Bits, Bears, and Beyond in Banff: Part Deux

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You might remember that about one month ago, Nicole blogged about the conference Beyond i.i.d. in information theory and told us about bits, bears, and beyond in Banff. I was very pleased that Nicole did so, because this conference has become one of my favorites in recent years (ok, it’s my favorite). You can look at her post to see what is meant by “Beyond i.i.d.” The focus of the conference includes cutting-edge topics in quantum Shannon theory, and the conference still has a nice “small world” feel to it (for example, the most recent edition and the first one featured a music session from participants). Here is a picture of some of us having a fun time:

Will Matthews, Felix Leditzky, me, and Nilanjana Datta (facing away) singing

Will Matthews, Felix Leditzky, me, and Nilanjana Datta (facing away) singing “Jamaica Farewell”.

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:

Screen Shot 2015-09-12 at 3.55.17 PM

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 p(a,b) 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 p(a,b,e) characterizing all three of their movements, he notices that it factors as p(a,b,e) = p(a|e) p(b|e) p(e). 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 a variable), the probability distribution reduces to p(b|e) p(e), because p(a|e) is a conditional probability distribution. A characteristic of a Markov probability distribution is that one could reproduce the original distribution p(a,b,e) simply by acting on the e variable of p(b|e) p(e) with the conditional probability distribution p(a|e). 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 p(a|e). 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 p(a|e). 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 p(a,b,e) 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 p(a|e) p(b|e) p(e). 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.

The enigma of Robert Hooke

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In 1675, Robert Hooke published the “true mathematical and mechanical form” for the shape of an ideal arch.  However, Hooke wrote the theory as an anagram,

abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux.

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

Hooke-Arch

An excerpt from Hooke’s manuscript “A description of helioscopes, and some other instruments”.

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 equation y = x3.

A figure from Wren's design of St. Paul's Cathedral. (Courtesy of the British Museum)

A figure from Wren’s design of St. Paul’s Cathedral. (Courtesy of the British Museum)

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.

Beware global search and replace!

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I’m old enough to remember when cutting and pasting were really done with scissors and glue (or Scotch tape). When I was a graduate student in the late 1970s, few physicists typed their own papers, and if they did they left gaps in the text, to be filled in later with handwritten equations. The gold standard of technical typing was the IBM Correcting Selectric II typewriter. Among its innovations was the correction ribbon, which allowed one to remove a typo with the touch of a key. But it was especially important for scientists that the Selectric could type mathematical characters, including Greek letters.

IBM Selectric typeballs

IBM Selectric typeballs

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.

Quantum Information meets Quantum Matter

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“Quantum Information meets Quantum Matter”, it sounds like the beginning of a perfect romance story. It is probably not the kind that makes an Oscar movie, but it does get many physicists excited, physicists including Bei, Duanlu, Xiaogang and me. Actually we find the story so compelling that we decided to write a book about it, and it all started one day in 2011 when Bei popped the question ‘Do you want to write a book about it?’ during one of our conversations.

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.

Kitaev, Moore, Read share Dirac Medal!

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Since its founding 30 years ago, the Dirac Medal has been one of the most prestigious honors in theoretical physics. Particle theorists and string theorists have claimed most of the medals, but occasionally other fields break through, as when Haldane, Kane, and Zhang shared the 2012 Dirac Medal for their pioneering work on topological insulators. I was excited to learn today that the 2015 Dirac Medal has been awarded to Alexei Kitaev, Greg Moore, and Nick Read “for their interdisciplinary contributions which introduced  concepts of conformal field theory and non-abelian quasiparticle statistics in condensed matter systems and  applications of these ideas to quantum computation.”

Left to right: Alexei Kitaev, Greg Moore and Nicholas Read.

Left to right: Alexei Kitaev, Greg Moore, and Nick Read.

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!”

Bits, bears, and beyond in Banff

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Another conference about entropy. Another graveyard.

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.

