Last May, I received an email entitled “Amy Poehler’s Smart Girls.” The actress, I read, had co-founded the Smart Girls organization to promote confidence and creativity in preteens and teens. Smart Girls was creating a webseries hosted by Megan Amram, author of *Science…for Her!* The book parodies women’s magazines and ridicules stereotypes of women as unsuited for science.

Megan would host the webseries, “Experimenting with Megan,” in character as an airhead. She planned to interview “kick-ass lady scientists/professors/doctors” in a parody of a talk show. Would I, the email asked, participate?

I’m such a straitlaced fogey, I never say “kick-ass.” I’m such a workaholic, I don’t watch webshows. I’ve not seen Parks and Recreation, the TV series that starred Amy Poehler and for which Megan wrote. The Hollywood bug hasn’t bitten me, though I live 30 minutes from Studio City.

But I found myself in a studio the next month. Young men and women typed on laptops and chattered in the airy, bright waiting lounge. Beyond a doorway lay the set, enclosed by fabric-covered walls that prevented sounds from echoing. Script-filled binders passed from hand to hand, while makeup artists, cameramen, and gophers scurried about.

Disney’s Mouseketeers couldn’t have exuded more enthusiasm or friendliness than the “Experimenting” team. “Can I bring you a bottle of water?” team members kept asking me and each other. “Would you like a chair?” The other women who interviewed that day—two biologist postdocs—welcomed me into their powwow. Each of us, we learned, is outnumbered by men at work. None of us wears a lab coat, despite stereotypes of scientists as white-coated. Each pours herself into her work: One postdoc was editing a grant proposal while off-set.

I watched one interview, in which Megan asked why biologists study fruit flies instead of “cuter” test subjects. Then I stepped on-set beside her. I perched on an armchair that almost swallowed my 5’ 3.5” self.* Textbooks, chemistry flasks, and high-heeled pumps stood on the bookshelves behind Megan.

The room quieted. A clapperboard clapped: “Take one.” Megan thanked me for coming, then launched into questions.

Megan hadn’t warned me what she’d ask. We began with “Do you like me?” and “What is the ‘information’ [in ‘quantum information theory’], and do you ever say, ‘Too much information’?” Each question rode hot on the heels of the last. The barrage reminded me of interviews for not-necessarily-scientific scholarships. Advice offered by one scholarship-committee member, the year before I came to Caltech, came to mind: Let loose. Act like an athlete tearing down the field, the opposing team’s colors at the edges of your vision. Savor the challenge.

I savored it. I’d received instructions to play the straight man, answering Megan’s absurdity with science. To “Too much information?” I parried that we can never know enough. When I mentioned that quantum mechanics describes electrons, Megan asked about the electricity she feels upon seeing Chris Hemsworth. (I hadn’t heard of Chris Hemsworth. After watching the interview online, a friend reported that she’d enjoyed the reference to *Thor*. “What reference to *Thor*?” I asked. Hemsworth, she informed me, plays the title character.) I dodged Chris Hemsworth; caught “electricity”; and stretched to superconductors, quantum devices whose charges can flow forever.

Academic seminars conclude with question-and-answer sessions. If only those Q&As zinged with as much freshness and flexibility as Megan’s.

The “Experimenting” approach to stereotype-blasting diverges from mine. High-heeled pumps, I mentioned, decorated the set. The “Experimenting” team was parodying the stereotype of women as shoe-crazed. “Look at this stereotype!” the set shouts. “Isn’t it ridiculous?”

As a woman who detests high heels and shoe shopping, I prefer to starve the stereotype of justification. I’ve preferred reading to shopping since before middle school, when classmates began frequenting malls. I feel more comfortable demonstrating, through silence, how little shoes interest me. I’d rather offer no reason for anyone to associate me with shoes.**

I scarcely believe that I appear just after a “sexy science” tagline and a hot-or-not quiz. Before my interview on her quantum episode, Megan discussed the relationship between atoms and Adams. Three guests helped her, three Hollywood personalities named “Adam.”*** Megan held up cartoons of atoms, and photos of Adams, and asked her guests to rate their hotness. I couldn’t have played Megan’s role, couldn’t imagine myself in her (high-heeled) shoes.

But I respect the “Experimenting” style. Megan’s character serves as a foil for the interviewee I watched. Megan’s ridiculousness underscored the postdoc’s professionalism and expertise.

According to online enthusiasm, “Experimenting” humor resonates with many viewers. So diverse is the community that needs introducing to STEM, diverse senses of humor have roles to play. So deep run STEM’s social challenges, multiple angles need attacking.

Just as diverse perspectives can benefit women-in-STEM efforts, so can diverse perspectives benefit STEM. Which is why STEM needs women, Adams, shoe-lovers, shoe-haters…and experimentation.

*With gratitude to the “Experimenting” team for the opportunity to contribute to its cause. The live-action interview appears here (beginning at 2:42),** and a follow-up personality quiz appears here.*

*If you’re 5′ 3.5″, every half-inch matters.

**Except when I blog about how little I wish to associate with shoes.

***Megan introduced her guests as “Adam Shankman, Adam Pally, and an intern that we made legally change his name to Adam to be on the show.” The “intern” is Adam Rymer, president of Legendary Digital Networks. Legendary owns Amy Poehler’s Smart Girls.

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“Quantum physics seems to inspire creative minds, so we can’t wait to see what this year’s contest will bring,” says Scientific American Editor in Chief and competition judge Mariette DiChristina.

