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Inside the Secret Math Society Known Simply as Nicolas Bourbaki




Antoine Chambert-Loir’s initiation into one of math’s oldest secret societies began with a phone call. “They told me Bourbaki would like me to come and see if I’d work with them,” he said.

Chambert-Loir accepted, and for a week in September 2001 he spent seven hours a day reading math texts out loud and discussing them with the members of the group, whose identities are unknown to the rest of the world.

He was never officially asked to join, but on the last day he was given a long-term task — to finish a manuscript the group had been working on since 1975. When Chambert-Loir later received a report on the meeting he saw that he was listed as a “membrifié,” indicating he was part of the group. Ever since, he’s helped advance an almost Sisyphean tradition of math writing that predates World War II.

The group is known as “Nicolas Bourbaki” and is usually referred to as just Bourbaki. The name is a collective pseudonym borrowed from a real-life 19th-century French general who never had anything to do with mathematics. It’s unclear why they chose the name, though it may have originated in a prank played by the founding mathematicians as undergraduates at the École Normale Supérieure (ENS) in Paris.

“There was some custom to play pranks on first-years, and one of those pranks was to pretend that some General Bourbaki would arrive and visit the school and maybe give a totally obscure talk about mathematics,” said Chambert-Loir, a mathematician at the University of Paris who has acted as a spokesperson for the group and is its one publicly identified member.

Bourbaki began in 1934, the initiative of a small number of recent ENS alumni. Many of them were among the best mathematicians of their generation. But as they surveyed their field, they saw a problem. The exact nature of that problem is also the subject of myth.

In one telling, Bourbaki was a response to the loss of a generation of mathematicians to World War I, after which the group’s founders wanted to find a way to preserve what math knowledge remained in Europe.

“There is a story that young French mathematicians were not seen as a government priority during [the] First World War and many were sent to war and died there,” said Sébastien Gouëzel of the University of Rennes, who is not publicly identified with the group but, like many mathematicians, is familiar with its activities.

In a more prosaic but probably also more likely rendering, the original Bourbaki members were simply dissatisfied with the field’s textbooks and wanted to create better ones. “I think at the beginning it was just for that very concrete matter,” Chambert-Loir said.

Whatever their motivation, the founders of Bourbaki began to write. Yet instead of writing textbooks, they ended up creating something completely novel: free-standing books that explained advanced mathematics without reference to any outside sources.

The first Bourbaki text was meant to be about differential geometry, which reflected the tastes of some of the group’s early members, luminaries like Henri Cartan and André Weil. But the project quickly expanded, since it’s hard to explain one mathematical idea without involving many others.

“They realized that if they wanted to do this cleanly, they needed [ideas from other areas], and Bourbaki grew and grew into something huge,” Gouëzel said.

The most distinctive feature of Bourbaki was the writing style: rigorous, formal and stripped to the logical studs. The books spelled out mathematical theorems from the ground up without skipping any steps — exhibiting an unusual degree of thoroughness among mathematicians.

“In Bourbaki, essentially, there are no gaps,” Gouëzel said. “They are super precise.”

But that precision comes at a cost: Bourbaki books can be hard to read. They don’t offer a contextualizing narrative that explains where concepts come from, instead letting the ideas speak for themselves.

“Essentially, you give no comment about what you do or why you do it,” Chambert-Loir said. “You state stuff and prove it, and that’s it.”

Bourbaki joined its distinctive writing style to a distinctive writing process. Once a member produces a draft, the group gathers in person, reads it aloud and suggests notes for revision. They repeat these steps until there is unanimous agreement that the text is ready for publication. It’s a long process that can take a decade or more to complete.

This focus on collaboration is also where the group’s insistence on anonymity comes from. They keep membership secret to reinforce the notion that the books are a pure expression of mathematics as it is, not an individual’s take on the topic. It’s also an ethic that can seem out of step with aspects of modern math culture.

“It’s sort of hard to imagine a group of young academics right now, people without permanent lifelong positions, devoting a huge amount of time to something they’ll never get credit for,” said Lillian Pierce of Duke University. “This group took this on in a sort of selfless way.”