BIRS

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 p1 of facing down a black bear, a probability p2 of facing down a grizzly, and so on. These probabilities form a distribution {pi} 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 HS, of the distribution:

Number of bits required per day → HS({pi}).

The Shannon entropy describes many asymptotic properties of i.i.d. variables. Similarly, the von Neumann entropy HvN 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.

Entropy zoo

“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.

Presentation

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.

John

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.

Cemetery

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

The mentors that shape us

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Three years and three weeks ago I started my first blog. I wasn’t quite sure what to call it, so I went to John for advice. I had several names in mind, but John quickly zeroed in on one: Quantum Frontiers. The url was available, the name was simple and to the point, it had the word quantum in it, and it was appropriate for a blog that was to provide a vantage point from which the public could view the frontiers of quantum science. But there was a problem; we had no followers and when I first asked John if he would write something for the blog, he had said: I don’t know… I will see…maybe some day… let me think about it. The next day John uploaded More to come, the first real post on Quantum Frontiers after the introductory Hello quantum world! We had agreed on a system in order to keep the quality of the posts above some basic level: we would send each other our posts for editing before we made them public. That way, we could catch any silly typos and have a second pair of eyes do some fact-checking. So, when John sent me his first post, I went to task editing away typos. But the power that comes with being editor-in-chief corrupts. So, when I saw the following sentence in More to come…

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.

Doi-Inthananon-Thailand

Doi-Inthananon temple in Chiang Mai, Thailand. A breathtaking city, host of this year’s international math olympiad.

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-b is 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!

Ant-Man and the Quantum Realm

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ant-man-2015-marvel-movieIt was the first week of August last summer and I was at LAX for a trip to North Carolina as a guest speaker at Project Scientist’s STEM summer camp for young women. I had booked an early morning flight and had arrived at my gate with time to spare, so I decided to get some breakfast. I walked by a smart-looking salad bar and thought: Today is the day. Moving past the salad bar, I ordered a juicy cheeseburger with fries at the adjacent McDonald’s. Growing up in Greece, eating McDonald’s was a rare treat; as was playing video games with my brothers and reading comic books late at night. Yet, through a weird twist of fate, it was these last two guilty pleasures that were to become my Hadouken!, my Hulk, Smash! of choice for breaking down the barriers between the world of quantum mechanics and the everyday reality of our super-normal, super-predictable lives.

I finished my burger, stuffing the last few delicious fries in my mouth, when my phone buzzed – I had a call from The Science and Entertainment Exchange, a non-profit organization funded by the National Academy of Sciences, whose purpose is to bring leading scientists in contact with Hollywood in order to elevate the level of science in the movies. I was to report to Atlanta, GA for a movie consult on a new superhero movie: Ant-Man. As I listened to the director of the Exchange discuss the assignment, I grumbled to myself: Why can’t I be the guy who works on Thor? Who is this Ant-Man anyways? But, in typical Hollywood fashion, the call had a happy ending: “Paul Rudd is Ant-Man. He may be at the meeting, but I can’t promise anything.” Marvel would cover my expenses, so my reluctant reply went something like this:

Hell yeah.

The meeting was in three days time. I would finish my visit to Queens University in Charlotte, NC and take the next flight out to Atlanta. But first, some fun and games were in order. As part of my visit to Project Scientist’s camp, I was invited to teach quantum mechanics to a group of forty girls, ages 8-12, all of whom were interested in science, engineering and mathematics and many of whom did not have the financial means to pursue these interests outside of the classroom. So, I went to Queens University with several copies of MinecraftEDU, the educational component to one of the most popular video games of all time: Minecraft. As I described in “Can a game teach kids quantum mechanics”, I spent the summer of 2013 designing qCraft, a modification (mod) to Minecraft that allows players to craft blocks imbued with quantum superpowers such as quantum superposition and quantum entanglement. The mod, developed in collaboration with Google and TeachersGaming, became popular, amassing millions of downloads around the world. But it is one thing to look at statistics as a measure of success, and another to look into the eyes of these girls as they lost themselves in a game (qCraft is free to download and comes with an accompanying curriculum) designed to teach them the heady concepts that inspired Richard Feynman to quip: If you think you understand quantum theory, you don’t.

