significantly sharpened by going down to three.

- N. David Mermin, “Quantum mysteries revisited,”
*American Journal of Physics,***58**(1990), 731–734. DOI:10.1119/1.16503. - N. David Mermin, “Hidden variables and the two theorems of John Bell,”
*Reviews of Modern Physics***65**(1993), 803–815. arXiv:1802.10119.

For reasons that made sense at the time, I gathered all the homework problems together at the end into a chapter called, well, “Exercises”. And now I keep getting spam invitations to conferences and special issues of journals no one has ever heard of, asking me to share my pivotal work “in the field of Exercises”.

]]>In a 1964 interview, the physicist Karl Darrow calls the story “impossible to check”. And in another interview, Robert Mulliken (not to be confused with Robert Millikan) shares the story of Lunn having “sent a paper to the Physical Review which was turned down and which anticipated the quantum mechanics”. Mulliken heard the story from the physical chemist William Draper Harkins. Similarly, Leonard Loeb told Thomas Kuhn that Lunn “was probably a misunderstood genius, and who was completely frustrated, because his one great paper with his one great idea was turned down by a journal”.

Lunn did apparently try to present what sounds like a grandiose paper (“Relativity, quantum theory, and the wave theories of light and gravitation”) at the American Physical Society meeting in April 1923, but his paper was only “read by title”. The abstract ran as follows:

This paper is a preliminary report on a theory originally sought in order to meet the recognized need for a reconciliation between wave theory and quantum phenomena; its scope of adaptation proves to be quite wide. It includes (1) a wave theory of gravitation in quantitative connection with optical, electronic, and radioactivity data; (2) a related general suggestion of a theory connecting molecular properties with properties of matter in bulk; (3) alternatives for some of the current features in the theories of atomic structure; (4) a new interpretation and deduction of formulas for series and band spectra, using in lieu of the quantum condition a substitute directly related to long familiar physical notions; (5) a modification of Lagrangian dynamics which promises to be of service in the study of complex atomic and molecular structures; (6) a non-quantum theory of specific heat and black radiation. Results so far reached deal mostly with problems approachable by elementary methods or approximate computations. A set of formulas has been obtained which yield computation of the electron constants $e$, $h$, $m$ and mass ratios, assuming from observation only the Rydberg constant, velocity of light, gravitation constant, and Faraday constant, with results in each case in practical agreement with measured values.

Darrow says, “I know that in 1924 he wanted to give a twenty or a thirty minute paper before the American Physical Society in Washington, but then authorities of the Society refused him more than ten minutes”.

Lunn’s abstract in the 1924 proceedings has a similar explain-everything atmosphere:

Relativity, the quantum phenomena, and a kinematic geometry of matter and radiation.A. C. LUNN, University of Chicago. The theory indicated in an earlier paper (Phys. Rev. 21, 711, 1923), has since been developed, extended in scope, and so ordered as to permit of treatment as a deductive space-time geometry. It unites the treatment of the quantum phenomena with the rest of physical theory in a way that yields to illustration by familiar physical images. It resolves into matters of choice a number of hitherto controversial alternatives in the interpretation of phenomena, and allows freedom of use of a range of concrete types of representation including many other concepts commonly discarded. Among special topics more recently found to affiliate with the scheme may be mentioned the Stark and Zeeman effects and fine structure, resonance potentials, and the intensity and distribution of general x-radiation. Improvements have been made in the setting of the formulas connecting $e$, $h$, and $m$ with pre-electron data. A program has emerged for the foundation of a trial mathematical chemistry by determination of types of atoms, valence, number of isotopes, atomic weights, and spectrum levels.

I can easily imagine a paper with that attempted scope being incomprehensible to whoever had the task of evaluating it, and so any really good morsels within it would have been lost.

**UPDATE (4 November):** I wrote to the *Physical Review* offices on the chance that they had more information and received this reply from Robert Garisto, the Managing Editor of *Physical Review Letters.*

]]>Thank you for your query. Our records from the early 20th century are fragmentary. I am not sure if we have any from before 1930, much less a complete set that could answer your question.

But I see that Arthur C. Lunn published 7 papers in the Physical Review from 1912-1922. So he was a known author to the editors. Those were different times, and while it is possible that he submitted a paper that was rejected and never published elsewhere, for what it’s worth, it strikes me as unlikely.

* * *

I saw the Count lying within the box upon the earth, some of which the rude falling from the cart had scattered over him. He was deathly pale, just like a waxen image, and the red eyes glared with the horrible vindictive look which I knew too well.

As I looked, the eyes saw the sinking sun, and the look of hate in them turned to triumph.

But, on the instant, came the sweep and flash of Jonathan’s great knife. I shrieked as I saw it shear through the throat; whilst at the same moment Mr. Morris’s bowie knife plunged into the heart. And a voice rang out across the sky: “KUKRI AND BOWIE COMBO MOVE, MOTHERFUUUCKER!”