Bourbaki quickly had an impact on mathematics. Some of the first books, published in the 1940s and ’50s, invented vocabulary that is now standard — terms like “injective,” “surjective” and “bijective,” which are used to describe properties of a map between two sets.

This was the first of two main periods in which Bourbaki was especially influential. The second came in the 1970s when the group published a series of books on Lie groups and Lie algebras that is “unanimously considered a masterpiece,” Chambert-Loir said.

Today, the influence of the group’s books has waned. It’s best known instead for the Bourbaki Seminars, a series of high-profile lectures on the most important recent results in math, held in Paris. When Bourbaki invited Pierce to give one in June 2017, she recognized that the talk would take a lot of time to prepare, but she also knew that due to the seminar’s status in the field, “it’s an invitation you have to accept.”

Even while organizing (and attending) the public lectures, members of Bourbaki don’t disclose their identities. Pierce recalls that during her time in Paris she went out to lunch “with a number of people who it seemed fair to assume were part of Bourbaki, but in the spirit of things I didn’t try to hear their last names.”

According to Pierce, the anonymity is maintained only in a “spirit of fun” these days. “There is no rigor to the secrecy,” she said.

Though its seminars are now more influential than its books, Bourbaki — which has about 10 members currently — is still producing texts according to its founding principles. And Chambert-Loir, 49, is nearing the end of his time with the group, since custom has it that members step down when they turn 50.

Even as he prepares to leave, the project he was handed at the end of his first week remains unfinished. “For 15 years I patiently typed it into LaTeX, made corrections, then we read everything aloud year after year,” he said.

It could easily be half a century from the time the work began to when it’s completed.  That’s a long time by modern publishing standards, where papers land online even in draft form. But then again, maybe it isn’t so long when the product is meant to stand forever.

Correction: November 9, 2020

Lillian Pierce gave her Bourbaki Seminar talk in June 2017, not July 2017. The article has been revised accordingly.



Random walks




A college professor of mine proposed a restaurant venture to our class. He taught statistical mechanics, the physics of many-particle systems. Examples range from airplane fuel to ice cubes to primordial soup. Such systems contain 1024 particles each—so many particles that we couldn’t track them all if we tried. We can gather only a little information about the particles, so their actions look random.

So does a drunkard’s walk. Imagine a college student who (outside of the pandemic) has stayed out an hour too late and accepted one too many red plastic cups. He’s arrived halfway down a sidewalk, where he’s clutching a lamppost, en route home. Each step has a 50% chance of carrying him leftward and a 50% chance of carrying him rightward. This scenario repeats itself every Friday. On average, five minutes after arriving at the lamppost, he’s back at the lamppost. But, if we wait for a time T, we have a decent chance of finding him a distance sqrt{T} away. These characteristic typify a simple random walk.

Random walks crop up across statistical physics. For instance, consider a grain of pollen dropped onto a thin film of water. The water molecules buffet the grain, which random-walks across the film. Robert Brown observed this walk in 1827, so we call it Brownian motion. Or consider a magnet at room temperature. The magnet’s constituents don’t walk across the surface, but they orient themselves according random-walk mathematics. And, in quantum many-particle systems, information can spread via a random walk. 

So, my statistical-mechanics professor said, someone should open a restaurant near MIT. Serve lo mein and Peking duck, and call the restaurant the Random Wok.

This is the professor who, years later, confronted another alumna and me at a snack buffet.

“You know what this is?” he asked, waving a pastry in front of us. We stared for a moment, concluded that the obvious answer wouldn’t suffice, and shook our heads.

“A brownie in motion!”

Not only pollen grains undergo Brownian motion, and not only drunkards undergo random walks. Many people random-walk to their careers, trying out and discarding alternatives en route. We may think that we know our destination, but we collide with a water molecule and change course.

Such is the thrust of Random Walks, a podcast to which I contributed an interview last month. Abhigyan Ray, an undergraduate in Mumbai, created the podcast. Courses, he thought, acquaint us only with the successes in science. Stereotypes cast scientists as lone geniuses working in closed offices and silent labs. He resolved to spotlight the collaborations, the wrong turns, the lessons learned the hard way—the random walks—of science. Interviewees range from a Microsoft researcher to a Harvard computer scientist to a neurobiology professor to a genomicist.

