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 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.

Here’s one way to get out of a black hole!

Two weeks ago I attended an exciting workshop at Stanford, organized by the It from Qubit collaboration, which I covered enthusiastically on Twitter. Many of the talks at the workshop provided fodder for possible blog posts, but one in particular especially struck my fancy. In explaining how to recover information that has fallen into a black hole (under just the right conditions), Juan Maldacena offered a new perspective on a problem that has worried me for many years. I am eagerly awaiting Juan’s paper, with Douglas Stanford and Zhenbin Yang, which will provide more details.


My cell-phone photo of Juan Maldacena lecturing at Stanford, 22 March 2017.

Almost 10 years ago I visited the Perimeter Institute to attend a conference, and by chance was assigned an office shared with Patrick Hayden. Patrick was a professor at McGill at that time, but I knew him well from his years at Caltech as a Sherman Fairchild Prize Fellow, and deeply respected him. Our proximity that week ignited a collaboration which turned out to be one of the most satisfying of my career.

To my surprise, Patrick revealed he had been thinking about  black holes, a long-time passion of mine but not previously a research interest of his, and that he had already arrived at a startling insight which would be central to the paper we later wrote together. Patrick wondered what would happen if Alice possessed a black hole which happened to be highly entangled with a quantum computer held by Bob. He imagined Alice throwing a qubit into the black hole, after which Bob would collect the black hole’s Hawking radiation and feed it into his quantum computer for processing. Drawing on his knowledge about quantum communication through noisy channels, Patrick argued that  Bob would only need to grab a few qubits from the radiation in order to salvage Alice’s qubit successfully by doing an appropriate quantum computation.


Alice tosses a qubit into a black hole, which is entangled with Bob’s quantum computer. Bob grabs some Hawking radiation, then does a quantum computation to decode Alice’s qubit.

This idea got my adrenaline pumping, stirring a vigorous dialogue. Patrick had initially assumed that the subsystem of the black hole ejected in the Hawking radiation had been randomly chosen, but we eventually decided (based on a simple picture of the quantum computation performed by the black hole) that it should take a time scaling like M log M (where M is the black hole mass expressed in Planck units) for Alice’s qubit to get scrambled up with the rest of her black hole. Only after this scrambling time would her qubit leak out in the Hawking radiation. This time is actually shockingly short, about a millisecond for a solar mass black hole. The best previous estimate for how long it would take for Alice’s qubit to emerge (scaling like M3), had been about 1067 years.

This short time scale aroused memories of discussions with Lenny Susskind back in 1993, vividly recreated in Lenny’s engaging book The Black Hole War. Because of the black hole’s peculiar geometry, it seemed conceivable that Bob could distill a copy of Alice’s qubit from the Hawking radiation and then leap into the black hole, joining Alice, who could then toss her copy of the qubit to Bob. It disturbed me that Bob would then hold two perfect copies of Alice’s qubit; I was a quantum information novice at the time, but I knew enough to realize that making a perfect clone of a qubit would violate the rules of quantum mechanics. I proposed to Lenny a possible resolution of this “cloning puzzle”: If Bob has to wait outside the black hole for too long in order to distill Alice’s qubit, then when he finally jumps in it may be too late for Alice’s qubit to catch up to Bob inside the black hole before Bob is destroyed by the powerful gravitational forces inside. Revisiting that scenario, I realized that the scrambling time M log M, though short, was just barely long enough for the story to be self-consistent. It was gratifying that things seemed to fit together so nicely, as though a deep truth were being affirmed.


If Bob decodes the Hawking radiation and then jumps into the black hole, can he acquire two identical copies of Alice’s qubit?

Patrick and I viewed our paper as a welcome opportunity to draw the quantum information and quantum gravity communities closer together, and we wrote it with both audiences in mind. We had fun writing it, adding rhetorical flourishes which we hoped would draw in readers who might otherwise be put off by unfamiliar ideas and terminology.

In their recent work, Juan and his collaborators propose a different way to think about the problem. They stripped down our Hawking radiation decoding scenario to a model so simple that it can be analyzed quite explicitly, yielding a pleasing result. What had worried me so much was that there seemed to be two copies of the same qubit, one carried into the black hole by Alice and the other residing outside the black hole in the Hawking radiation. I was alarmed by the prospect of a rendezvous of the two copies. Maldacena et al. argue that my concern was based on a misconception. There is just one copy, either inside the black hole or outside, but not both. In effect, as Bob extracts his copy of the qubit on the outside, he destroys Alice’s copy on the inside!