— Bram Stoker, more or less

]]>Yudkowsky clearly intends to argue that the scientific community is broken and his brand of Rationalism(TM) is superior, but what he’s actually done is take all the weaknesses that physicists have when discussing quantum foundations and present them in a more concentrated form. There’s the accepting whatever mathematical formulation you learn first as the ultimate truth, the reliance upon oversimplified labels and third-hand accounts rather than studying what the pioneers themselves wrote, the general unwillingness to get out of the armchair and go even so far as the library…

Let’s open with Yudkowsky’s “If Many Worlds Had Come First“, where a fake version of Hugh Everett trades places with a fake version of Niels Bohr. Now, to me this sounds like a bizarrely overcomplicated rhetorical exercise, but if it is to be done correctly, then the fictional Bohr should espouse the view of the historical Everett and vice versa. But because it’s showboating, that’s not what we get.

First, we get some bizarre revisionism:

Macroscopic decoherence, a.k.a. many-worlds, was first proposed in a 1957 paper by Hugh Everett III.

No, decoherence was introduced by Zeh in 1970. And it took another decade for the idea to take off, thanks to Zurek coming along with work that was (a) fairly interpretation-neutral in its presentation and (b) originally inspired by trying to clarify Bohr rather than dethrone him. Nowadays, the theory of decoherence is recognized as a calculational tool that anybody can use regardless of their preferred interpretation of quantum physics, because it’s just applying the standard math to a particular class of situations, and all interpretations agree on the standard math. Using “macroscopic decoherence” as a synonym for “many-worlds interpretation” makes no logical sense.

Then, we get a transparent attempt to make Everett look good while academia looks bad:

Crushed, Everett left academic physics, invented the general use of Lagrange multipliers in optimization problems, and became a multimillionaire.

The whole point of Lagrange multipliers was always “optimization problems” (minimizing or maximizing a functional in the calculus of variations is optimization); this should be “operations research” or “management science”. More fundamentally, though, one could with equal justice say that Everett lacked the temperament to argue for his ideas, antagonized and scorned those who most closely agreed with him, and died miserable. See what I did there?

It wasn’t until 1970, when Bryce DeWitt (who coined the term “many-worlds”) wrote an article for Physics Today, that the general field was first informed of Everett’s ideas.

False. Everett’s paper was discussed at the Chapel Hill conference in 1957, where Feynman came down pretty harsh on it. And in 1959, Everett met with more people than just Bohr when he visited Copenhagen. In 1962, Everett presented his interpretation at a conference at the Xavier University of Cincinnati, with prominent physicists like Wigner in attendance. And of course, all this is in addition to the fact that Everett’s ’57 paper was published in the *Reviews of Modern Physics,* one of the most prominent journals of the physics profession. Why didn’t more people care until the ’70s? (shrug) It answered no specific question about a concrete physics problem, and as noted above, quite possibly Everett himself was just not the man to sell it. My own impression of that paper was that it had enough places where it just assumed the math works out that it needed at least one more round of revision (to be clear about the problems, if not to solve them, since nobody has done that yet). But I too am judging it with the benefit of hindsight.

And suppose that no one had proposed collapse theories until 1957.

Bohr did not propose a collapse theory.

Now, I actually stumbled across “If Many Worlds Had Come First” tonight while looking for something else, Yudkowsky’s “explanation” of Bell’s theorem. It’s a muddle of percentages that make my eyes glaze over, and quantum information theory is *my job.* Why he does it that way, I have no idea. Bell’s original argument from 1964 is actually *easier* to follow, and Yudkowsky name-drops the GHZ state, so he seems to be *aware* of more recent developments that made the point even simpler. Perhaps he wanted to make the mathematics “elementary”, but by not using (Mermin’s improvement of) the GHZ argument, he brings in needless trig functions and introduces a whole heap of angles that look completely arbitrary. It’s a mess.

What does Bell’s Theorem plus its experimental verification tell us, exactly?

My favorite phrasing is one I encountered in D. M. Appleby: “Quantum mechanics is inconsistent with the classical assumption that a measurement tells us about a property previously possessed by the system.”

OK. Let’s dig in. That paper isn’t about Bell’s theorem; it’s about the *Bell–Kochen–Specker theorem,* another result in the same area also proved by Bell (and independently by the team of Kochen and Specker). It has a similar upshot, but its assumptions are more abstract and harder to justify physically.

But it gets better. In his very next paper, Appleby writes,

If I am asked to accept Bohr as the authoritative voice of final truth, then I cannot assent. But if his writings are approached in a more flexible spirit, as a source of insights which are not the less seminal for being obscure, they suggest some interesting questions. I do not know if this line of thought will be fruitful. But I feel it is worth pursuing.

Not quite the message that Yudkowsky would want to convey. But it was there for him to read, written in 2004, years before LessWrong even existed.

I’ll admit, I probably wouldn’t have noticed that or gone on at length about it, were it not for the fact that Appleby is a collaborator of mine.

]]>J. J. Thomson: (points at atom) pudding

]]>“Oh, but it’s a source of inspiration!”

So, you’ve never been to a writers’ workshop, spent 30 minutes with the staff on the school literary magazine, seen the original “You’re the man now, dog!” scene, or had any other exposure to the thousand and one gimmicks invented over the centuries to get people to put one word after another.

“It provides examples for teaching the art of critique!”

Why not teach with examples, just hear me out here, by actual humans?

“Students can learn to write by rewriting the output!”

Am I the only one who finds passing off an edit of an unattributable mishmash as one’s own work to be, well, flagrantly unethical?

“You’re just yelling at a cloud! What’s next, calling for us to reject modernity and embrace tradition?”

I’d rather we built our future using the best parts of our present rather than the worst.

]]>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?”

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