Rock-paper-scissors, granite-clock-idea

I have a soft spot for lamassu. Ten-foot-tall statues of these winged bull-men guarded the entrances to ancient Assyrian palaces. Show me lamassu, or apkallu—human-shaped winged deities—or other reliefs from the Neo-Assyrian capital of Nineveh, and you’ll have trouble showing me the door.

Assyrian art fills a gallery in London’s British Museum. Lamassu flank the gallery’s entrance. Carvings fill the interior: depictions of soldiers attacking, captives trudging, and kings hunting lions. The artwork’s vastness, its endurance, and the contact with a three-thousand-year-old civilization floor me. I tore myself away as the museum closed one Sunday night.


I visited the British Museum the night before visiting Jonathan Oppenheim’s research group at University College London (UCL). Jonathan combines quantum information theory with thermodynamics. He and others co-invented thermodynamic resource theories (TRTs), which Quantum Frontiers regulars will know of. TRTs are quantum-information-theoretic models for systems that exchange energy with their environments.

Energy is conjugate to time: Hamiltonians, mathematical objects that represent energy, represent also translations through time. We measure time with clocks. Little wonder that one can study quantum clocks using a model for energy exchanges.

Mischa Woods, Ralph Silva, and Jonathan used a resource theory to design an autonomous quantum clock. “Autonomous” means that the clock contains all the parts it needs to operate, needs no periodic winding-up, etc. When might we want an autonomous clock? When building quantum devices that operate independently of classical engineers. Or when performing a quantum computation: Computers must perform logical gates at specific times.


Wolfgang Pauli and others studied quantum clocks, the authors recall. How, Pauli asked, would an ideal clock look? Its Hamiltonian, \hat{H}_{\rm C}, would have eigenstates | E \rangle. The labels E denote possible amounts of energy.

The Hamiltonian would be conjugate to a “time operator” \hat{t}. Let | \theta \rangle denote an eigenstate of \hat{t}. This “time state” would equal an even superposition over the | E \rangle’s. The clock would occupy the state | \theta \rangle at time t_\theta.

Imagine measuring the clock, to learn the time, or controlling another system with the clock. The interaction would disturb the clock, changing the clock’s state. The disturbance wouldn’t mar the clock’s timekeeping, if the clock were ideal. What would enable an ideal clock to withstand the disturbances? The ability to have any amount of energy: E must stretch from - \infty to \infty. Such clocks can’t exist.

Approximations to them can. Mischa, Ralph, and Jonathan designed a finite-size clock, then characterized how accurately the clock mimics the ideal. (Experts: The clock corresponds to a Hilbert space of finite dimensionality d. The clock begins in a Gaussian state that peaks at one time state | \theta \rangle. The finite-width Gaussian offers more stability than a clock state.)

Disturbances degrade our ability to distinguish instants by measuring the clock. Imagine gazing at a kitchen clock through blurry lenses: You couldn’t distinguish 6:00 from 5:59 or 6:01. Disturbances also hinder the clock’s ability to implement processes, such as gates in a computation, at desired instants.

Mischa & co. quantified these degradations. The errors made by the clock, they found, decay inverse-exponentially with the clock’s size: Grow the clock a little, and the errors shrink a lot.


Time has degraded the lamassu, but only a little. You can distinguish feathers in their wings and strands in their beards. People portray such artifacts as having “withstood the flow of time,” or “evaded,” or “resisted.” Such portrayals have never appealed to me. I prefer to think of the lamassu as surviving not because they clash with time, but because they harmonize with it. The prospect of harmonizing with time—whatever that means—has enticed me throughout my life. The prospect partially underlies my research into time—perhaps childishly, foolishly—I recognize if I remove my blurry lenses before gazing in the mirror.

The creation of lasting works, like lamassu, has enticed me throughout my life. I’ve scrapbooked, archived, and recorded, and tended memories as though they were Great-Grandma’s cookbook. Ancient civilizations began alluring me at age six, partially due to artifacts’ longevity. No wonder I study the second law of thermodynamics.

Yet doing theoretical physics makes no sense from another perspective. The ancient Egyptians sculpted granite, when they could afford it. Gudea, king of the ancient city-state of Lagash, immortalized himself in diorite. I fashion ideas, which lack substance. Imagine playing, rather than rock-paper-scissors, granite-diorite-idea. The idea wouldn’t stand a chance.

Would it? Because an idea lacks substance, it can manifest in many forms. Plato’s cave allegory has manifested as a story, as classroom lectures, on handwritten pages, on word processors and websites, in cartloads of novels, in the film The Matrix, in one of the four most memorable advertisements I received from colleges as a high-school junior, and elsewhere. Plato’s allegory has survived since about the fourth century BCE. King Ashurbanipal’s lion-hunt reliefs have survived for only about 200 years longer.

The lion-hunt reliefs—and lamassu—exude a grandness, a majesty that’s attracted me as their longevity has. The nature of time and the perfect clock have as much grandness. Leaving the British Museum’s Assyrian gallery at 6 PM one Sunday, I couldn’t have asked for a more fitting location, 24 hours later, than in a theoretical-physics conversation.