You can find my episode on Instagram, Apple Podcasts, Google Podcasts, and Spotify. We discuss the bridging of disciplines; the usefulness of a liberal-arts education in physics; Quantum Frontiers; and the delights of poking fun at my PhD advisor, fellow blogger and Institute for Quantum Information and Matter director John Preskill


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How Dynamical Quantum Memories Forget




Lukasz Fidkowski1, Jeongwan Haah2, and Matthew B. Hastings3,2

1Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
2Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA
3Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA

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Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the “mixed” phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems $—$ i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears $—$ purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.

► BibTeX data

► References

[1] Y. Li, X. Chen, and M. P. A. Fisher, “Quantum zeno effect and the many-body entanglement transition,” Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134.

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[4] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, “Unitary-projective entanglement dynamics,” Phys. Rev. B 99, 224307 (2019).

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[6] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, “Quantum error correction in scrambling dynamics and measurement-induced phase transition,” Phys. Rev. Lett. 125, 030505 (2019), arXiv:1903.05124.

[7] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, “Self-organized error correction in random unitary circuits with measurement,” (2020), arXiv:2002.12385.

[8] F. G. Brandao, A. W. Harrow, and M. Horodecki, “Local random quantum circuits are approximate polynomial-designs,” Commun. Math. Phys. 346, 397–434 (2016), arXiv:1208.0692.

[9] A. Harrow and S. Mehraban, “Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates,” (2018), arXiv:1809.06957.

[10] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, “Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates,” (2020), arXiv:2002.09524.

[11] S. Bravyi, “Lagrangian representation for fermionic linear optics,” Quantum Inf. and Comp. 5, 216 (2005), arXiv:quant-ph/​0404180.

[12] M. J. Gullans and D. A. Huse, “Scalable probes of measurement-induced criticality,” Phys. Rev. Lett. 125, 070606 (2020) 125, 070606 (2020b), arXiv:1910.00020.

[13] X. Cao, A. Tilloy, and A. D. Luca, “Entanglement in a fermion chain under continuous monitoring,” SciPost Phys. 7, 24 (2019), arXiv:1804.04638.

[14] X. Chen, Y. Li, M. P. A. Fisher, and A. Lucas, “Emergent conformal symmetry in nonunitary random dynamics of free fermions,” Phys. Rev. Research 2, 033017 (2020), arXiv:2004.09577.

[15] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, “Entanglement phase transitions in measurement-only dynamics,” (2020), arXiv:2004.09560.

[16] A. Nahum and B. Skinner, “Entanglement and dynamics of diffusion-annihilation processes with majorana defects,” Phys. Rev. Research 2, 023288 (2020), arXiv:1911.11169.

[17] M. B. Hastings, “Random unitaries give quantum expanders,” Physical Review A 76, 032315 (2007), arXiv:0706.0556.

[18] Y. Li and M. P. A. Fisher, “Statistical mechanics of quantum error-correcting codes,” (2020), arXiv:2007.03822 [quant-ph].

[19] E. S. Meckes, The random matrix theory of the classical compact groups, Vol. 218 (Cambridge University Press, 2019).

[20] K. M. R. Audenaert, “A sharp fannes-type inequality for the von neumann entropy,” J. Phys. A 40, 8127–8136 (2007), quant-ph/​0610146.

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[23] A. Nahum, P. Serna, A. M. Somoza, and M. Ortuño, “Loop models with crossings,” Phys. Rev. B 87, 184204 (2013).

Cited by

[1] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani, “Entanglement phase transitions in measurement-only dynamics”, arXiv:2004.09560.

[2] Adam Nahum, Sthitadhi Roy, Brian Skinner, and Jonathan Ruhman, “Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory”, arXiv:2009.11311.

[3] Michael J. Gullans, Stefan Krastanov, David A. Huse, Liang Jiang, and Steven T. Flammia, “Quantum coding with low-depth random circuits”, arXiv:2010.09775.

[4] Jason Iaconis, Andrew Lucas, and Xiao Chen, “Measurement-induced phase transitions in quantum automaton circuits”, arXiv:2010.02196.

[5] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli, “Topological order and criticality in (2+1)D monitored random quantum circuits”, arXiv:2011.06595.