A panel of judges will select the winners and runner-ups in two categories: **Open** and **Youth**. The public will also vote and decide the **People’s Choice Prize** from entries shortlisted across both categories. Winners will receive a trophy, a cash prize and a one-year digital subscription to ScientificAmerican.com. The winner of the Open category will also be featured on ScientificAmerican.com.

The quantum world offers lots of scope for enthralling characters and mind-blowing plot twists, according to Artur Ekert, director of the Centre for Quantum Technologies and co-inventor of quantum cryptography. “A writer has plenty to play with when science allows things to be in two places – or even two universes – at once,” he says. “The result might be funny, tense or even confusing. But it certainly won’t be boring.” Artur is one of the Open category judges.

Another judge is Colin Sullivan, editor of Futures, Nature’s own science-themed fiction strand. “Science fiction is a powerful and innovative genre,” Colin says. “We are excited to see what kinds of stories quantum physics can inspire.”

The 2015 Quantum Shorts contest is also supported by scientific partners around the world. The Institute for Quantum Information and Matter is proud to sponsor this competition, along with our friends at the Centre for Engineered Quantum Systems, an Australian Research Council Centre of Excellence, the Institute for Quantum Computing at the University of Waterloo, and the Joint Quantum Institute, a research partnership between the University of Maryland and the National Institute of Standards and Technology.

Submissions to Quantum Shorts 2015 are limited to 1000 words and can be entered into the Quantum Shorts competition via the website at http://shorts.quantumlah.org, which also features a full set of rules and guidelines.

For more information about the organizer and partners, please visit the competition website at http://shorts.quantumlah.org.

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The thank-you slide of my presentation remained onscreen, and the question-and-answer session had begun. I was presenting a seminar about thermodynamic resource theories (TRTs), models developed by quantum-information theorists for small-scale exchanges of heat and work. The audience consisted of condensed-matter physicists who studied graphene and photonic crystals. I was beginning to regret my topic’s abstractness.

The question-asker pointed at a listener.

“This is an experimentalist,” he continued, “your arch-nemesis. What implications does your theory have for his lab? Does it have any? Why should he care?”

I could have answered better. I apologized that quantum-information theorists, reared on the rarefied air of Dirac bras and kets, had developed TRTs. I recalled the baby steps with which science sometimes migrates from theory to experiment. I could have advocated for bounding, with idealizations, efficiencies achievable in labs. I should have invoked the connections being developed with fluctuation results, statistical mechanical theorems that have withstood experimental tests.

The crowd looked unconvinced, but I scored one point: The experimentalist was not my arch-nemesis.

“My new friend,” I corrected the questioner.

His question has burned in my mind for two years. Experiments have inspired, but not guided, TRTs. TRTs have yet to drive experiments. Can we strengthen the connection between TRTs and the natural world? If so, what tools must resource theorists develop to predict outcomes of experiments? If not, are resource theorists doing physics?

I explore answers to these questions in a paper released today. Ian Durham and Dean Rickles were kind enough to request a contribution for a book of conference proceedings. The conference, “Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge” took place at the University of Cambridge (including a graveyard thereof), thanks to FQXi (the Foundational Questions Institute).

What, I asked my advisor, does one write for conference proceedings?

“Proceedings are a great opportunity to get something off your chest,” John said.

That seminar Q&A had sat on my chest, like a pet cat who half-smothers you while you’re sleeping, for two years. Theorists often justify TRTs with experiments.^{*} Experimentalists, an argument goes, are probing limits of physics. Conventional statistical mechanics describe these regimes poorly. To understand these experiments, and to apply them to technologies, we must explore TRTs.

Does that argument not merit testing? If experimentalists observe the extremes predicted with TRTs, then the justifications for, and the timeliness of, TRT research will grow.

You’ve read the paper’s introduction, the first eight paragraphs of this blog post. (Who wouldn’t want to begin a paper with a mortifying anecdote?) Later in the paper, I introduce TRTs and their role in *one-shot statistical mechanics*, the analysis of work, heat, and entropies on small scales. I discuss whether TRTs can be realized and whether physicists should care. I identify eleven opportunities for shifting TRTs toward experiments. Three opportunities concern what merits realizing and how, in principle, we can realize it. Six adjustments to TRTs could improve TRTs’ realism. Two more-out-there opportunities, though less critical to realizations, could diversify the platforms with which we might realize TRTs.

One opportunity is the physical realization of thermal embezzlement. TRTs, like thermodynamic laws, dictate how systems can and cannot evolve. Suppose that a state cannot transform into a state : . An ancilla , called a catalyst, might facilitate the transformation: . Catalysts act like engines used to extract work from a pair of heat baths.

Engines degrade, so a realistic transformation might yield , wherein resembles . For certain definitions of “resembles,”** TRTs imply, one can extract arbitrary amounts of work by negligibly degrading . Detecting the degradation—the work extraction’s cost—is difficult. Extracting arbitrary amounts of work at a difficult-to-detect cost contradicts the spirit of thermodynamic law.

The spirit, not the letter. Embezzlement seems physically realizable, in principle. Detecting embezzlement could push experimentalists’ abilities to distinguish between close-together states and . I hope that that challenge, and the chance to violate the spirit of thermodynamic law, attracts researchers. Alternatively, theorists could redefine “resembles” so that doesn’t rub the law the wrong way.