With thanks to Jonathan, to Álvaro Martín-Alhambra, and to Mischa for their hospitality at UCL; to Ada Cohen for the “Art history of ancient Egypt and the ancient Near East” course for which I’d been hankering for years; to my brother, for transmitting the ancient-civilizations bug; and to my parents, who fed the infection with museum visits.

Click here for a follow-up to the quantum-clock paper.


The word dominates chapter one of Richard Holmes’s book The Age of WonderHolmes writes biographies of Romantic-Era writers: Mary Wollstonecraft, Percy Shelley, and Samuel Taylor Coleridge populate his bibliography. They have cameos in Age. But their scientific counterparts star.

“Their natural-philosopher” counterparts, I should say. The word “scientist” emerged as the Romantic Era closed. Romanticism, a literary and artistic movement, flourished between the 1700s and the 1800s. Romantics championed self-expression, individuality, and emotion over convention and artificiality. Romantics wondered at, and drew inspiration from, the natural world. So, Holmes argues, did Romantic-Era natural philosophers. They explored, searched, and innovated with Wollstonecraft’s, Shelley’s, and Coleridge’s zest.

Age of Wonder

Holmes depicts Wilhelm and Caroline Herschel, a German brother and sister, discovering the planet Uranus. Humphry Davy, an amateur poet from Penzance, inventing a lamp that saved miners’ lives. Michael Faraday, a working-class Londoner, inspired by Davy’s chemistry lectures.

Joseph Banks in paradise.

So Holmes entitled chapter one.

Banks studied natural history as a young English gentleman during the 1760s. He then sailed around the world, a botanist on exploratory expeditions. The second expedition brought Banks aboard the HMS Endeavor. Captain James Cook steered the ship to Brazil, Tahiti, Australia, and New Zealand. Banks brought a few colleagues onboard. They studied the native flora, fauna, skies, and tribes.

Banks, with fellow botanist Daniel Solander, accumulated over 30,000 plant samples. Artist Sydney Parkinson drew the plants during the voyage. Parkinson’s drawings underlay 743 copper engravings that Banks commissioned upon returning to England. Banks planned to publish the engravings as the book Florilegium. He never succeeded. Two institutions executed Banks’s plan more than 200 years later.

Banks’s Florilegium crowns an exhibition at the University of California at Santa Barbara (UCSB). UCSB’s Special Research Collections will host “Botanical Illustrations and Scientific Discovery—Joseph Banks and the Exploration of the South Pacific, 1768–1771” until May 2018. The exhibition features maps of Banks’s journeys, biographical sketches of Banks and Cook, contemporary art inspired by the engravings, and the Florilegium.

online poster

The exhibition spotlights “plants that have subsequently become important ornamental plants on the UCSB campus, throughout Santa Barbara, and beyond.” One sees, roaming Santa Barbara, slivers of Banks’s paradise.

2 bouganvilleas

In Santa Barbara resides the Kavli Institute for Theoretical Physics (KITP). The KITP is hosting a program about the physics of quantum information (QI). QI scientists are congregating from across the world. Everyone visits for a few weeks or months, meeting some participants and missing others (those who have left or will arrive later). Participants attend and present tutorials, explore beyond their areas of expertise, and initiate research collaborations.

A conference capstoned the program, one week this October. Several speakers had founded subfields of physics: quantum error correction (how to fix errors that dog quantum computers), quantum computational complexity (how quickly quantum computers can solve hard problems), topological quantum computation, AdS/CFT (a parallel between certain gravitational systems and certain quantum systems), and more. Swaths of science exist because of these thinkers.


One evening that week, I visited the Joseph Banks exhibition.

Joseph Banks in paradise.

I’d thought that, by “paradise,” Holmes had meant “physical attractions”: lush flowers, vibrant colors, fresh fish, and warm sand. Another meaning occurred to me, after the conference talks, as I stood before a glass case in the library.

Joseph Banks, disembarking from the Endeavour, didn’t disembark onto just an island. He disembarked onto terra incognita. Never had he or his colleagues seen the blossoms, seed pods, or sprouts before him. Swaths of science awaited. What could the natural philosopher have craved more?

QI scientists of a certain age reminisce about the 1990s, the cowboy days of QI. When impactful theorems, protocols, and experiments abounded. When they dangled, like ripe fruit, just above your head. All you had to do was look up, reach out, and prove a pineapple.


Typical 1990s quantum-information scientist

That generation left mine few simple theorems to prove. But QI hasn’t suffered extinction. Its frontiers have advanced into other fields of science. Researchers are gaining insight into thermodynamics, quantum gravity, condensed matter, and chemistry from QI. The KITP conference highlighted connections with quantum gravity.

…in paradise.

What could a natural philosopher crave more?


Artwork commissioned by the UCSB library: “Sprawling Neobiotic Chimera (After Banks’ Florilegium),” by Rose Briccetti

Most KITP talks are recorded and released online. You can access talks from the conference here. My talk, about quantum chaos and thermalization, appears here. 

With gratitude to the KITP, and to the program organizers and the conference organizers, for the opportunity to participate. 

