Consider a disk in the complex plane. If one applies an affine-linear map to this disk, one obtains

For maps that are merely holomorphic instead of affine-linear, one has some variants of this assertion, which I am recording here mostly for my own reference:

Theorem 1 (Holomorphic images of disks)Let be a disk in the complex plane, and be a holomorphic function with .

- (i) (Open mapping theorem or inverse function theorem) contains a disk for some . (In fact there is even a holomorphic right inverse of from to .)
- (ii) (Bloch theorem) contains a disk for some absolute constant and some . (In fact there is even a holomorphic right inverse of from to .)
- (iii) (Koebe quarter theorem) If is injective, then contains the disk .
- (iv) If is a polynomial of degree , then contains the disk .
- (v) If one has a bound of the form for all and some , then contains the disk for some absolute constant . (In fact there is holomorphic right inverse of from to .)

Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”; an equivalent form of this result also appears in Lemma 4 of this paper of Miller. The proof is simple:

*Proof:* (Proof of (iv)) Let , then we have a lower bound for the log-derivative of at :

The constant in (iv) is completely sharp: if and is non-zero then contains the disk

but avoids the origin, thus does not contain any disk of the form . This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form for an absolute constant .Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:

*Proof:* (Proof of (v)) From the Cauchy inequalities one has for , hence by Taylor’s theorem with remainder for . By Rouche’s theorem, this implies that the function has a unique zero in for any , if is a sufficiently small absolute constant. The claim follows.

Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):

*Proof:* (Proof of (ii)) By shrinking slightly if necessary we may assume that extends analytically to the closure of the disk . Let be the constant in (v) with ; we will prove (iii) with replaced by . If we have for all then we are done by (v), so we may assume without loss of generality that there is such that . If for all then by (v) we have

Here is another classical result stated by Alexander (and then proven by Kakeya and by Szego, but also implied to a classical theorem of Grace and Heawood) that is broadly compatible with parts (iii), (iv) of the above theorem:

Proposition 2Let be a disk in the complex plane, and be a polynomial of degree with for all . Then is injective on .

The radius is best possible, for the polynomial has non-vanishing on , but one has , and lie on the boundary of .

If one narrows slightly to then one can quickly prove this proposition as follows. Suppose for contradiction that there exist distinct with , thus if we let be the line segment contour from to then . However, by assumption we may factor where all the lie outside of . Elementary trigonometry then tells us that the argument of only varies by less than as traverses , hence the argument of only varies by less than . Thus takes values in an open half-plane avoiding the origin and so it is not possible for to vanish.

To recover the best constant of requires some effort. By taking contrapositives and applying an affine rescaling and some trigonometry, the proposition can be deduced from the following result, known variously as the Grace-Heawood theorem or the complex Rolle theorem.

Proposition 3 (Grace-Heawood theorem)Let be a polynomial of degree such that . Then contains a zero in the closure of .

This is in turn implied by a remarkable and powerful theorem of Grace (which we shall prove shortly). Given two polynomials of degree at most , define the *apolar* form by

Theorem 4 (Grace’s theorem)Let be a circle or line in , dividing into two open connected regions . Let be two polynomials of degree at most , with all the zeroes of lying in and all the zeroes of lying in . Then .

(Contrapositively: if , then the zeroes of cannot be separated from the zeroes of by a circle or line.)

Indeed, a brief calculation reveals the identity

where is the degree polynomial The zeroes of are for , so the Grace-Heawood theorem follows by applying Grace’s theorem with equal to the boundary of .The same method of proof gives the following nice consequence:

Theorem 5 (Perpendicular bisector theorem)Let be a polynomial such that for some distinct . Then the zeroes of cannot all lie on one side of the perpendicular bisector of . For instance, if , then the zeroes of cannot all lie in the halfplane or the halfplane .

I’d be interested in seeing a proof of this latter theorem that did not proceed via Grace’s theorem.

Now we give a proof of Grace’s theorem. The case can be established by direct computation, so suppose inductively that and that the claim has already been established for . Given the involvement of circles and lines it is natural to suspect that a Möbius transformation symmetry is involved. This is indeed the case and can be made precise as follows. Let denote the vector space of polynomials of degree at most , then the apolar form is a bilinear form . Each translation on the complex plane induces a corresponding map on , mapping each polynomial to its shift . We claim that the apolar form is invariant with respect to these translations:

Taking derivatives in , it suffices to establish the skew-adjointness relation but this is clear from the alternating form of (1).Next, we see that the inversion map also induces a corresponding map on , mapping each polynomial to its inversion . From (1) we see that this map also (projectively) preserves the apolar form:

More generally, the group of Möbius transformations on the Riemann sphere acts projectively on , with each Möbius transformation mapping each to , where is the unique (up to constants) rational function that maps this a map from to (its divisor is ). Since the Möbius transformations are generated by translations and inversion, we see that the action of Möbius transformations projectively preserves the apolar form; also, we see this action of on also moves the zeroes of each by (viewing polynomials of degree less than in as having zeroes at infinity). In particular, the hypotheses and conclusions of Grace’s theorem are preserved by this Möbius action. We can then apply such a transformation to move one of the zeroes of to infinity (thus making a polynomial of degree ), so that must now be a circle, with the zeroes of inside the circle and the remaining zeroes of outside the circle. But then By the Gauss-Lucas theorem, the zeroes of are also inside . The claim now follows from the induction hypothesis.