The paper’s broadness evokes a caveat of Arthur Eddington’s. In 1927, Eddington presented Gifford Lectures entitled *The Nature of the Physical World*. Being a physicist, he admitted, “I have much to fear from the expert philosophical critic.” Specializing in TRTs, I have much to fear from the expert experimental critic. The paper is intended to point out, and to initiate responses to, the lack of physical realizations of TRTs. Some concerns are practical; some, philosophical. I expect and hope that the discussion will continue…preferably with more cooperation and charity than during that Q&A.

If you want to continue the discussion, drop me a line.

^{*}So do theorists-in-training. I have.

**A definition that involves the trace distance.

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The Beyond i.i.d. series has shaped a lot of the research going on in this area and has certainly affected my own research directions. The first Beyond i.i.d. was held in Cambridge, UK in January 2013, organized by Nilanjana Datta and Renato Renner. It had a clever logo, featuring cyclists of various sorts biking one after another, the first few looking the same and the ones behind them breaking out of the i.i.d. pattern:

It was also at the Cambridge edition that the famous entropy zoo first appeared, which has now been significantly updated, based on recent progress in the area. The next Beyond i.i.d. happened in Singapore in May 2014, organized by Marco Tomamichel, Vincent Tan, and Stephanie Wehner. (Stephanie was a recent “quantum woman” for her work on a loophole-free Bell experiment.)

The tradition continued this past summer in beautiful Banff, Canada. I hope that it goes on for a long time. At least I have next year’s to look forward to, which will be in beachy Barcelona in the summertime, (as of now) planned to be just one week before Alexander Holevo presents the Shannon lecture in Barcelona at the ISIT 2016 conference (by the way, this is the first time that a quantum information theorist has won the prestigious Shannon award).

So why am I blabbing on and on about the Beyond i.i.d. conference if Nicole already wrote a great summary of the Banff edition this past summer? Well, she didn’t have room in her blog post to cover one of my favorite topics that was discussed at my favorite conference, so she graciously invited me to discuss it here.

The driving question of my new favorite topic is “*What is the right notion of a quantum Markov chain?*” The past year or so has seen some remarkable progress in this direction. To motivate it, let’s go back to bears, and specifically bear attacks (as featured in Nicole’s post). In Banff, the locals there told us that they had never heard of a bear attacking a group of four or more people who hike together. But let’s suppose that Alice, Bob, and Eve ignore this advice and head out together for a hike in the mountains. Also, in a different group are 50 of Alice’s sisters, but the park rangers are focusing their binoculars on the group of three (Alice, Bob, and Eve), observing their movements, because they are concerned that a group of three will run into trouble.

In the distance, there is a clever bear observing the movements of Alice, Bob, and Eve, and he notices some curious things. If he looks at Alice and Bob’s movements alone, they appear to take each step randomly, but for the most part together. That is, their steps appear correlated. He records their movements for some time and estimates a probability distribution that characterizes their movements. However, if he considers the movements of Alice, Bob, and Eve all together, he realizes that Alice and Bob are really taking their steps based on what Eve does, who in turn is taking her steps completely at random. So at this point the bear surmises that Eve is the mediator of the correlations observed between Alice and Bob’s movements, and when he writes down an estimate for the probability distribution characterizing all three of their movements, he notices that it factors as . That is, the bear sees that the distribution forms a *Markov chain*.

“*What is an important property of such a Markov chain?*“, asks the bear. Well, neglecting Alice’s movements (summing over the variable), the probability distribution reduces to , because is a conditional probability distribution. A characteristic of a Markov probability distribution is that one could reproduce the original distribution simply by acting on the variable of with the conditional probability distribution . So the bear realizes that it would be possible for Alice to be lost and subsequently replaced by Eve calling in one of Alice’s sisters, such that nobody else would notice anything different from before — it would appear as if the movements of all three were unchanged once this replacement occurs. Salivating at his realization, the bear takes Eve briefly aside without any of the others noticing. The bear explains that he will not eat Eve and will instead eat Alice if Eve can call in one of Alice’s sisters and direct her movements to be chosen according to the distribution . Eve, realizing that her options are limited (ok, ok, maybe there are other options…), makes a deal with the bear. So the bear promptly eats Alice, and Eve draws in one of Alice’s sisters, whom Eve then directs to walk according to the distribution . This process repeats, going on and on, and all the while, the park rangers, focusing exclusively on the movements on the party of three, don’t think anything of what’s going because they observe that the joint distribution describing the movements of “Alice,” Bob, and Eve never seems to change (let’s assume that the actions of the bear and Eve are very fast :) ). So the bear is very satisfied after eating Alice and some of her sisters, and Eve is pleased not to be eaten, at the same time never having cared too much for Alice or any of her sisters.

A natural question arises: “*What could Alice and Bob do to prevent this terrible situation from arising, in which Alice and so many of her sisters get eaten without the park rangers noticing anything?*” Well, Alice and Bob could attempt to coordinate their movements independently of Eve’s. Even better, before heading out on a hike, they could make sure to have brought along several entangled pairs of particles (and perhaps some bear spray). If Alice and Bob choose their movements according to the outcomes of measurements of the entangled pairs, then it would be impossible for Alice to be eaten and the park rangers not to notice. That is, the distribution describing their movements could never be described by a Markov chain distribution of the form . Thus, in such a scenario, as soon as the bear attacks Alice and then Eve replaces her with one of her sisters, the park rangers would immediately notice something different about the movements of the party of three and then figure out what is going on. So at least Alice could save her sisters…

What is the lesson here? A similar scenario is faced in quantum key distribution. Eve and other attackers (such as a bear) might try to steal what is there in Alice’s system and then replace it with something else, in an attempt to go undetected. If the situation is described by classical physics, this would be possible if Eve had access to a “hidden variable” that dictates the actions of Alice and Bob. But according to Bell’s theorem or the monogamy of entanglement, it is impossible for a “hidden variable” strategy to mimic the outcomes of measurements performed on sufficiently entangled particles.