Standing back at Stanford

T-shirt 1

This T-shirt came to mind last September. I was standing in front of a large silver-colored table littered with wires, cylinders, and tubes. Greg Bentsen was pointing at components and explaining their functions. He works in Monika Schleier-Smith’s lab, as a PhD student, at Stanford.

Monika’s group manipulates rubidium atoms. A few thousand atoms sit in one of the cylinders. That cylinder contains another cylinder, an optical cavity, that contains the atoms. A mirror caps each of the cavity’s ends. Light in the cavity bounces off the mirrors.

Light bounces off your bathroom mirror similarly. But we can describe your bathroom’s light accurately with Maxwellian electrodynamics, a theory developed during the 1800s. We describe the cavity’s light with quantum electrodynamics (QED). Hence we call the lab’s set-up cavity QED.

The light interacts with the atoms, entangling with them. The entanglement imprints information about the atoms on the light. Suppose that light escaped from the cavity. Greg and friends could measure the light, then infer about the atoms’ quantum state.

A little light leaks through the mirrors, though most light bounces off. From leaked light, you can infer about the ensemble of atoms. You can’t infer about individual atoms. For example, consider an atom’s electrons. Each electron has a quantum property called a spin. We sometimes imagine the spin as an arrow that points upward or downward. Together, the electrons’ spins form the atom’s joint spin. You can tell, from leaked light, whether one atom’s spin points upward. But you can’t tell which atom’s spin points upward. You can’t see the atoms for the ensemble.

Monika’s team can. They’ve cut a hole in their cylinder. Light escapes the cavity through the hole. The light from the hole’s left-hand edge carries information about the leftmost atom, and so on. The team develops a photograph of the line of atoms. Imagine holding a photograph of a line of people. You can point to one person, and say, “Aha! She’s the xkcd fan.” Similarly, Greg and friends can point to one atom in their photograph and say, “Aha! That atom has an upward-pointing spin.” Monika’s team is developing single-site imaging.


Aha! She’s the xkcd fan.

Monika’s team plans to image atoms in such detail, they won’t need for light to leak through the mirrors. Light leakage creates problems, including by entangling the atoms with the world outside the cavity. Suppose you had to diminish the amount of light that leaks from a rubidium cavity. How should you proceed?

Tell the mirrors,

T-shirt 2

You should lengthen the cavity. Why? Imagine a photon, a particle of light, in the cavity. It zooms down the cavity’s length, hits a mirror, bounces off, retreats up the cavity’s length, hits the other mirror, and bounces off. The photon repeats this process until a mirror hit fails to generate a bounce. The mirror transmits the photon to the exterior; the photon leaks out. How can you reduce leaks? By preventing photons from hitting mirrors so often, by forcing the photons to zoom longer, by lengthening the cavity, by shifting the mirrors outward.

So Greg hinted, beside that silver-colored table in Monika’s lab. The hint struck a chord: I recognized the impulse to

T-shirt 3

The impulse had led me to Stanford.

Weeks earlier, I’d written my first paper about quantum chaos and information scrambling. I’d sat and read and calculated and read and sat and emailed and written. I needed to stand up, leave my cavity, and image my work from other perspectives.

Stanford physicists had written quantum-chaos papers I admired. So I visited, presented about my work, and talked. Patrick Hayden introduced me to a result that might help me apply my result to another problem. His group helped me simplify a mathematical expression. Monika reflected that a measurement scheme I’d proposed sounded not unreasonable for cavity QED.

And Greg led me to recognize the principle behind my visit: Sometimes, you have to

T-shirt 4

to move forward.

With gratitude to Greg, Monika, Patrick, and the rest of Monika’s and Patrick’s groups for their time, consideration, explanations, and feedback. With thanks to Patrick and Stanford’s Institute for Theoretical Physics for their hospitality.

Entropy Avengers

As you already know if you read my rare (but highly refined!) blog samples, I have spent a big chunk of my professorial career teaching statistical mechanics. And if you teach statistical mechanics, there is pretty much one thing you obsess about: entropy.

So you can imagine my joy of finally seeing a fully anti-entropic superhero appearing on my facebook account (physics enthusiasts out there – the project is seeking support on Kickstarter):

Apart from the plug for Assa Auerbach’s project (which, for full disclosure, I have just supported), I would like to use this as an excuse to share my lessons about entropy. With the same level of seriousness. Here they are, in order of increasing entropy.

1. Cost of entropy. Entropy is always marketed as a very palpable thing. Disorder. In class, however, it is calculated via an enumeration of the ‘microscopic states of the system’. For an atomic gas I know how to calculate the entropy (throw me at the blackboard in the middle of the night, no problem. Bosons or Fermions – anytime!) But how can the concept be applied to our practical existence? I have a proposal:

Quantify entropy by the cost (in $’s) of cleaning up the mess!

Examples can be found at all scales. For anything household-related, we should use the H_k constant. H_k=$25/hour for my housekeeper. You break a glass – it takes about 10 minutes to clean. That puts the entropy of the wreckage at $4.17. Having a birthday party takes about 2 hours to clean up: $50 entropy.