girls_qcraftMy visit to Charlotte was so wonderful that I nearly decided to cancel my trip to Atlanta in order to stay with the girls and their mentors until the end of the week. But Mr. Rudd deserved the very best science could offer, so I boarded my flight and resolved to bring Project Scientist to Caltech the next summer. On my way to Atlanta, I used the in-flight WiFi to do some research on Ant-Man. He was part of the original Avengers, a founding member, in fact (nice!) His name was Dr. Hank Pym. He had developed a particle, aptly named after himself, which allowed him to shrink the space between atoms (well now…) He embedded vials of that particle in a suit that allowed him to shrink to the size of an ant (of course, that makes sense.) In short, he was a mad, mad scientist. And I was called in to help his successor, Scott Lang (Paul Rudd’s character), navigate his way through the physics of shrinking. Holy guacamole Ant-man! How does one shrink the space between atoms? As James Kakalios, author of The Physics of Superheroes, puts it in a recent article on Nate Silver’s FiveThirtyEight:

We’re made of atoms, and the neighboring atoms are all touching each other. One method of changing your size that’s out: Just squeeze the atoms closer together.
…
So the other option: What determines the size of atoms anyway?
…
We can calculate this with quantum mechanics, and it turns out to be the ratio of fundamental constants: Planck’s constant and the mass of an electron and the charge of the electron and this and that. The thing that all these constants have in common is that they’re constant. They don’t change.

Wonderful. Just peachy. How am I supposed to come up with a way that will allow Ant-Man to shrink to the size of an ant, if one of the top experts in movie science magic thinks that our best bet is to somehow change fundamental constants of nature?

Enter Pinewood Studios

The flight was longer than I expected, which gave me time to think. A limo was waiting for me at the airport; I was to be taken directly to Pinewood Studios, luggage in hand and all. Once I was at my destination, I was escorted to the 3rd floor of a nondescript building, accessible only through special authorization (nice touch, Marvel). I was shown to what seemed like the main conference room, an open area with a large conference table. I expected that I would have to wait an hour before anyone showed up, so I started fiddling with my phone, making myself look busy. The next time I looked up, Paul Rudd was tapping my shoulder, dressed in sweats after what seemed like a training session for The 300.

The Meeting

Within minutes, the director (Peyton Reed), the writers, producers, VFX specialists, computer playback experts (I became friends with their supervisor, Mr. Matthew Morrissey, who went to great lengths to represent the atoms on-screen as clouds of probability, in the s, p, d, f orbital configurations you see flashing in quantum superposition behind Hank Pym at his lab) and everyone else involved with the movie was in the room. I sat at the head of the long table with Paul next to me. He asked most of the questions along with the director, but at the time I didn’t know Paul was involved with writing the script. We discussed a lot of things, but what got everyone excited was when I told them that the laws of physics as we know them may break down as we delve deeper and deeper into the quantum realm. You see, all of the other superheroes, no matter how strong and super, had powers that conformed to the laws of physics (stretching them from time to time, but never breaking them). But if someone could go to a place where the laws of physics as we know them were not yet formed, at a place where the arrow of time was broken and the fabric of space was not yet woven, the powers of such a master of the quantum realm would only be constrained by their ability to come back to the same (or similar) reality from which they departed. All the superheroes of Marvel and DC Comics combined would stand no chance against Ant-Man with a malfunctioning regulator…

The Quantum Realm

The birth of the term itself is an interesting story. Brad Winderbaum, co-producer for the movie, emailed me a couple of days after the meeting with the following request: Could I come up with a term describing Ant-Man going to the “microverse”? The term “microverse” carried legal baggage, so something fresh was needed. I offered “going nano”, “going quantum”, “going atomic”, or… “quantum realm”. I didn’t know how small the “microverse” scale was supposed to be in a writer’s mind (in a physicist’s mind it is exactly 10^{-6} meters – one thousandth of a millimeter), hence the many options. The reply was quick:

Thanks Spiros! Quantum Realm is a pretty great term.