Since we never have perfectly entangled particles or ones whose distributions exactly factor as Markov chains, it would be ideal to quantify, for a given three-party quantum state of Alice, Bob, and Eve, how well one could recover from the loss of the Alice system by Eve performing a recovery channel on her system alone. This would help us to better understand approximate cases that we expect to appear in realistic scenarios. At the same time, we could have a more clear understanding of what constitutes an *approximate quantum Markov chain*.

Now due to recent results of Fawzi and Renner, we know that this quantification of quantum non-Markovianity is possible by using the conditional quantum mutual information (CQMI), a fundamental measure of information in quantum information theory. We already knew that the CQMI is non-negative when evaluated for any three-party quantum state, due to the strong subadditivity inequality, but now we can say more than that: If the CQMI is small, then Eve can recover well from the loss of Alice, implying that the reduced state of Alice and Bob’s system could not have been too entangled in the first place. Relatedly, if Eve cannot recover well from the loss of Alice, then the CQMI cannot be small. The CQMI is the quantity underlying the squashed entanglement measure, which in turn plays a fundamental role in characterizing the performance of realistic quantum key distribution systems.

Since the original results of Fawzi and Renner appeared on the arXiv, this topic has seen much activity in “quantum information land.” Here are some papers related to this topic, which have appeared in the past year or so (apologies if I missed your paper!):

Renyi generalizations of the conditional quantum mutual information

Fidelity of recovery, geometric squashed entanglement, and measurement recoverability

Renyi squashed entanglement, discord, and relative entropy differences

Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps

Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution

Multipartite quantum correlations and local recoverability

Monotonicity of quantum relative entropy and recoverability

Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

The Fidelity of Recovery is Multiplicative

Universal recovery map for approximate Markov chains

Recoverability in quantum information theory

Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map

Universal recovery from a decrease of quantum relative entropy

Some of the papers are admittedly that of myself and my collaborators, but *hey!*, please forgive me, I’ve been excited about the topic. We now know simpler proofs of the original Fawzi–Renner results and extensions of them that apply to the quantum relative entropy as well. Since the quantum relative entropy is such a fundamental quantity in quantum information theory, some of the above papers provide sweeping ramifications for many foundational statements in quantum information theory, including entanglement theory, quantum distinguishability, the Holevo bound, quantum discord, multipartite information measures, etc. Beyond i.i.d. had a day of talks dedicated to the topic, and I think we will continue seeing further developments in this area.

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abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux.

Its solution was never published in his lifetime. What was the secret hiding in these series of letters?

The arch is one of the fundamental building blocks in architecture. Used in bridges, cathedrals, doorways, etc., arches provide an aesthetic quality to the structures they dwell within. Their key utility comes from their ability to support weight above an empty space, by distributing the load onto the abutments at its feet. A dome functions much like an arch, except a dome takes on a three-dimensional shape whereas an arch is two-dimensional. Paradoxically, while being the backbone of the many edifices, arches and domes themselves are extremely delicate: a single misplaced component along its curve, or an improper shape in the design would spell doom for the entire structure.

The Romans employed the rounded arch/dome—in the shape of a semicircle/hemisphere–in their bridges and pantheons. The Gothic architecture favored the pointed arch and the ribbed vault in their designs. However, neither of these arch forms were adequate for the progressively grander structures and more ambitious cathedrals sought in the 17th century. Following the great fire of London in 1666, a massive rebuilding effort was under way. Among the new public buildings, the most prominent was to be St. Paul’s Cathedral with its signature dome. A modern theory of arches was sorely needed: what is the perfect shape for an arch/dome?

Christopher Wren, the chief architect of St. Paul’s Cathedral, consulted Hooke on the dome’s design. To quote from the cathedral’s website [1]:

The two half-sections [of the dome] in the study employ a formula devised by Robert Hooke in about 1671 for calculating the curve of a parabolic dome and reducing its thickness. Hooke had explored this curve the three-dimensional equivalent of the ‘hanging chain’, or catenary arch: the shape of a weighted chain which, when inverted, produces the ideal profile for a self-supporting arch. He thought that such a curve derived from the equationy=x^{3}.

How did Hooke came about the shape for the dome? It wasn’t until after Hooke’s death his executor provided the unencrypted solution to the anagram [2]

Ut pendet continuum flexile, sic stabit contiguum rigidum inversum

which translates to

As hangs a flexible cable so, inverted, stand the touching pieces of an arch.

In other words, the ideal shape of an arch is exactly that of a freely hanging rope, only upside down. Hooke understood that the building materials could withstand only compression forces and not tensile forces, in direct contrast to a rope that could resist tension but would buckle under compression. The mathematics describing the arch and the cable are in fact identical, save for a minus sign. Consequently, you could perform a real-time simulation of an arch using a piece of string!

Bonus: Robert published the anagram in his book describing helioscopes, simply to “fill up the vacancy of the ensuring page” [3]. On that very page among other claims, Hooke also wrote the anagram “*ceiiinosssttuu*” in regards to “the true theory of elasticity”. Can you solve this riddle?