Another insight which my combined experience as professor and parent has produced:

2. Conjecture: Babies are maximally efficient topological entropy machines. If you raised a 1 year-old you know exactly what I mean. You can at least guess why maximum efficiency. But why topological? A baby sauntering through the house leaves a string of destruction behind itself. The baby is a mess-creation string-operator! If you start lagging behind, doom will emerge – hence the maximum efficiency. By the way, the only strategy viable is to undo the damage as it happens. But this blog post is about entropy, not about parenting.

In fact, this allows us to establish a conversion of entropy measured in k_B units, to its, clearly more natural, measure in dollar units. A baby eats about 1000kCal/day=4200kJ/day. To fully deal with the consequences, we need a housekeeper to visit about once a week. 4200kJ/day times 7 days=29400 kJoules. These are consumed at T=300K. So an entropy of S=Q/T~100J/K, which is also S~6 \times 10^{24} (Q/k_B T) in dimensionless units, converts to S~$120, which is the cost of our weekly housekeeper visit. This gives a value of $ 10^{-23} per entropy of a two-level system. Quite a reasonable bang for the buck, don’t you think?

3. My conjecture (2) fails. The second law of thermodynamics is an inequality. Entropy \geq Q/T. Why does the conjecture fail? Babies are not ‘maximal’. Consider presidents. Consider the mess that the government can make. It is at the scale of trillions per year. $ 10^{12}. Using the rigorous conversion rule established above, this corresponds to 10^{35} two-level systems. Which happens to quite precisely match the combined number of electrons present in the human bodies of all our military personnel. But the mess, however, is created by very few individuals.

Given the large amounts of taxpayer money we dish out to deal with entropy in the world, Auerbach’s book is bound to make a big impact. In fact, maybe Max the demon would one day be nominated for the national medal of freedom, or at least be inducted into the National Academy of Sciences.

Glass beads and weak-measurement schemes

Richard Feynman fiddled with electronics in a home laboratory, growing up. I fiddled with arts and crafts.1 I glued popsicle sticks, painted plaques, braided yarn, and designed greeting cards. Of the supplies in my family’s crafts box, I adored the beads most. Of the beads, I favored the glass ones.

I would pour them on the carpet, some weekend afternoons. I’d inherited a hodgepodge: The beads’ sizes, colors, shapes, trimmings, and craftsmanship varied. No property divided the beads into families whose members looked like they belonged together. But divide the beads I tried. I might classify them by color, then subdivide classes by shape. The color and shape groupings precluded me from grouping by size. But, by loosening my original classification and combining members from two classes, I might incorporate trimmings into the categorization. I’d push my classification scheme as far as I could. Then, I’d rake the beads together and reorganize them according to different principles.

Why have I pursued theoretical physics? many people ask. I have many answers. They include “Because I adored organizing craft supplies at age eight.” I craft and organize ideas.


I’ve blogged about the out-of-time-ordered correlator (OTOC), a signature of how quantum information spreads throughout a many-particle system. Experimentalists want to measure the OTOC, to learn how information spreads. But measuring the OTOC requires tight control over many quantum particles.

I proposed a scheme for measuring the OTOC, with help from Chapman University physicist Justin Dressel. The scheme involves weak measurements. Weak measurements barely disturb the systems measured. (Most measurements of quantum systems disturb the measured systems. So intuited Werner Heisenberg when formulating his uncertainty principle.)

I had little hope for the weak-measurement scheme’s practicality. Consider the stereotypical experimentalist’s response to a stereotypical experimental proposal by a theorist: Oh, sure, we can implement that—in thirty years. Maybe. If the pace of technological development doubles. I expected to file the weak-measurement proposal in the “unfeasible” category.

But experimentalists started collaring me. The scheme sounds reasonable, they said. How many trials would one have to perform? Did the proposal require ancillas, helper systems used to control the measured system? Must each ancilla influence the whole measured system, or could an ancilla interact with just one particle? How did this proposal compare with alternatives?

I met with a cavity-QED2 experimentalist and a cold-atoms expert. I talked with postdocs over skype, with heads of labs at Caltech, with grad students in Taiwan, and with John Preskill in his office. I questioned an NMR3 experimentalist over lunch and fielded superconducting-qubit4 questions in the sunshine. I read papers, reread papers, and powwowed with Justin.

I wouldn’t have managed half so well without Justin and without Brian Swingle. Brian and coauthors proposed the first OTOC-measurement scheme. He reached out after finding my first OTOC paper.

According to that paper, the OTOC is a moment of a quasiprobability.5 How does that quasiprobability look, we wondered? How does it behave? What properties does it have? Our answers appear in a paper we released with Justin this month. We calculate the quasiprobability in two examples, prove properties of the quasiprobability, and argue that the OTOC motivates generalizations of quasiprobability theory. We also enhance the weak-measurement scheme and analyze it.

Amidst that analysis, in a 10 x 6 table, we classify glass beads.


We inventoried our experimental conversations and distilled them. We culled measurement-scheme features analogous to bead size, color, and shape. Each property labels a row in the table. Each measurement scheme labels a column. Each scheme has, I learned, gold flecks and dents, hues and mottling, an angle at which it catches the light.