Et voilà. Ant-Man was going to the quantum realm, a place where time and space dissolve and the only thing strong enough to anchor Scott Lang to reality is… You have to watch the movie to see what that is – it was an off-the-cuff remark I made at the meeting… At the end of the meeting, Paul, Peyton and the others thanked me and asked me if I could fly to San Francisco the next week for the first week of shooting. There, I would have met Michael Douglas and Evangeline Lilly, but I declined the generous offer. It was the week of Innoworks Academy at Caltech, an award-winning summer camp for middle school kids on the free/reduced lunch program. As the camp’s adviser, I resolved to never miss a camp as long as I was in the country and San Francisco is in the same state as Caltech. My mom would be proud of my decision (I hope), though an autograph from Mr. Douglas would have fetched me a really good hug.

The Movie

I just watched the movie yesterday (it is actually good!) and the feeling was surreal. Because I had no idea what to expect. Because I never thought that the people in that room would take what I was saying seriously enough to use it in the movie. I never got a copy of the script and during the official premiere three weeks ago, I was delivering a lecture on the future of quantum computing in a monastery in Madrid, Spain. When I found out that Kevin Feige, president of Marvel Studios, said this at a recent interview, my heart skipped several beats:

But the truth is, there is so much in Ant-Man: introducing a new hero, introducing a very important part of technology in the Marvel universe, the Pym particles. Ant-Man getting on the Avengers’ radar in this film and even – this is the weirdest part, you shouldn’t really talk about it because it won’t be apparent for years – but the whole notion of the quantum realm and the whole notion of going to places that are so out there, they are almost mind-bendingly hard to fathom. It all plays into Phase Three.

The third phase of the Marvel Cinematic Universe is about to go quantum and all I can think of is: I better start reading comic books again. But first, I have to teach a bunch of 8-12 year-old girls quantum physics through Minecraft. It is, after all, the final week of Project Scientist here at Caltech this summer and the theme is coding. With quantum computers at the near horizon, these young women need to learn how to program Asimov’s laws of quantum robotics into our benevolent quantum A.I. overlords. These young women are humanity’s new hope…

Of Supersoakers and squeezed states

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“BBs,” the lecturer said. I was sitting in the center of my row of seats, the two yards between me and the whiteboard empty. But I fancied I hadn’t heard correctly. “You know, like in BB guns?”

I had heard correctly. I nodded.

“Did you play with BB guns when you were a kid?”

I nodded again.

“I had BB guns,” the lecturer ruminated. “I had to defend myself from my brothers.”

I nodded more vigorously. My brother and I love each other, but we’ve crossed toy pistols.

“Photons are like BBs, like bullets.”

Light, the lecturer continued, behaves like BBs under certain conditions. Under other conditions, light behaves differently. Different behaviors correspond to different species of light. Some species, we can approximate with classical (nonquantum*) physics. Some species, we can’t.

Kids begged less for BB guns, in my experience, than for water guns. I grew up in Florida, where swimming season stretches from April till September. To reload a BB gun, you have to fetch spent BBs. But, toting a Supersoaker, you swim in ammunition.

Water guns brought to mind water waves, which resemble a species of classical light. If BBs resemble photons, I mused, what about Supersoaker sprays? Water balloons?

I resolved to draw as many parallels as I could between species of light and childhood weapons.

