Category Archives: Quantum mechanics

Friends Don’t Let Friends Learn Probability from Yudkowsky

Suppose I said, “I have this clock that I really like. It’s a very nice clock. So, I am going to measure everything I can in terms of the times registered on this clock.”

“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.
Continue reading Friends Don’t Let Friends Learn Probability from Yudkowsky

The Parable of the Muffins

Let’s try to make a profound statement about reality by thinking hard about baked goods.

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 $X$?
Continue reading The Parable of the Muffins

The Lunn–Schrödinger Equation?

Wikipedia claims that Arthur C. Lunn discovered what we now call the Schrödinger equation some years before Schrödinger. I wonder if there is more to say about this than what the references cited there provide (they have the feel of being faithful recollections, but are light on specifics).

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.

Friends Don’t Let Friends Learn Physics From Yudkowsky

With the demise of Reddit, we have lost /r/SneerClub, the Internet’s hot spot for mocking those who proclaim allegiance to capital-R Rationality and related ideologies like longtermism. Somewhere in between the discussions of heavy stuff like sexual harassment in Effective Altruism culture and total frivolity were the rambles about science. I thought I would pull a couple such comments out of the archives and edit them into something shaped like a blog post. So, consider this your Attention Conservation Notice: if you’d rather not work through a self-admittedly rough explanation of how Eliezer Yudkowsky’s claims about quantum physics are just silly, exit now.

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…
Continue reading Friends Don’t Let Friends Learn Physics From Yudkowsky

A Picture for the Mind: the Bloch Ball

Now and then, stories will pop up in the news about the latest hot new thing in quantum computers. If the story makes any attempt to explain why quantum computing is special or interesting, it often recycles a remark along the lines of, “A quantum bit can be both 0 and 1 simultaneously.” This, well, ehhhhh… It’s rather like saying that Boston is at both the North Pole and the South Pole simultaneously. Something important has been lost. I figured I should take a stab at explaining what. Our goal today is to develop a mental picture for a qubit, the basic unit that quantum computers are typically regarded as built out of. To be more precise, we will develop a mental picture for the mathematics of a qubit, not for how to implement one in the lab. There are many ways to do so, and getting into the details of any one method would, for our purposes today, be a distraction. Instead, we will be brave and face the issue on a more abstract level.

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.

Bloch ball, with the center point and the points where the axes intersect the outer sphere marked with dots

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)

New Textbook

Copies of a textbook surrounded by Oaxacan carved wooden animals

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

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

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

ERRATA:

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

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

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

Thoughts on “relational Quantum Mechanics”

Recently, the far-flung QBism discussion group nominally centered at UMass Boston has been conversing about Carlo Rovelli’s relational interpretation of quantum mechanics. Trying to think all this through halfway clearly, I wrote some notes. They don’t seem to be moving in the direction of a paper, and they’re too chatty for the arXiv even by my standards, so this seems the best place to host them.

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.

Canonical Probabilities by Directly Quantizing Thermodynamics

I’ve had this derivation kicking around a while, and today seemed like as good a day as any to make a fuller write-up of it:

  • 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.

More on Bohr

This post carries further on in the vein of my earlier writings on how the way most physicists talk about “the Copenhagen interpretation of quantum mechanics” is largely ahistorical.

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!

Underappreciated

Some time ago, I had one of those odd little thoughts that could be the spark of an essay. But in this particular case, the point I wanted to make felt like it could be made most clearly by demonstration, rather than explication. So, I wrote a concise report on “An Underappreciated Exchange in the Bohr–Einstein Debate.” Judging by the modest splash of positive e-mail that I received after posting it, I think I layered the whimsy and the serious point adequately well.

My 2019 in Science

First, of course, there was the doubt and the pain.

But we’ve already covered that.

Let’s talk about the papers I managed to get out the door and into public view. In retrospect, the list is pleasingly not insubstantial:

There was also From Gender to Gleason, my review of Adam Becker’s book What is Real? (2018). By the time I was done, it was as lengthy as a paper, but the arXiv isn’t really a host for book reviews, so I just posted it here at Sunclipse and moved on.