[1] https://www.stpauls.co.uk/history-collections/the-collections/architectural-archive/wren-office-drawings/5-designs-for-the-dome-c16871708

[2] Written in Latin, the ‘u’ and ‘v’ are the same letter.

[3] In truth, Hooke was likely trying to avoid being scooped by his contemporaries, notably Issac Newton.

This article was inspired by my visit to the Huntington Library. I would like to thank Catherine Wehrey for the illustrations and help with the research.

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It wasn’t easy. Many different typeballs were available, to support various fonts and special characters. Typing a displayed equation or in-line equation usually involved swapping back and forth between typeballs to access all the needed symbols. Most physics research groups had staff who knew how to use the IBM Selectric and spent much of their time typing manuscripts.

Though the IBM Selectric was used by many groups, typewriters have unique personalities, as forensic scientists know. I had a friend who claimed he had learned to recognize telltale differences among documents produced by various IBM Selectric machines. That way, whenever he received a referee report, he could identify its place of origin.

Manuscripts did not evolve through 23 typeset versions in those days, as one of my recent papers did. Editing was arduous and frustrating, particularly for a lowly graduate student like me, who needed to beg Blanche to set aside what she was doing for Steve Weinberg and devote a moment or two to working on my paper.

It was tremendously liberating when I learned to use TeX in 1990 and started typing my own papers. (Not LaTeX in those days, but Plain TeX embellished by a macro for formatting.) That was a technological advance that definitely improved my productivity. An earlier generation had felt the same way about the Xerox machine.

But as I was reminded a few days ago, while technological advances can be empowering, they can also be dangerous when used recklessly. I was editing a very long document, and decided to make a change. I had repeatedly used $x$ to denote an n-bit string, and thought it better to use $\vec x$ instead. I was walking through the paper with the replace button, changing each $x$ to $\vec x$ where the change seemed warranted. But I slipped once, and hit the “Replace All” button instead of “Replace.” My computer curtly informed me that it had made the replacement 1011 times. Oops …

This was a revocable error. There must have been a way to undo it (though it was not immediately obvious how). Or I could have closed the file without saving, losing some recent edits but limiting the damage.

But it was late at night and I was tired. I panicked, immediately saving and LaTeXing the file. It was a mess.

Okay, no problem, all I had to do was replace every \vec x with x and everything would be fine. Except that in the original replacement I had neglected to specify “Match Case.” In 264 places $X$ had become $\vec x$, and the new replacement did not restore the capitalization. It took hours to restore every $X$ by hand, and there are probably a few more that I haven’t noticed yet.

Which brings me to the cautionary tale of one of my former graduate students, Robert Navin. Rob’s thesis had two main topics, scattering off vortices and scattering off monopoles. On the night before the thesis due date, Rob made a horrifying discovery. The crux of his analysis of scattering off vortices concerned the singularity structure of a certain analytic function, and the chapter about vortices made many references to the poles of this function. What Rob realized at this late stage is that these singularities are actually branch points, not poles!

What to do? It’s late and you’re tired and your thesis is due in a few hours. Aha! Global search and replace! Rob replaced every occurrence of “pole” in his thesis by “branch point.” Problem solved.

Except … Rob had momentarily forgotten about that chapter on monopoles. Which, when I read the thesis, had been transformed into a chapter on monobranch points. His committee accepted the thesis, but requested some changes …

Rob Navin no longer does physics, but has been very successful in finance. I’m sure he’s more careful now.

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This idea quickly sparked enthusiasm among the rest of us, who have all been working in this interdisciplinary area and are witness to its rising power. In fact Xiao-Gang has had the same idea of book writing for some time. So now here we are, four years later, posting the first version of the book on arXiv last week. (arXiv link)

The book is a condensed matter book on the topic of strongly interacting many-body systems, with a special focus on the emergence of topological order. This is an exciting topic, with new developments everyday. We are not trying to cover the whole picture, but rather to present just one perspective – the quantum information perspective – of the story. Quantum information ideas, like entanglement, quantum circuit, quantum codes are becoming ever more popular nowadays in condensed matter study and have lead to many important developments. On the other hand, they are not usually taught in condensed matter courses or covered by condensed matter books. Therefore, we feel that writing a book may help bridge the gap.

We keep the writing in a self-consistent way, requiring minimum background in quantum information and condensed matter. The first part introduces concepts in quantum information that is going to be useful in the later study of condensed matter systems. (It is by no means a well-rounded introduction to quantum information and should not be read in that way.) The second part moves onto explaining one major topic of condensed matter theory, the local Hamiltonians and their ground states, and contains introduction to the most basic concepts in condensed matter theory like locality, gap, universality, etc. The third part then focuses on the emergence of topological order, first presenting a historical and intuitive picture of topological order and then building a more systematic approach based on entanglement and quantum circuit. With this framework established, the fourth part studies some interesting topological phases in 1D and 2D, with the help of the tensor network formalism. Finally part V concludes with the outlook of where this miraculous encounter of quantum information and condensed matter would take us – the unification between information and matter.

We hope that, with such a structure, the book is accessible to both condensed matter students / researchers interested in this quantum information approach and also quantum information people who are interested in condensed matter topics. And of course, the book is also limited by the perspective we are taking. Compared to a standard condensed matter book, we are missing even the most elementary ingredient – the free fermion. Therefore, this book is not to be read as a standard textbook on condensed matter theory. On the other hand, by presenting a new approach, we hope to bring the readers to the frontiers of current research.