Under scrutiny, the Supersoaker analogy held little water (sorry). A Supersoaker releases water in a stream, rather than in a coherent wave. By coherent, I mean that the wave has a well-defined wavelength: The distance from the first crest to the second equals the distance from the second to the third, and so on. I can’t even identify crests in the Supersoaker photo below.

http://facstaff.gpc.edu/~pgore/PhysicalScience/Waves.html, http://www.mlive.com/living/grand-rapids/index.ssf/2009/07/happy_birthday_super_soaker_mo.html

Coherent waves vs. Supersoaker not-really-waves

Maybe Supersoaker sprays resemble incoherent light? Incoherent light is a mixture of waves of all different wavelengths. Classical physics approximates incoherent light, examples of which include sunlight. If you tease apart sunlight into coherent components, you’ll find waves with short wavelengths (such as ultraviolet rays), waves with medium (such as light we can see), and waves with long (such as microwaves). You can’t ascribe just one wavelength to incoherent light, just as I seemed unable to ascribe a wavelength to Supersoaker sprays.

But Supersoaker sprays differ from incoherent light in other respects. I’d expect triggers, for instance, to introduce nonlinearity into the spray’s dynamics. Readers who know more than I about fluid mechanics can correct me.

http://www.parentdish.com/2010/07/14/water-balloon-volley-game/

Though far-reaching and forceful, Supersoakers weigh down combatants and are difficult to hide. If you need ammunition small enough for a sneak attack, I recommend water balloons. Water balloons resemble squeezed states, which form a quantum class of light related to the Uncertainty Principle.

Werner Heisenberg proposed that, the more you know about a quantum particle’s position, the less you can know about its momentum, and vice versa. Let’s represent your uncertainty about the position by Δx and your uncertainty about the momentum by Δp. The product of these uncertainties can’t dip below some number, represented by ћ/2:

\Delta_x \Delta_p \geq \frac{\hbar}{2}.

Neither uncertainty, for example, can equal zero. Heisenberg’s proposal has evolved into more rigorous, more general forms. But the story remains familiar: The lesser the “spread in the possible values” of some property (like position), the greater the “spread in the possible values” of another property (like momentum).

Imagine plotting the possible positions along a graph’s horizontal axis and the possible momenta along the vertical. The points that could characterize our quantum system form a blob of area ћ/2. Doesn’t the blob resemble a water balloon?

Imagine squeezing a water balloon along one direction. The balloon bulges out along another. Now, imagine squeezing most of the quantum uncertainty along one direction in the diagram. You’ve depicted a squeezed state.

http://www.rp-photonics.com/squeezed_states_of_light.html

Depiction of a squeezed state

Not all childhood weapons contain water or BBs, and not all states of light contain photons.** A vacuum is a state that consists of zero photons. Classical physics suggests that the vacuum is empty and lacks energy. A sliver of energy, called zero-point energy, pervades each quantum vacuum. The Uncertainty Principle offers one reason why.

The vacuum reminds me of the silent treatment. Silence sounds empty, but it can harbor malevolence as quantum vacua harbor energy. Middle-school outcasts beware zero-point malice.

Retreating up Memory Lane, I ran out of analogies between classes of light and childhood weapons. Children play with lasers (with laser pointers and laser-tag guns), and lasers emit (approximately) coherent light. But laser light’s resemblance to laser light doesn’t count as an analogy. The class of incoherent light includes thermal states. (Non-experts, I’m about to spew jargon. If you have the energy, I recommend Googling the italicized terms. If you haven’t, feel free to skip to the next paragraph.) Physicists model much of the natural world with thermal states. To whichever readers identify childhood weapons that resemble them, I offer ten points. I offer 20 for mimicry of solitons or solitary waves, and 25 for that of parametric down-conversion or photon antibunching.

But if sunshine and Supersoakers lure you away from your laptop, I can’t object. Happy summer.

With thanks to Bassam Helou for corrections and discussions.

*Pardon my simplifying inaccuracy. Some nonquantum physics is nonclassical.

**More precisely, not all Fock states correspond to particle numbers n > 0. Alternatively: Not all states of light correspond to positive expectation values \langle \hat{n} \rangle > 0 of the particle-number operator \hat{n}.