I’ve kept most of the glass beads that fascinated me at age eight. Some of the beads have dispersed to necklaces, picture frames, and eyeglass leashes. I moved the remnants, a few years ago, to a compartmentalized box. Doesn’t it resemble the table?


That’s why I work at the IQIM.


1I fiddled in a home laboratory, too, in a garage. But I lived across the street from that garage. I lived two rooms from an arts-and-crafts box.

2Cavity QED consists of light interacting with atoms in a box.

3Lots of nuclei manipulated with magnetic fields. “NMR” stands for “nuclear magnetic resonance.” MRI machines, used to scan brains, rely on NMR.

4Superconducting circuits are tiny, cold quantum circuits.

5A quasiprobability resembles a probability but behaves more oddly: Probabilities range between zero and one; quasiprobabilities can dip below zero. Think of a moment as like an average.

With thanks to all who questioned me; to all who answered questions of mine; to my wonderful coauthors; and to my parents, who stocked the crafts box.

The entangled fabric of space

We live in the information revolution. We translate everything into vast sequences of ones and zeroes. From our personal email to our work documents, from our heart rates to our credit rates, from our preferred movies to our movie preferences, all things information are represented using this minimal {0,1} alphabet which our digital helpers “understand” and process. Many of us physicists are now taking this information revolution at heart and embracing the “It from qubit” motto. Our dream: to understand space, time and gravity as emergent features in a world made of information – quantum information.

Over the past two years, I have been obsessively trying to understand this profound perspective more rigorously. Recently, John Preskill and I have taken a further step in this direction in the recent paper: quantum code properties from holographic geometries. In it, we make progress in interpreting features of the holographic approach to quantum gravity in the terms of quantum information constructs. 

In this post I would like to present some context for this work through analogies which hopefully help intuitively convey the general ideas. While still containing some technical content, this post is not likely to satisfy those readers seeking a precise in-depth presentation. To you I can only recommend the masterfully delivered lecture notes on gravity and entanglement by Mark Van Raamsdonk.  

Entanglement as a cat’s cradle


A cat’s cradle serves as a crude metaphor for quantum mechanical entanglement. The full image provides a complete description of the string and how it is laced in a stable configuration around the two hands. However, this lacing does not describe a stable configuration of half the string on one hand. The string would become disentangled and fall if we were to suddenly remove one of the hands or cut through the middle.

Of all the concepts needed to explain emergent spacetime, maybe the most difficult is that of quantum entanglement. While the word seems to convey some kind of string wound up in a complicated way, it is actually a quality which may describe information in quantum mechanical systems. In particular, it applies to a system for which we have a complete description as a whole, but are only capable of describing certain statistical properties of its parts. In other words, our knowledge of the whole loses predictive power when we are only concerned with the parts. Entanglement entropy is a measure of information which quantifies this.

While our metaphor for entanglement is quite crude, it will serve the purpose of this post. Namely, to illustrate one of the driving premises for the holographic approach to quantum gravity, that the very structure of spacetime is emergent and built up from entanglement entropy.

Knit and crochet your way into the manifolds

But let us bring back our metaphors and try to convey the content of this identification. For this, we resort to the unlikely worlds of knitting and crochet. Indeed, by a planned combination of individual loops and stitches, these traditional crafts are capable of approximating any kind of surface (2D Riemannian surface would be the technical term).

Here I have presented some examples with uniform curvature R: flat in green, positive curvature (ball) in yellow and negative curvature (coral reef) in purple. While actual practitioners may be more interested in getting the shape right on hats and socks for loved ones, for us the point is that if we take a step back, these objects built of simple loops, hooks and stitches could end up looking a lot like the smooth surfaces that a physicist might like to use to describe 2D space. This is cute, but can we push this metaphor even further?

Well, first of all, although the pictures above are only representing 2D surfaces, we can expect that a similar approach should allow approximating 3D and even higher dimensional objects (again the technical term is Riemannian manifolds). It would just make things much harder to present in a picture. These woolen structures are, in fact, quite reminiscent of tensor networks, a modern mathematical construct widely used in the field of quantum information. There too, we combine basic building blocks (tensors) through simple operations (tensor index contraction) to build a more complex composite object. In the tensor network world, the structure of the network (how its nodes are connected to other nodes) generically defines the entanglement structure of the resulting object.


This regular tensor network layout was used to describe hyperbolic space which is similar to the purple crochet. However, they apriori look quite dissimilar due to the use of the Poincaré disk model where tensors further from the center look smaller. Another difference is that the high degree of regularity is achieved at the expense of having very few tensors per curvature radius (as compared to its purple crochet cousin). However, planarity and regularity don’t seem to be essential so the crochet probably provides a better intuitive picture.

Roughly speaking, tensor networks are ingenious ways of encoding (quantum) inputs into (quantum) outputs. In particular, if you enter some input at the boundary of your tensor network, the tensors do the work of processing that information throughout the network so that if you ask for an output at any one of the nodes in the bulk of the tensor network, you get the right encoded answer. In other words, the information we input into the tensor network begins its journey at the dangling edges found at the boundary of the network and travels through the bulk edges by exploiting them as information bridges between the nodes of the network.