The most important thing I want to say here is: this arXiv version is * NOT* the final version. We posted it so that we can gather feedbacks from our colleagues. Therefore, it is not yet ready for junior students to read in order to learn the subject. On the other hand, if you are a researcher in a related field, please send us criticism, comments, suggestions, or whatever comes to your mind. We will be very grateful for that! (One thing we already learned (thanks Burak!) is that we forgot to put in all the references on conditional mutual information. That will be corrected in a later version, together with everything else.) The final version will be published by Springer as part of their “Quantum Information Science and Technology” series.

I guess it is quite obvious that me writing on the blog of the Institute for Quantum Information and Matter (IQIM) about this book titled “Quantum Information meets Quantum Matter” (QIQM) is not a simple coincidence. The romance story between the two emerged in the past decade or so and has been growing at a rate much beyond expectations. Our book is merely an attempt to record some aspects of the beginning. Let’s see where it will take us.

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I have written before about the exciting day in April 1997 when Alesha and I met, and I heard for the first time about the thrilling concept of a topological quantum computer. I’ll take the liberty of drawing a quote from that post, which seems particularly relevant today:

Over coffee at the Red Door Cafe that afternoon, we bonded over our shared admiration for a visionary paper by Greg Moore and Nick Read about non-abelian anyons in fractional quantum Hall systems, though neither of us fully understood the paper (and I still don’t). Maybe, we mused together, non-abelian anyons are not just a theorist’s dream … It was the beginning of a beautiful friendship.

As all physics students know, fundamental particles in three spatial dimensions come in two varieties, bosons and fermions, but in two spatial dimensions more exotic possibilities abound, dubbed “anyons” by Wilczek. Anyons have an exotic spin, a fraction of an electron’s spin, and corresponding exotic statistics — when one anyon is carried around another, their quantum state picks up a nontrivial topological phase. (I had some fun discussions with Frank Wilczek in 1981 as he was developing the theory of anyons. In some of his writings Frank has kindly credited me for suggesting to him that a robust spin-statistics connection should hold in two dimensions, so that fractional spin is necessarily accompanied by fractional statistics. The truth is that my understanding of this point was murky at best back then.) Not long after Wilczek’s paper, Bert Halperin recognized the relevance of anyons to the strange fractional quantum Hall states that had recently been discovered; these support particle-like objects carrying a fraction of the electron’s electric charge, which Halperin recognized to be anyons.

Non-abelian anyons are even more exotic. In a system with many widely separated non-abelian anyons, there are a vast number of different ways for the particles to “fuse” together, giving rise to many possible quantum states, all of which are in principle distinguishable but in practice are hard to tell apart. Furthermore, by “braiding” the anyons (performing a sequence of particle exchanges, so the world lines of the anyons trace out a braid in three-dimensional spacetime), this state can be manipulated, coherently processing the quantum information encoded in the system.

Others (including me) had mused about non-abelian anyons before Moore and Read came along, but no one had proposed a plausible story for how such exotic objects would arise in a realistic laboratory setting. As collaborators, Moore and Read complemented one another perfectly. Greg was, and is, one of the world’s leading experts on conformal field theory. Nick was, and is, one of the world’s leading experts on the fractional quantum Hall effect. Together, they realized that one of the already known fractional quantum Hall states (at filling factor 5/2) is a good candidate for a topological phase supporting non-abelian anyons. This was an inspired guess, most likely correct, though we still don’t have smoking gun experimental evidence 25 years later. Their paper is a magical and rare combination of mathematical sophistication with brilliant intuition.

Alexei arrived at his ideas about non-abelian anyons coming from a different direction, though I suspect he drew inspiration from the earlier deep contributions of Moore and Read. He was trying to imagine a physical system that could store and process a quantum state reliably. Normally quantum systems are very fragile — just looking at the system alters its state. To prevent a quantum computer from making errors, we need to isolate the information processed by the computer from the environment. A system of non-abelian anyons has just the right properties to make this possible; it carries lots of information, but the environment can’t read (or damage) that information when it looks at the particles one at a time. That’s because the information is not encoded in the individual particles, but instead in subtle collective properties shared by many particles at once.

Alexei and I had inspiring discussions about topological quantum computing when we first met at Caltech in April 1997, which continued at a meeting in Torino, Italy that summer, where we shared a bedroom. I was usually asleep by the time he came to bed, because he was staying up late, typing his paper.

Alexei did not think it important to publish his now renowned 1997 paper in a journal — he was content for the paper to be accessible on the arXiv. But after a few years I started to get worried … in my eyes Alexei was becoming an increasingly likely Nobel Prize candidate. Would it cause a problem if his most famous paper had never been published? Just to be safe, I arranged for it to appear in Annals of Physics in 2003, where I was on the editorial board at the time. Frank Wilczek, then the editor, was delighted by this submission, which has definitely boosted the journal’s impact factor! (“Fault-tolerant quantum computation by anyons” has 2633 citations as of today, according to Google Scholar.) Nobelists are ineligible for the Dirac Medal, but some past medalists have proceeded to greater glory. It could happen again, right?