A group led by Jianwei Pan and Chao-Yang Lu, based mainly at USTC in Hefei, China, announced today that it achieved BosonSampling with 40-70 detected photons—up to and beyond the limit where a classical supercomputer could feasibly verify the results. (Technically, they achieved a variant called Gaussian BosonSampling: a generalization of what I called Scattershot BosonSampling in a 2013 post on this blog.)

For more, see also Emily Conover’s piece in *Science News*, or Daniel Garisto’s in *Scientific American*, both of which I consulted on. (Full disclosure: I was one of the reviewers for the Pan group’s *Science* paper, and will be writing the Perspective article to accompany it.)

The new result follows the announcement of 14-photon BosonSampling by the same group a year ago. It represents the second time quantum supremacy has been reported, following Google’s celebrated announcement from last year, and the first time it’s been done using photonics rather than superconducting qubits.

As the co-inventor of BosonSampling (with Alex Arkhipov), obviously I’m gratified about this.

For anyone who regards it as boring or obvious, here and here is Gil Kalai, on this blog, telling me why BosonSampling would never scale beyond 8-10 photons. (He wrote that, if aliens forced us to try, then much like with the Ramsey number R(6,6), our only hope would be to attack the aliens.) Here’s Kalai making a similar prediction, on the impossibility of quantum supremacy by BosonSampling or any other means, in his plenary address to the International Congress of Mathematicians two years ago.

Even if we set aside the quantum computing skeptics, many colleagues told me they thought experimental BosonSampling was a dead end, because of photon losses and the staggering difficulty of synchronizing 50-100 single-photon sources. They said that a convincing demonstration of quantum supremacy would have to await the arrival of quantum fault-tolerance—or at any rate, some hardware platform more robust than photonics. I always agreed that they might be right. Furthermore, even if 50-photon BosonSampling *was* possible, after Google reached the supremacy milestone first with superconducting qubits, it wasn’t clear if anyone would still bother. Even when I learned a year ago about the USTC group’s intention to go for it, I was skeptical, figuring I’d believe it when I saw the paper.

Obviously the new result isn’t dispositive. Maybe Gil Kalai really WON, BY A LOT, before Hugo Chávez hacked the Pan group’s computers to make it seem otherwise? (Sorry, couldn’t resist.) Nevertheless, as someone whose intellectual origins are close to pure math, it’s strange and exciting to find myself in a field where, once in a while, the world itself gets to weigh in on a theoretical disagreement.

Since excitement is best when paired with accurate understanding, please help yourself to the following FAQ, which I might add more to over the next couple days.

**What is BosonSampling?** You must be new here! In increasing order of difficulty, here’s an *MIT News* article from back in 2011, here’s the Wikipedia page, here are my PowerPoint slides, here are my lecture notes from Rio de Janeiro, here’s my original paper with Arkhipov…

**What is quantum supremacy?** Roughly, the use of a programmable or configurable quantum computer to solve *some* well-defined computational problem much faster than we know how to solve it with any existing classical computer. “Quantum supremacy,” a term coined by John Preskill in 2012, does *not* mean useful QC, or scalable QC, or fault-tolerant QC, all of which remain outstanding challenges. For more, see my Supreme Quantum Supremacy FAQ, or (e.g.) my recent Lytle Lecture for the University of Washington.

**If Google already announced quantum supremacy a year ago, what’s the point of this new experiment?** To me, at least, quantum supremacy seems important enough to do at least twice! Also, as I said, this represents the first demonstration that quantum supremacy is possible *via photonics*. Finally, as the authors point out, the new experiment has one big technical advantage compared to Google’s: namely, many more possible output states (~10^{30} of them, rather than a mere ~9 quadrillion). This makes it infeasible to calculate the whole probability distribution over outputs and store it on a gigantic hard disk (after which one could easily generate as many samples as one wanted), which is what IBM proposed doing in its response to Google’s announcement.

**Is BosonSampling a form of universal quantum computing?** No, we don’t even think it can simulate universal *classical* computing! It’s designed for exactly one task: namely, demonstrating quantum supremacy and refuting Gil Kalai. It *might* have some other applications besides that, but if so, they’ll be icing on the cake. This is in contrast to Google’s Sycamore processor, which in principle *is* a universal quantum computer, just with a severe limit on the number of qubits (53) and how many layers of gates one can apply to them (about 20).