To reach this conclusion, several ideas are invoked. First, we analyze the problem in the case where we understand quantum gravity best, the case of a negatively curved spacetime called anti-de Sitter space.  In effect, this trick allows us to trap a black hole inside a bottle, which is very advantageous because we can study the physics of the black hole by considering what happens on the walls of the bottle. Second, we envision Bob’s quantum computer as another black hole which is entangled with Alice’s black hole. When two black holes in anti-de Sitter space are entangled, the resulting geometry has a “wormhole” which connects together the interiors of the two black holes. Third, we chose the entangled pair of black holes to be in a very special quantum state, called the “thermofield double” state. This just means that the wormhole connecting the black holes is as short as possible. Fourth, to make the analysis even simpler, we suppose there is just one spatial dimension, which makes it easier to draw a picture of the spacetime. Now each wall of the bottle is just a point in space, with the left wall lying outside Bob’s side of the wormhole, and the right wall lying outside Alice’s side.

An important property of the wormhole is that it is not traversable. That is, when Alice throws her qubit into her black hole and it enters her end of the wormhole, the qubit cannot emerge from the other end. Instead it is stuck inside, unable to get out on either Alice’s side or Bob’s side. Most ways of manipulating the black holes from the outside would just make the wormhole longer and exacerbate the situation, but in a clever recent paper Ping Gao, Daniel Jafferis, and Aron Wall pointed out an exception. We can imagine a quantum wire connecting the left wall and right wall, which simulates a process in which Bob extracts a small amount of Hawking radiation from the right wall (that is, from Alice’s black hole), and carefully deposits it on the left wall (inserting it into Bob’s quantum computer). Gao, Jafferis, and Wall find that this procedure, by altering the trajectories of Alice’s and Bob’s walls, can actually make the wormhole traversable!


(a) A nontraversable wormhole. Alice’s qubit, thrown into the black hole, never reaches Bob. (b) Stealing some Hawking radiation from Alice’s side and inserting it on Bob’s side makes the wormhole traversable. Now Alice’s qubit reaches Bob, who can easily “decode” it.

This picture gives us a beautiful geometric interpretation of the decoding protocol that Patrick and I had described. It is the interaction between Alice’s wall and Bob’s wall that brings Alice’s qubit within Bob’s grasp. By allowing Alice’s qubit to reach Bob at the other end of the wormhole, that interaction suffices to perform Bob’s decoding task, which is especially easy in this case because Bob’s quantum computer was connected to Alice’s black hole by a short wormhole when she threw her qubit inside.


If, after a delay, Bob’s jumps into the black hole, he might find Alice’s qubit inside. But if he does, that qubit cannot be decoded by Bob’s quantum computer. Bob has no way to attain two copies of the qubit.

And what if Bob conducts his daring experiment, in which he decodes Alice’s qubit while still outside the black hole, and then jumps into the black hole to check whether the same qubit is also still inside? The above spacetime diagram contrasts two possible outcomes of Bob’s experiment. After entering the black hole, Alice might throw her qubit toward Bob so he can catch it inside the black hole. But if she does, then the qubit never reaches Bob’s quantum computer, and he won’t be able to decode it from the outside. On the other hand, Alice might allow her qubit to reach Bob’s quantum computer at the other end of the (now traversable) wormhole. But if she does, Bob won’t find the qubit when he enters the black hole. Either way, there is just one copy of the qubit, and no way to clone it. I shouldn’t have been so worried!

Granted, we have only described what happens in an oversimplified model of a black hole, but the lessons learned may be more broadly applicable. The case for broader applicability rests on a highly speculative idea, what Maldacena and Susskind called the ER=EPR conjecture, which I wrote about in this earlier blog post. One consequence of the conjecture is that a black hole highly entangled with a quantum computer is equivalent, after a transformation acting only on the computer, to two black holes connected by a short wormhole (though it might be difficult to actually execute that transformation). The insights of Gao-Jafferis-Wall and Maldacena-Stanford-Yang, together with the ER=EPR viewpoint, indicate that we don’t have to worry about the same quantum information being in two places at once. Quantum mechanics can survive the attack of the clones. Whew!