In the figure representing the cat’s cradle, these dangling input edges can be though of as the fingers holding the wool. Now, if we partition these edges into two disjoint sets (say, the fingers on the left hand and the fingers on the right hand, respectively), there will be some amount of entanglement between them. How much? In general, we cannot say, but under certain assumptions we find that it is proportional to the minimum cut through the network. Imagine you had an incredible number of fingers holding your wool structure. Now separate these fingers arbitrarily into two subsets L and R (we may call them left hand and right hand, although there is nothing right or left handy about them). By pulling left hand and right hand apart, the wool might stretch until at some point it breaks. How many threads will break? Well, the question is analogous to the entanglement one. We might expect, however, that a minimal number of threads break such that each hand can go its own way. This is what we call the minimal cut. In tensor networks, entanglement entropy is always bounded above by such a minimal cut and it has been confirmed that under certain conditions entanglement also reaches, or approximates, this bound. In this respect, our wool analogy seems to be working out.


Holography, in the context of black holes, was sparked by a profound observation of Jacob Bekenstein and Stephen Hawking, which identified the surface area of a black hole horizon (in Planck units) with its entropy, or information content:BHentropyF1

S_{BH} = \frac{k A_{BH}}{4\ell_p^2} .

Here, S_{BH} is the entropy associated to the black hole, A_{BH} is its horizon area, \ell_p is the Planck length and k is Boltzmann’s constant.
Why is this equation such a big deal? Well, there are many reasons, but let me emphasize one. For theoretical physicists, it is common to get rid of physical units by relating them through universal constants. For example, the theory of special relativity allows us to identify units of distance with units of time through the equation d=ct using the speed of light c. General relativity further allows us to identify mass and energy through the famous E=mc^2. By considering the Bekenstein-Hawking entropy, units of area are being swept away altogether! They are being identified with dimensionless units of information (one square meter is roughly 1.4\times10^{69} bits according to the Bousso bound).

Initially, the identification of area and information was proposed to reconcile black holes with the laws of thermodynamics. However, this has turned out to be the main hint leading to the holographic principle, wherein states that describe a certain volume of space in a theory of quantum gravity can also be thought of as being represented at the lower dimensional boundary of the given volume. This idea, put forth by Gerard ‘t Hooft, was later given a more precise interpretation by Leonard Susskind and subsequently by Juan Maldacena through the celebrated AdS/CFT correspondence. I will not dwell in the details of the AdS/CFT correspondence as I am not an expert myself. However, this correspondence gave S. Ryu and T. Takayanagi  (RT) a setting to vastly generalize the identification of area as an information quantity. They proposed identifying the area of minimal surfaces on the bulk (remember the minimal cut?) with entanglement entropy in the boundary theory.

Roughly speaking, if we were to split the boundary into two regions, left L and right R it should be possible to also partition the bulk in a way that each piece of the bulk has either L or R in its boundary. Ryu and Takayanagi proposed that the area of the smallest surface \chi_R=\chi_L which splits the bulk in this way would be proportional to the entanglement entropy between the two parts

S_L = S_R = \frac{|\chi_L|}{4G} =\frac{|\chi_R|}{4G}.

It turns out that some quantum field theory states admit such a geometric interpretation. Many high energy theory colleagues have ideas about when this is possible and what are the necessary conditions. By far the best studied setting for this holographic duality is AdS/CFT, where Ryu and Takayanagi first checked their proposal. Here, the entanglement features of  the lowest energy state of a conformal field theory are matched to surfaces in a hyperbolic space (like the purple crochet and the tensor network presented). However, other geometries have been shown to match the RT prediction with respect to the entanglement properties of different states. The key point here is that the boundary states do not have any geometry per se. They just manifest different amounts of entanglement when partitioned in different ways.


The holographic program suggests that bulk geometry emerges from the entanglement properties of the boundary state. Spacetime materializes from the information structure of the boundary instead of being a fundamental structure as in general relativity. Am I saying that we should strip everything physical, including space in favor of ones and zeros? Well, first of all, it is not just me who is pushing this approach. Secondly, no one is claiming that we should start making all our physical reasoning in terms of ones and zeros.

Let me give an example. We know that the sea is composed mostly of water molecules. The observation of waves that travel, superpose and break can be labeled as an emergent phenomenon. However, to a surfer, a wave is much more real than the water molecules composing it and the fact that it is emergent is of no practical consequence when trying to predict where a wave will break. A proficient physicist, armed with tools from statistical mechanics (there are more than 10^{25} molecules per liter), could probably derive a macroscopic model for waves from the microscopic theory of particles. In the process of learning what the surfer already understood, he would identify elements of the  microscopic theory which become irrelevant for such questions. Such details could be whether the sea has an odd or even number of molecules or the presence of a few fish.

In the case of holography, each square meter corresponds to 1.4\times10^{69} bits of entanglement. We don’t even have words to describe anything close to this outrageously large exponent which leaves plenty of room for emergence. Even taking all the information on the internet – estimated at 10^{22} bits (10 zettabits) – we can’t even match the area equivalent of the smallest known particle. The fact that there are so many orders of magnitude makes it difficult to extrapolate our understanding of the geometric domain to the information domain and vice versa. This is precisely the realm where techniques such as those from statistical mechanics successfully get rid of irrelevant details.