On Being a Quantum Physicist in Autumn 2019

(a friendly warning for police violence, transphobia and philosophy of physics)

The way I see it, the two big Why? questions about quantum mechanics are, first, why do we use the particular mathematical apparatus of quantum theory, as opposed to any alternative we might imagine? And second, why do we only find it necessary to work with the full perplexities of quantum physics some of the time? These two questions are related. In order to understand how imprecise measurements might wash out quantum weirdness, we need to characterize which features of quantum theory really are fundamentally weird. And this, in turn, requires separating deep principles from convenient conventions and illuminating the true core of the physics. My own research has focused on the first question, but the second is never too far from my mind.

Of course, I have a lot on my mind these days, but I don’t think I’m special in that regard.

If you ask me, a “quantum system” can be any part of nature that is subject to an agent’s inquiry. A “quantum measurement” is, in principle, any action that an agent takes upon a quantum system. The road between Boston’s City Hall and the Holocaust Memorial is a quantum system. When the police use their bicycles as battering rams against queer kids and street medics, running towards the trouble is a quantum measurement. Being threatened with pepper spray, while secoondhand exposure already stings the eye and throat, one human thrown to the pavement in the intersection in front of you while another arrest happens on the sidewalk just behind you, is an outcome of that measurement. Unsurprisingly, textbooks provide little guidance on casting that event into the algebraic formalism of density matrices, and in the moment, other types of expertise are more immediately useful.

I first encountered quantum physics in a serious way during the spring of my second year at university — 2003, that would have been. I did not particularly care about the conceptual or philosophical “foundations” of it until the summer of 2010. The interval in between encompassed six semesters of quantum mechanics and subjects dependent upon it, along with my first attempts to find a research problem in the area. Once my curiosity had been provoked, it took the better part of a year to find an “interpretation” of quantum mechanics that was at all satisfying, and longer than that to make the transition from “this is how a member of that school would answer that question” to “this is what I declare myself”. Part of that transition was my discovery that I could put my own stamp on the ideas: The concepts and the history provoked new mathematical questions, which I could approach with a background that nobody else had.

The interpretation I adopted was the QBism of Chris Fuchs and Rüdiger Schack, later joined by N. David Mermin.

QBism is

an interpretation of quantum mechanics in which the ideas of agent and experience are fundamental. A “quantum measurement” is an act that an agent performs on the external world. A “quantum state” is an agent’s encoding of her own personal expectations for what she might experience as a consequence of her actions. Moreover, each measurement outcome is a personal event, an experience specific to the agent who incites it. Subjective judgments thus comprise much of the quantum machinery, but the formalism of the theory establishes the standard to which agents should strive to hold their expectations, and that standard for the relations among beliefs is as objective as any other physical theory.

That’s how we put it in the FAQ. Any physicist who is weird enough to endorse an interpretation of quantum mechanics will naturally get inquiries about it. Many of these, we get often enough that we try to compile good answers together into a nicely portable package — with the proviso that the quantum is a project, and some answers are not final because if physics were easy, we’d be done by now.

There’s a question which seems particularly suited to answering in the blog format, though: “Why don’t you believe in the Many Worlds Interpretation?”
Continue reading On Being a Quantum Physicist in Autumn 2019

Concerning Wigner’s Former Roommate

I attended a workshop on the mini-genre of Extended Wigner’s Friend “paradoxes” but did not think that I’d write much on the topic myself. And, indeed, the comment I eventually produced is mostly bibliography.

B. C. Stacey, “On QBism and Assumption (Q)” [arXiv:1907.03805].

I correct two misapprehensions, one historical and one conceptual, in the recent literature on extensions of the Wigner’s Friend thought-experiment. Perhaps fittingly, both concern the accurate description of some quantum physicists’ beliefs by others.

Also available via SciRate.

On Reconstructing the Quantum

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

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

Also available via SciRate.