**Is BosonSampling at least a step toward universal quantum computing?** I think so! In 2000, Knill, Laflamme, and Milburn (KLM) famously showed that pure, non-interacting photons, passing through a network of beamsplitters, are capable of universal QC, provided we assume one extra thing: namely, the ability to measure the photons at intermediate times, and change which beamsplitters to apply to the remaining photons depending on the outcome. In other words, “BosonSampling plus adaptive measurements equals universality.” Basically, KLM is the holy grail that experimental optics groups around the world have been working toward for 20 years, with BosonSampling just a more achievable pit stop along the way.

**Are there any applications of BosonSampling?** We don’t know yet. There are proposals in the literature to apply BosonSampling to vibronic spectra in quantum chemistry, finding dense subgraphs, and other problems, but I’m not yet convinced that these proposals will yield real speedups over the best we can do with classical computers, for a task of practical interest that involves estimating specific numbers (as opposed to sampling tasks, where BosonSampling almost certainly *does* yield exponential speedups, but which are rarely the thing practitioners directly care about). [See this comment for further discussion of the issues regarding dense subgraphs.] In a completely different direction, one could try to use BosonSampling to generate cryptographically certified random bits, along the lines of my proposal from 2018, much like one could with qubit-based quantum circuits.

**How hard is it to simulate BosonSampling on a classical computer?** As far as we know today, the difficulty of simulating a “generic” BosonSampling experiment increases roughly like 2^{n}, where n is the number of detected photons. It *might* be easier than that, particularly when noise and imperfections are taken into account; and at any rate it might be easier to spoof the statistical tests that one applies to verify the outputs. I and others managed to give some theoretical evidence against those possibilities, but just like with Google’s experiment, it’s conceivable that some future breakthrough will change the outlook and remove the case for quantum supremacy.

**Do you have any amusing stories?** When I refereed the *Science* paper, I asked why the authors directly verified the results of their experiment only for up to 26-30 photons, relying on plausible extrapolations beyond that. While directly verifying the results of n-photon BosonSampling takes ~2^{n} time for any known classical algorithm, I said, surely it should be possible with existing computers to go up to n=40 or n=50? A couple weeks later, the authors responded, saying that they’d now verified their results up to n=40, but it burned $400,000 worth of supercomputer time so they decided to stop there. This was by far the most expensive referee report I ever wrote!

Also: when Covid first started, and facemasks were plentiful in China but almost impossible to get in the US, Chao-Yang Lu, one of the leaders of the new work and my sometime correspondent on the theory of BosonSampling, decided to mail me a box of 200 masks (I didn’t ask for it). I don’t think that influenced my later review, but it was appreciated nonetheless.

Huge congratulations to the whole team for their accomplishment!

I’m at another Zoom conference this week, QCD Meets Gravity. This year it’s hosted by Northwestern.

QCD Meets Gravity is a conference series focused on the often-surprising links between quantum chromodynamics on the one hand and gravity on the other. By thinking of gravity as the “square” of forces like the strong nuclear force, researchers have unlocked new calculation techniques and deep insights.

Last year’s conference was very focused on one particular topic, trying to predict the gravitational waves observed by LIGO and VIRGO. That’s still a core topic of the conference, but it feels like there is a bit more diversity in topics this year. We’ve seen a variety of talks on different “squares”: new theories that square to other theories, and new calculations that benefit from “squaring” (even surprising applications to the Navier-Stokes equation!) There are talks on subjects from String Theory to Effective Field Theory, and even a talk on a very different way that “QCD meets gravity”, in collisions of neutron stars.

With still a few more talks to go, expect me to say a bit more next week, probably discussing a few in more detail. (Several people presented exciting work in progress!) Until then, I should get back to watching!

As Emily was kind enough to point out earlier, my new book is out on the arXiv!

It’s arXived by agreement with the wonderful Cambridge University Press, who will publish it in April 2021. You can pre-order it now, as the ideal festive gift for any friend who enjoys deferred gratification.

You can read some blurb about the book on my page about it or CUP’s page.

I’ve been writing about entropy and diversity on this blog for nearly ten years now. The book collects together and connects together many of the stories I’ve told.

One of the reasons I wanted to write a book is to show how all these stories fit together, which can be much harder to discern from scattered blog posts over a long period. So if you’ve seen my posts on entropy or diversity and not really known what to make of it all, perhaps the book will provide an answer.

Here’s the list of chapters, with links to some related Café posts (plus one from Azimuth) — often from back when my ideas were in a less developed form.

Finally, a puzzle: what’s shown on the cover of the book?

There’s a new entry in today’s arXiv listings:

Entropy and Diversity: The Axiomatic Approach, a viii + 442 pages book to be published by Cambridge University Press in April 2021.