High energy theorists and people with a background in general relativity tend to picture things in a continuum language. For example, part of their daily butter are Riemannian or Lorentzian manifolds which are respectively used to describe space and spacetime. In contrast, most of information theory is usually applied to deal with discrete elements such as bits, elementary circuit gates, etc. Nevertheless, I believe it is fruitful to straddle this cultural divide to the benefit of both parties. In a way, the convergence we are seeking is analogous to the one achieved by the kinetic theory of gases, which allowed the unification of thermodynamics with classical mechanics.

So what did we do?

The remarkable success of the geometric RT prediction to different bulk geometries such as the BTZ black holes and the generality of the entanglement result for its random tensor network cousins emboldened us to take the RT prescription beyond its usual domain of application. We considered applying it to arbitrary Riemannian manifolds that are space-like and that can be approximated by a smoothly knit fabric.

Furthermore, we went on to consider the implications that such assumptions would have when the corresponding geometries are interpreted as error-correcting codes. In fact, our work elaborates on the perspective of A. Almheiri, X. Dong and D. Harlow (ADH) where quantum error-correcting code properties of AdS/CFT were laid out; it is hard to overemphasize the influence of this work. Our work considers general geometries and identifies properties a code associated to a specific holographic geometry should satisfy.

In the cat cradle/fabric metaphor for holography, the fingers at the boundary constitute the boundary theory without gravity and the resulted fabric represents a bulk geometry in the corresponding bulk gravitational theory. Bulk observables may be represented in different ways on the boundary, but not arbitrarily. This raises the question of which parts of the bulk correspond to which parts of the boundary. In general, there is not a one to one mapping. However, if we partition the boundary in two parts L and R, we expect to be able to split the bulk into two corresponding regions  {\mathcal E}[L]  and  {\mathcal E}[R]. This is the content of the entanglement wedge hypothesis, which is our other main assumption.  In our metaphor, one could imagine that we pull the left fingers up and the right fingers down (taking care not to get hurt). At some point, the fabric breaks through \chi_R into two pieces. In the setting we are concerned with, these pieces maintain part of the original structure, which tells us which bulk information was available in one piece of the boundary and which part was available in the other.

Although we do not produce new explicit examples of such codes, we worked our way towards developing a language which translates between the holographic/geometric perspective and the coding theory perspective. We specifically build upon the language of operator algebra quantum error correction (OAQEC) which allows individually focusing on different parts of the logical message. In doing so we identified several coding theoretic bounds and quantities, some of which we found to be applicable beyond the context of holography. A particularly noteworthy one is a strengthening of the quantum Singleton bound, which defines a trade-off between how much logical information can be packed in a code, how much physical space is used for encoding this information and how well-protected the information is from erasures.

One of the central observations of ADH highlights how quantum codes have properties from both classical error-correcting codes and secret sharing schemes. On the one hand, logical encoded information should be protected from loss of small parts of the carrier, a property quantified by the code distance. On the other hand, the logical encoded information should not become accessible until a sufficiently large part of the carrier is available to us. This is quantified by the threshold of a corresponding secret sharing scheme. We call this quantity price as it identifies how much of the carrier we would need before someone could reconstruct the message faithfully. In general, it is hard to balance these two competing requirements; a statement which can be made rigorous. This kind of complementarity has long been recognized in quantum cryptography. However, we found that according to holographic predictions, codes admitting a geometric interpretation achieve a remarkable optimality in the trade-off between these features.

Our exploration of alternative geometries is rewarded by the following guidelines


In uberholography, bulk observables are accessible in a Cantor type fractal shaped subregion of the boundary. This is illustrated on the Poincare disc presentation of negatively curved bulk.

  • Hyperbolic geometries predict a fixed polynomial scaling for code distance. This is illustrated by a feature we call uberholography. We use this name because there is an excess of holography wherein bulk observables can be represented on intricate subsets of the boundary which have fractal dimension even smaller than the boundary itself.
  • Hyperbolic geometries suggest the possibility of decoding procedures which are local on the boundary geometry. This property may be connected to the locality of the corresponding boundary field theory.
  • Flat and positive curvature geometries may lead to codes with better parameters in terms of distance and rates (ratio of logical information to physical information). A hemisphere reaches optimum parameters, saturating coding bounds.


    Seven iterations of a ternary Cantor set (dark line) on the unit interval. Each iteration is obtained by punching holes from the previous one and the set obtained in the limit is a fractal.

Current day quantum computers are far from the number of qubits required to invoke an emergent geometry. Nevertheless, it is exhilarating to take a step back and consider how the properties of the codes, and information in general, may be interpreted geometrically. On the other hand, I find that the quantum code language we adapt to the context of holography might eventually serve as a useful tool in distinguishing which boundary features are relevant or irrelevant for the emergent properties of the holographic dual. Ours is but one contribution in a very active field. However, the one thing I am certain about is that these are exciting times to be doing physics.