Holography and the MERA

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The AdS/MERA correspondence has been making the rounds of the blogosphere with nice posts by Scott Aaronson and Sean Carroll, so let’s take a look at the topic here at Quantum Frontiers.

The question of how to formulate a quantum theory of gravity is a long-standing open problem in theoretical physics. Somewhat recently, an idea that has gained a lot of traction (and that Spiros has blogged about before) is emergence. This is the idea that space and time may emerge from some more fine-grained quantum objects and their interactions. If we could understand how classical spacetime emerges from an underlying quantum system, then it’s not too much of a stretch to hope that this understanding would give us insight into the full quantum nature of spacetime.

One type of emergence is exhibited in holography, which is the idea that certain (D+1)-dimensional systems with gravity are exactly equivalent to D-dimensional quantum theories without gravity. (Note that we’re calling time a dimension here. For example, you would say that on a day-to-day basis we experience D = 4 dimensions.) In this case, that extra +1 dimension and the concomitant gravitational dynamics are emergent phenomena.

A nice aspect of holography is that it is explicitly realized by the AdS/CFT correspondence. This correspondence proposes that a particular class of spacetimes—ones that asymptotically look like anti-de Sitter space, or AdS—are equivalent to states of a particular type of quantum system—a conformal field theory, or CFT. A convenient visualization is to draw the AdS spacetime as a cylinder, where time marches forward as you move up the cylinder and different slices of the cylinder correspond to snapshots of space at different instants of time. Conveniently, in this picture you can think of the corresponding CFT as living on the boundary of the cylinder, which, you should note, has one less dimension than the “bulk” inside the cylinder.

board

Even within this nice picture of holography that we get from the AdS/CFT correspondence, there is a question of how exactly do CFT, or boundary quantities map onto quantities in the AdS bulk. This is where a certain tool from quantum information theory called tensor networks has recently shown a lot of promise.

A tensor network is a way to efficiently represent certain states of a quantum system. Moreover, they have nice graphical representations which look something like this:

 mera

Beni discussed one type of tensor network in his post on holographic codes. In this post, let’s discuss the tensor network shown above, which is known as the Multiscale Entanglement Renormalization Ansatz, or MERA.

The MERA was initially developed by Guifre Vidal and Glen Evenbly as an efficient approximation to the ground state of a CFT. Roughly speaking, in the picture of a MERA above, one starts with a simple state at the centre, and as you move outward through the network, the MERA tells you how to build up a CFT state which lives on the legs at the boundary. The MERA caught the eye of Brian Swingle, who noticed that it looks an awfully lot like a discretization of a slice of the AdS cylinder shown above. As such, it wasn’t a preposterously big leap to suggest a possible “AdS/MERA correspondence.” Namely, perhaps it’s more than a simple coincidence that a MERA both encodes a CFT state and resembles a slice of AdS. Perhaps the MERA gives us the tools that are required to construct a map between the boundary and the bulk!

So, how seriously should one take the possibility of an AdS/MERA correspondence? That’s the question that my colleagues and I addressed in a recent paper. Essentially, there are several properties that a consistent holographic theory should satisfy in both the bulk and the boundary. We asked whether these properties are still simultaneously satisfied in a correspondence where the bulk and boundary are related by a MERA.

What we found was that you invariably run into inconsistencies between bulk and boundary physics, at least in the simplest construals of what an AdS/MERA correspondence might be. This doesn’t mean that there is no hope for an AdS/MERA correspondence. Rather, it says that the simplest approach will not work. For a good correspondence, you would need to augment the MERA with some additional structure, or perhaps consider different tensor networks altogether. For instance, the holographic code features a tensor network which hints at a possible bulk/boundary correspondence, and the consistency conditions that we proposed are a good list of checks for Beni and company as they work out the extent to which the code can describe holographic CFTs. Indeed, a good way to summarize how our work fits into the picture of quantum gravity alongside holography and tensors networks is by saying that it’s nice to have good signposts on the road when you don’t have a map.