Alesha and I have now been close friends and collaborators for 18 years, but I have actually known Greg and Nick even longer. I taught at Harvard for a few years in the early 1980s, at a time when an amazingly talented crew of physics graduate students roamed the halls, of whom Andy Cohen, Jacques Distler, Ben Grinstein, David Kaplan, Aneesh Manohar, Ann Nelson, and Phil Nelson among others all made indelible impressions. But there was something special about Greg. The word that comes to mind is *intensity*. Few students exhibit as much drive and passion for physics as Greg did in those days. He’s calmer now, but still pretty intense. I met Nick a few years later when we tried to recruit him to the Caltech faculty. Luring him to southern California turned out to be a lost cause because he didn’t know how to drive a car. I suppose he’s learned by now?* Whenever I’ve spoken to Nick in the years since then, I’ve always been dazzled by his clarity of thought.

Non-abelian anyons are at a pivotal stage, with lots of experimental hints supporting their existence, but still no ironclad evidence. I feel confident this will change in the next few years. These are exciting times!

And guess what? This occasion gives me another opportunity to dust off one of my poems!

Anyon, Anyon

Anyon, anyon, where do you roam?

Braid for a while before you go home.

Though you’re condemned just to slide on a table,

A life in 2D also means that you’re able

To be of a type neither Fermi nor Bose

And to know left from right — that’s a kick, I suppose.

You and your buddy were made in a pair

Then wandered around, braiding here, braiding there.

You’ll fuse back together when braiding is through

We’ll bid you adieu as you vanish from view.

Alexei exhibits a knack for persuading

That someday we’ll crunch quantum data by braiding,

With quantum states hidden where no one can see,

Protected from damage through top-ology.

Anyon, anyon, where do you roam?

Braid for a while, before you go home.

*Note added: Nick confirms, “Yes, I’ve had a driving license since 1992, and a car since 1994!”

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Last year, I blogged about the University of Cambridge cemetery visited by participants in the conference “Eddington and Wheeler: Information and Interaction.” We’d lectured each other about entropy–a quantification of decay, of the march of time. Then we marched to an overgrown graveyard, where scientists who’d lectured about entropy decades earlier were decaying.

This July, I attended the conference “Beyond i.i.d. in information theory.” The acronym “i.i.d.” stands for “independent and identically distributed,” which requires its own explanation. The conference took place at BIRS, the Banff International Research Station, in Canada. Locals pronounce “BIRS” as “burrs,” the spiky plant bits that stick to your socks when you hike. (I had thought that one pronounces “BIRS” as “beers,” over which participants in quantum conferences debate about the Measurement Problem.) Conversations at “Beyond i.i.d.” dinner tables ranged from mathematical identities to the hiking for which most tourists visit Banff to the bears we’d been advised to avoid while hiking. So let me explain the meaning of “i.i.d.” in terms of bear attacks.

*The BIRS conference center. Beyond here, there be bears.*

Suppose that, every day, exactly one bear attacks you as you hike in Banff. Every day, you have a probability p_{1} of facing down a black bear, a probability p_{2} of facing down a grizzly, and so on. These probabilities form a *distribution* {p_{i}} over the set of possible events (of possible attacks). We call the type of attack that occurs on a given day a *random variable*. The distribution associated with each day equals the distribution associated with each other day. Hence the variables are *identically distributed*. The Monday distribution doesn’t affect the Tuesday distribution and so on, so the distributions are *independent.*

Information theorists quantify efficiencies with which i.i.d. tasks can be performed. Suppose that your mother expresses concern about your hiking. She asks you to report which bear harassed you on which day. You compress your report into the fewest possible *bits*, or units of information. Consider the limit as the number of days approaches infinity, called the *asymptotic limit*. The number of bits required per day approaches a function, called the *Shannon* entropy H_{S}, of the distribution:

Number of bits required per day → H_{S}({p_{i}}).

The Shannon entropy describes many asymptotic properties of i.i.d. variables. Similarly, the *von Neumann entropy* H_{vN} describes many asymptotic properties of i.i.d. quantum states.

But you don’t hike for infinitely many days. The rate of black-bear attacks ebbs and flows. If you stumbled into grizzly land on Friday, you’ll probably avoid it, and have a lower grizzly-attack probability, on Saturday. Into how few bits can you compress a set of nonasymptotic, non-i.i.d. variables?

We answer such questions in terms of *ɛ-smooth α-Rényi entropies*, the *sandwiched Rényi relative entropy*, the *hypothesis-testing entropy*, and related beasts. These beasts form a zoo diagrammed by conference participant Philippe Faist. I wish I had his diagram on a placemat.

“Beyond i.i.d.” participants define these entropies, generalize the entropies, probe the entropies’ properties, and apply the entropies to physics. Want to quantify the efficiency with which you can perform an information-processing task or a thermodynamic task? An entropy might hold the key.

Many highlights distinguished the conference; I’ll mention a handful. If the jargon upsets your stomach, skip three paragraphs to Thermodynamic Thursday.

Aram Harrow introduced a resource theory that resembles entanglement theory but whose agents pay to communicate classically. Why, I interrupted him, define such a theory? The backstory involves a wager against quantum-information pioneer Charlie Bennett (more precisely, against an opinion of Bennett’s). For details, and for a quantum version of *The* *Princess and the Pea*, watch Aram’s talk.

Graeme Smith and colleagues “remove[d] the . . . creativity” from proofs that certain entropic quantities satisfy subadditivity. Subadditivity is a property that facilitates proofs and that offers physical insights into applications*.* Graeme & co. designed an algorithm for checking whether entropic quantity Q satisfies subadditivity. Just add water; no innovation required. How appropriate, conference co-organizer Mark Wilde observed. BIRS has the slogan “Inspiring creativity.”