Local operations and Chinese communications

The workshop spotlighted entanglement. It began in Shanghai, paused as participants hopped the Taiwan Strait, and resumed in Taipei. We discussed quantum operations and chaos, thermodynamics and field theory.1 I planned to return from Taipei to Shanghai to Los Angeles.

Quantum thermodynamicist Nelly Ng and I drove to the Taipei airport early. News from Air China curtailed our self-congratulations: China’s military was running an operation near Shanghai. Commercial planes couldn’t land. I’d miss my flight to LA.


Two quantum thermodynamicists in Shanghai

An operation?

Quantum information theorists use a mindset called operationalism. We envision experimentalists in separate labs. Call the experimentalists Alice, Bob, and Eve (ABE). We tell stories about ABE to formulate and analyze problems. Which quantum states do ABE prepare? How do ABE evolve, or manipulate, the states? Which measurements do ABE perform? Do they communicate about the measurements’ outcomes?

Operationalism concretizes ideas. The outlook checks us from drifting into philosophy and into abstractions difficult to apply physics tools to.2 Operationalism infuses our language, our framing of problems, and our mathematical proofs.

Experimentalists can perform some operations more easily than others. Suppose that Alice controls the magnets, lasers, and photodetectors in her lab; Bob controls the equipment in his; and Eve controls the equipment in hers. Each experimentalist can perform local operations (LO). Suppose that Alice, Bob, and Eve can talk on the phone and send emails. They exchange classical communications (CC).

You can’t generate entanglement using LOCC. Entanglement consists of strong correlations that quantum systems can share and that classical systems can’t. A quantum system in Alice’s lab can hold more information about a quantum system of Bob’s than any classical system could. We must create and control entanglement to operate quantum computers. Creating and controlling entanglement poses challenges. Hence quantum information scientists often model easy-to-perform operations with LOCC.

Suppose that some experimentalist Charlie loans entangled quantum systems to Alice, Bob, and Eve. How efficiently can ABE compute some quantity, exchange quantum messages, or perform other information-processing tasks, using that entanglement? Such questions underlie quantum information theory.


Taipei’s night market. Or Caltech’s neighborhood?

Local operations.

Nelly and I performed those, trying to finagle me to LA. I inquired at Air China’s check-in desk in English. Nelly inquired in Mandarin. An employee smiled sadly at each of us.

We branched out into classical communications. I called Expedia (“No, I do not want to fly to Manila”), United Airlines (“No flights for two days?”), my credit-card company, Air China’s American reservations office, Air China’s Chinese reservations office, and Air China’s Taipei reservations office. I called AT&T to ascertain why I couldn’t reach Air China (“Yes, please connect me to the airline. Could you tell me the number first? I’ll need to dial it after you connect me and the call is then dropped”).

As I called, Nelly emailed. She alerted Bob, aka Janet (Ling-Yan) Hung, who hosted half the workshop at Fudan University in Shanghai. Nelly emailed Eve, aka Feng-Li Lin, who hosted half the workshop at National Taiwan University in Taipei. Janet twiddled the magnets in her lab (investigated travel funding), and Feng-Li cooled a refrigerator in his.

ABE can process information only so efficiently, using LOCC. The time crept from 1:00 PM to 3:30.


Nelly Ng uses classical communications.

What could we have accomplished with quantum communication? Using LOCC, Alice can manipulate quantum states (like an electron’s orientation) in her lab. She can send nonquantum messages (like “My flight is delayed”) to Bob. She can’t send quantum information (like an electron’s orientation).

Alice and Bob can ape quantum communication, given entanglement. Suppose that Charlie strongly correlates two electrons. Suppose that Charlie gives Alice one electron and gives Bob the other. Alice can send one qubit–one unit of quantum information–to Bob. We call that sending quantum teleportation.

Suppose that air-traffic control had loaned entanglement to Janet, Feng-Li, and me. Could we have finagled me to LA quickly?

Quantum teleportation differs from human teleportation.


We didn’t need teleportation. Feng-Li arranged for me to visit Taiwan’s National Center for Theoretical Sciences (NCTS) for two days. Air China agreed to return me to Shanghai afterward. United would fly me to LA, thanks to help from Janet. Nelly rescued my luggage from leaving on the wrong flight.

Would I rather have teleported? I would have avoided a bushel of stress. But I wouldn’t have learned from Janet about Chinese science funding, wouldn’t have heard Feng-Li’s views about gravitational waves, wouldn’t have glimpsed Taiwanese countryside flitting past the train we rode to the NCTS.

According to some metrics, classical resources outperform quantum.


At Taiwan’s National Center for Theoretical Sciences

The workshop organizers have generously released videos of the lectures. My lecture about quantum chaos and fluctuation relations appears here and here. More talks appear here.

With gratitude to Janet Hung, Feng-Li Lin, and Nelly Ng; to Fudan University, National Taiwan University, and Taiwan’s National Center for Theoretical Sciences for their hospitality; and to Xiao Yu for administrative support.

Glossary and other clarifications:

1Field theory describes subatomic particles and light.

2Physics and philosophy enrich each other. But I haven’t trained in philosophy. I benefit from differentiating physics problems that I’ve equipped to solve from philosophy problems that I haven’t.