[6] Sarang Gopalakrishnan and Michael J. Gullans, “Entanglement and purification transitions in non-Hermitian quantum mechanics”, arXiv:2012.01435.

[7] Matteo Ippoliti and Vedika Khemani, “Postselection-free entanglement dynamics via spacetime duality”, arXiv:2010.15840.

[8] Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal, “Dimensional hybridity in measurement-induced criticality”, arXiv:2012.03857.

[9] Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig, “Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions”, arXiv:2012.04666.

[10] Shengqi Sang, Yaodong Li, Tianci Zhou, Xiao Chen, Timothy H. Hsieh, and Matthew P. A. Fisher, “Entanglement Negativity at Measurement-Induced Criticality”, arXiv:2012.00031.

The above citations are from SAO/NASA ADS (last updated successfully 2021-01-18 07:19:20). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-01-18 07:19:18).


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Autonomous Ticking Clocks from Axiomatic Principles




Mischa P. Woods

Institute for Theoretical Physics, ETH Zurich, Switzerland

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There are many different types of time keeping devices. We use the phrase $textit{ticking clock}$ to describe those which – simply put – “tick” at approximately regular intervals. Various important results have been derived for ticking clocks, and more are in the pipeline. It is thus important to understand the underlying models on which these results are founded. The aim of this paper is to introduce a new ticking clock model from axiomatic principles that overcomes concerns in the community about the physicality of the assumptions made in previous models. The ticking clock model in [1] achieves high accuracy, yet lacks the autonomy of the less accurate model in [2]. Importantly, the model we introduce here achieves the best of both models: it retains the autonomy of [2] while allowing for the high accuracies of [1]. What is more, [2] is revealed to be a special case of the new ticking clock model.

► BibTeX data

► References

[1] M. P. Woods, R. Silva, G. Pütz, S. Stupar, and R. Renner, “Quantum clocks are more accurate than classical ones,” (2018a), arXiv:1806.00491v2 [quant-ph].

[2] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Phys. Rev. X 7, 031022 (2017).

[3] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958).

[4] A. Peres, Am. J. Phys 48, 552 (1980).

[5] V. Bužek, R. Derka, and S. Massar, Phys. Rev. Lett. 82, 2207 (1999).

[6] P. Erker, “The Quantum Hourglass,” (2014), ETH Zürich.

[7] S. Ranković, Y.-C. Liang, and R. Renner, “Quantum clocks and their synchronisation | the Alternate Ticks Game,” (2015), arXiv:1506.01373v1 [quant-ph].

[8] M. P. Woods, R. Silva, and J. Oppenheim, Ann. Henri Poincaré (2018b), 10.1007/​s00023-018-0736-9.

[9] S. Khandelwal, M. P. Lock, and M. P. Woods, Quantum 4, 309 (2020).

[10] Y. Yang, L. Baumgärtner, R. Silva, and R. Renner, “Accuracy enhancing protocols for quantum clocks,” (2019), arXiv:1905.09707 [quant-ph].

[11] N. Yunger Halpern and D. T. Limmer, Phys. Rev. A 101, 042116 (2020).

[12] P. A. Hoehn, A. R. H. Smith, and M. P. E. Lock, “The Trinity of Relational Quantum Dynamics,” (2019), arXiv:1912.00033 [quant-ph].

[13] W. Pauli, in Handbuch der Physik, Vol. 24 (Springer, 1933) pp. 83–272.

[14] W. Pauli, Encyclopedia of Physics, Vol. 1 (Springer, Berlin, 1958) p. 60.

[15] Á. Rivas, S. F. Huelga, and M. B. Plenio, Reports on Progress in Physics 77, 094001 (2014).

[16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44 (Springer New York, New York, NY, 1983).

[17] R. Gandy and C. Yates, Mathematical Logic, Vol. 4 (Elsevier, 2001) (see page 267).

[18] A. Degasperis, L. Fonda, and G. C. Ghirardi, Il Nuovo Cimento A (1965-1970) 21, 471 (1974).

[19] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).

[20] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2007) . See section 3.3 Microscopic derivations. In particular, 3.3.1 Weak-coupling limit and 3.3.3 Singular-coupling limit.