Patrick Hayden applied one-shot entropies to AdS/CFT and emergent spacetime, enthused about elsewhere on this blog. Debbie Leung discussed approximations to Haar-random unitaries. Gilad Gour compared resource theories.

*Conference participants graciously tolerated my talk about thermodynamic resource theories. I closed my eyes to symbolize the ignorance quantified by entropy. Not really; the photo didn’t turn out as well as hoped, despite the photographer’s goodwill. But I could have closed my eyes **to** symbolize entropic ignorance.*

Thermodynamics and resource theories dominated Thursday. Thermodynamics is the physics of heat, work, entropy, and stasis. Resource theories are simple models for transformations, like from a charged battery and a Tesla car at the bottom of a hill to an empty battery and a Tesla atop a hill.

*My advisor’s** Tesla. No wonder I study thermodynamic resource theories.*

Philippe Faist, diagrammer of the Entropy Zoo, compared two models for thermodynamic operations. I introduced a generalization of resource theories for thermodynamics. Last year, Joe Renes of ETH and I broadened thermo resource theories to model exchanges of not only heat, but also particles, angular momentum, and other quantities. We calculated work in terms of the hypothesis-testing entropy. Though our generalization won’t surprise *Quantum Frontiers* diehards, the magic tricks in my presentation might.

At twilight on Thermodynamic Thursday, I meandered down the mountain from the conference center. Entropies hummed in my mind like the mosquitoes I slapped from my calves. Rising from scratching a bite, I confronted the Banff Cemetery. Half-wild greenery framed the headstones that bordered the gravel path I was following. Thermodynamicists have associated entropy with the passage of time, with deterioration, with a fate we can’t escape. I seem unable to escape from brushing past cemeteries at entropy conferences.

Not that I mind, I thought while scratching the bite in Pasadena. At least I escaped attacks by Banff’s bears.

*With thanks to the conference organizers and to BIRS for the opportunity to participate in “Beyond i.i.d. 2015.”*

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I was in awe of Wheeler. Some students thought he sucked.

I immediately changed it to…

I was in awe of Wheeler. Some students thought less of him.

And next, when I saw John write about himself,

Though I’m 59, few students seemed awed. Some thought I sucked. Maybe I did sometimes.

I massaged it into…

Though I’m 59, few students seemed awed. Some thought I was not as good. Maybe I wasn’t sometimes.

When John published the post, I read it again for any typos I might have missed. There were no typos. I felt useful! But when I saw that all mentions of *sucked *had been restored to their rightful place, I felt like an idiot. John did not fire a strongly-worded email back my way asking for an explanation as to my taking liberties with his own writing. He simply trusted that I would get the message in the comfort of my own awkwardness. It worked beautifully. John had set the tone for Quantum Frontier’s authentic voice with his very first post. It was to be personal, even if the subject matter was as scientifically hardcore as it got.

So when the time came for me to write my first post, I made it personal. I wrote about my time in Los Alamos as a postdoc, working on a problem in mathematical physics that almost broke me. It was Matt Hastings, an intellectual tornado, that helped me through these hard times. As my mentor, he didn’t say things like *Well done! Great progress!* *Good job, Spiro! *He said, *You can do this. *And when I finally did it, when I finally solved that damn problem, Matt came back to me and said: Beyond some typos, I cannot find any mistakes.* Good job, Spiro. *And it meant the world to me. The sleepless nights, the lonely days up in the Pajarito mountains of New Mexico, the times I had resolved to go work for my younger brother as a waiter in his first restaurant… those were the times that I had come upon a fork on the road and my mentor had helped me choose the path less traveled.

When the time came for me to write my next post, I ended by offering two problems for the readers to solve, with the following text as motivation:

This post is supposed to be an introduction to the insanely beautiful world of problem solving. It is not a world ruled by Kings and Queens. It is a world where commoners like you and me can become masters of their domain and even build an empire.

It has been way too long since my last “problem solving” post, so I leave you with a problem from this year’s International Math Olympiad, which took place in gorgeous Chiang Mai, Thailand. *FiverThirtyEight*‘s recent article about the dominance of the US math olympic team in this year’s competition, gives some context about the degree of difficulty of this problem:

Determine all triples (

a, b, c) of positive integers such that each of the numbers:ab-c,bc-a,ca-bis a power of two.Like Fermat’s Last Theorem, this problem is easy to describe and hard to solve. Only 5 percent of the competitors got full marks on this question, and nearly half (44 percent) got no points at all.

But, on the triumphant U.S. squad, four of the six team members nailed it.

In other words, only 1 in 20 kids in the competition solved this problem correctly and about half of the kids didn’t even know where to begin. For more perspective, each national team is comprised of the top 6 math prodigies in that country. In China, that means 6 out of something like 100 million kids. And only 3-4 of these kids solved the problem.

The coach of the US national team, Po-Shen Loh, a Caltech alum and an associate professor of mathematics at Carnegie Mellon University (give him tenure already) deserves some serious props. If you think this problem is too hard, I have this to say to you: Yes, it is. But, who cares? *You can do this.*

*Note: I will work out the solution in detail in an upcoming post, unless one of you solves it in the comments section before then!*

**Update:** Solution posted in comments below (in response to Anthony’s comment). Thank you all who posted some of the answers below. The solution is far from trivial, but I still wonder if an elegant solution exists that gives all four triples. Maybe the best solution is geometric? I hope one of you geniuses can figure that out!

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