Congratulations Tom! This is really something.

And even more impressively, this is the second published book that Tom has made freely available on the arXiv! His wonderful Basic Category Theory is there as well.

The Japanese take pride in ‘shinise’: businesses that have lasted for hundreds or even thousands of years. This points out an interesting alternative to the goal of profit maximization: maximizing the time of survival.

• Ben Dooley and Hisako Ueno, This Japanese shop is 1,020 years old. It knows a bit about surviving crises, *New York Times*, 2 December 2020.

Such enterprises may be less dynamic than those in other countries. But their resilience offers lessons for businesses in places like the United States, where the coronavirus has forced tens of thousands into bankruptcy.

“If you look at the economics textbooks, enterprises are supposed to be maximizing profits, scaling up their size, market share and growth rate. But these companies’ operating principles are completely different,” said Kenji Matsuoka, a professor emeritus of business at Ryukoku University in Kyoto.

“Their No. 1 priority is carrying on,” he added. “Each generation is like a runner in a relay race. What’s important is passing the baton.”

Japan is an old-business superpower. The country is home to more than 33,000 with at least 100 years of history — over 40 percent of the world’s total. Over 3,100 have been running for at least two centuries. Around 140 have existed for more than 500 years. And at least 19 claim to have been continuously operating since the first millennium.

(Some of the oldest companies, including Ichiwa, cannot definitively trace their history back to their founding, but their timelines are accepted by the government, scholars and — in Ichiwa’s case — the competing mochi shop across the street.)

The businesses, known as “shinise,” are a source of both pride and fascination. Regional governments promote their products. Business management books explain the secrets of their success. And entire travel guides are devoted to them.

Of course if some businesses try to maximize time of survival, they may be small compared to businesses that are mainly trying to become “big”—at least if size is not the best road to long-term survival, which apparently it’s not. So we’ll have short-lived dinosaurs tromping around, and, dodging their footsteps, long-lived mice.

The idea of different organisms pursuing different strategies is familiar in ecology, where people talk about r-selected and K-selected organisms. The former “emphasize high growth rates, typically exploit less-crowded ecological niches, and produce many offspring, each of which has a relatively low probability of surviving to adulthood.” The latter “display traits associated with living at densities close to carrying capacity and typically are strong competitors in such crowded niches, that invest more heavily in fewer offspring, each of which has a relatively high probability of surviving to adulthood.”

But the contrast between r-selection and K-selection seems different to me than the contrast between profit maximization and lifespan maximization. As far as I know, no organism except humans deliberately tries to maximize the lifetime of anything.

And amusingly, the theory of r-selection versus K-selection may also be nearing the end of its life:

When Stearns reviewed the status of the theory in 1992, he noted that from 1977 to 1982 there was an average of 42 references to the theory per year in the BIOSIS literature search service, but from 1984 to 1989 the average dropped to 16 per year and continued to decline. He concluded that r/K theory was a once useful heuristic that no longer serves a purpose in life history theory.

For newer thoughts, see:

• D. Reznick, M. J. Bryant and F. Bashey, r-and K-selection revisited: the role of population regulation in life-history evolution, *Ecology* **83** (2002) 1509–1520.

See also:

• Innan Sasaki, How to build a business that lasts more than 200 years—lessons from Japan’s shinise companies, *The Conversation*, 6 June 2019.

Among other things, she writes:

We also found there to be a dark side to the success of these age-old shinise firms. At least half of the 17 companies we interviewed spoke of hardships in maintaining their high social status. They experienced peer pressure not to innovate (and solely focus on maintaining tradition) and had to make personal sacrifices to maintain their family and business continuity.

As the vice president of Shioyoshiken, a sweets company established in 1884, told us:

In a shinise, the firm is the same as the family. We need to sacrifice our own will and our own feelings and what we want to do … Inheriting and continuing the household is very important … We do not continue the business because we particularly like that industry. The fact that our family makes sweets is a coincidence. What is important is to continue the household as it is.Innovations were sometimes discouraged by either the earlier family generation who were keen on maintaining the tradition, or peer shinise firms who cared about maintaining the tradition of the industry as a whole. Ultimately, we found that these businesses achieve such a long life through long-term sacrifice at both the personal and organisational level.

I spoke with Katie Breivik (Flatiron) today about a project to paint toy binary stars (from Breivik's model of how binaries form and evolve) onto toy spectroscopic targets (from Neige Frankel's model of how the Milky Way disk formed) to see how many binary stars and how many black-hole (or compact-object) binaries Adrian Price-Whelan (Flatiron) and I should be finding in the *APOGEE* survey. The project is simple in principle, but the matching up of differently simulated catalogs is a conceptual and administrative challenge! The hope for this project is that we can constrain something about the formation of black-hole binaries by the observation that *we don't find any* (or don't find very many) in *APOGEE*!