[21] P. F. Palmer, J. Math. Phys. 18, 527 (1977).

[22] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan, Rep. Math. Phys. 13, 149 (1978).

[23] S. Stupar, C. Klumpp, N. Gisin, and R. Renner, “Performance of Stochastic Clocks in the Alternate Ticks Game,” (2018), arXiv:1806.08812 [quant-ph].

[24] Y. Yang and R. Renner, “Ultimate limit on time signal generation,” (2020), arXiv:2004.07857 [quant-ph].

[25] I. Pikovski, M. Zych, F. Costa, and Č. Brukner, Nat. Phys. 11, 668 (2015).

[26] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).

[27] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).

[28] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed. (Cambridge University Press, USA, 2011).

[29] K. Kraus, A. Böhm, J. D. Dollard, and W. H. Wootters, eds., States, Effects, and Operations Fundamental Notions of Quantum Theory, Lecture Notes in Physics, Vol. 190 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1983).

[30] M.-D. Choi, Linear Algebra and its Applications 10, 285 (1975).

Cited by

[1] Antoine Rignon-Bret, Giacomo Guarnieri, John Goold, and Mark T. Mitchison, “Thermodynamics of precision in quantum nano-machines”, arXiv:2009.11303.

[2] G. J. Milburn, “The thermodynamics of clocks”, Contemporary Physics 61 2, 69 (2020).

[3] Emanuel Schwarzhans, Maximilian P. E. Lock, Paul Erker, Nicolai Friis, and Marcus Huber, “Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics”, arXiv:2007.01307.

The above citations are from SAO/NASA ADS (last updated successfully 2021-01-18 06:59:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-01-18 06:59:51).


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‘Ultrasound drill’ and nanodroplets break apart blood clots




Ultrasound drill
On target: artist’s impression of how an ultrasound drill and injection tube for nanodroplets could be used to break up blood clots in the body. (Courtesy: North Carolina State University)

A precision “ultrasound drill” combined with specially engineered nanodroplets could soon be used inside the body to break up stubborn, impenetrable blood clots – according to Leela Goel, Xiaoning Jiang  and colleagues at North Carolina State University. The team has done in vitro experiments demonstrating the technique, which if approved for clinical trials, could lead to promising new treatments for dangerous forms of thrombosis.

If blood clots do not break down quickly enough, they can retract over periods of several days, forming dense, non-porous clumps of cells. Each year, up to 600,000 people in the US alone can be affected by these clots – known as deep vein thromboses. In the past, their treatment has largely involved drugs that activate certain enzymes in the blood to break down the structures of the clots. However, the high drug doses and long treatment times required in this approach can cause significant damage in surrounding tissues.

More recently, a technique called sonothrombolysis has emerged. This uses ultrasound waves to cause microbubbles surrounding a clot to oscillate – enhancing both mechanical erosion and drug diffusion in the clot. However, this technique relies on large external ultrasound transducers, and cannot be used to treat veins that are obscured by ultrasound-blocking organs like the lungs or ribs.

Low boiling point

In their study, the North Carolina State team delivered nanodroplets to a clot created in an experimental apparatus. Because of their small size, the nandroplets easily penetrate retracted clots. Alongside the tube that delivers the nanodroplets is a catheter-based ultrasound “drill”, which produces precisely-aimed acoustic waves via a tiny, forward-viewing transducer.

The nanodroplets are specially engineered to have a low boiling point. The small amount of energy delivered by the drill is enough to vaporize the nanodroplets forming gas-filled microbubbles that rapidly expand and contract. These oscillations break down the clot through the process of cavitation – the creation of microscale streams and jets that weaken the clot’s mechanical structure. At the same time, the vibrations open up holes in the clot that enable enzyme-enhancing drugs to penetrate more easily. This enhances breakdown even further, while avoiding the need for high drug doses and long treatment times.

Over 30 min timescales, they found that clot masses could be reduced by around 40% – much more than the 17% for treatments that combine ultrasound, microbubbles, and enzyme-activating drugs. Although the team’s approach is still a long way from entering clinical practice, their results suggest that a breakthrough in the treatment of deep vein thromboses could be just over the horizon.

The technique is described in Microsystems & Nanoengineering.


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