Thanks to Juan, Douglas, and Lenny for ongoing discussions and correspondence which have helped me to understand their ideas (including a lucid explanation from Douglas at our Caltech group meeting last Wednesday). This story is still unfolding and there will be more to say. These are exciting times!

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.


My brother and I played the video game Sonic the Hedgehog on a Sega Dreamcast. The hero has spiky electric-blue fur and can run at the speed of sound.1 One of us, then the other, would battle monsters. Monster number one oozes onto a dark city street as an aquamarine puddle. The puddle spreads, then surges upward to form limbs and claws.2 The limbs splatter when Sonic attacks: Aqua globs rain onto the street.


The monster’s master, Dr. Eggman, has ginger mustachios and a body redolent of his name. He scoffs as the heroes congratulate themselves.

“Fools!” he cries, the pauses in his speech heightening the drama. “[That monster is] CHAOS…the GOD…of DE-STRUC-TION!” His cackle could put a Disney villain to shame.

Dr. Eggman’s outburst comes to mind when anyone asks what topic I’m working on.

“Chaos! And the flow of time, quantum theory, and the loss of information.”


Alexei Kitaev, a Caltech physicist, hooked me on chaos. I TAed his spring-2016 course. The registrar calls the course Ph 219c: Quantum Computation. I call the course Topics that Interest Alexei Kitaev.

“What do you plan to cover?” I asked at the end of winter term.

Topological quantum computation, Alexei replied. How you simulate Hamiltonians with quantum circuits. Or maybe…well, he was thinking of discussing black holes, information, and chaos.

If I’d had a tail, it would have wagged.

“What would you say about black holes?” I asked.


Sonic’s best friend, Tails the fox.

I fwumped down on the couch in Alexei’s office, and Alexei walked to his whiteboard. Scientists first noticed chaos in classical systems. Consider a double pendulum—a pendulum that hangs from the bottom of a pendulum that hangs from, say, a clock face. Imagine pulling the bottom pendulum far to one side, then releasing. The double pendulum will swing, bend, and loop-the-loop like a trapeze artist. Imagine freezing the trapeze artist after an amount t of time.

What if you pulled another double pendulum a hair’s breadth less far? You could let the pendulum swing, wait for a time t, and freeze this pendulum. This pendulum would probably lie far from its brother. This pendulum would probably have been moving with a different speed than its brother, in a different direction, just before the freeze. The double pendulum’s motion changes loads if the initial conditions change slightly. This sensitivity to initial conditions characterizes classical chaos.

A mathematical object F(t) reflects quantum systems’ sensitivities to initial conditions. [Experts: F(t) can evolve as an exponential governed by a Lyapunov-type exponent: \sim 1 - ({\rm const.})e^{\lambda_{\rm L} t}.] F(t) encodes a hypothetical process that snakes back and forth through time. This snaking earned F(t) the name “the out-of-time-ordered correlator” (OTOC). The snaking prevents experimentalists from measuring quantum systems’ OTOCs easily. But experimentalists are trying, because F(t) reveals how quantum information spreads via entanglement. Such entanglement distinguishes black holes, cold atoms, and specially prepared light from everyday, classical systems.

Alexei illustrated, on his whiteboard, the sensitivity to initial conditions.

“In case you’re taking votes about what to cover this spring,” I said, “I vote for chaos.”

We covered chaos. A guest attended one lecture: Beni Yoshida, a former IQIM postdoc. Beni and colleagues had devised quantum error-correcting codes for black holes.3 Beni’s foray into black-hole physics had led him to F(t). He’d written an OTOC paper that Alexei presented about. Beni presented about a follow-up paper. If I’d had another tail, it would have wagged.


Sonic’s friend has two tails.

Alexei’s course ended. My research shifted to many-body localization (MBL), a quantum phenomenon that stymies the spread of information. OTOC talk burbled beyond my office door.

At the end of the summer, IQIM postdoc Yichen Huang posted on Facebook, “In the past week, five papers (one of which is ours) appeared . . . studying out-of-time-ordered correlators in many-body localized systems.”

I looked down at the MBL calculation I was performing. I looked at my computer screen. I set down my pencil.


I marched to John Preskill’s office.


The bosses. Of different sorts, of course.

The OTOC kept flaring on my radar, I reported. Maybe the time had come for me to try contributing to the discussion. What might I contribute? What would be interesting?

We kicked around ideas.

“Well,” John ventured, “you’re interested in fluctuation relations, right?”

Something clicked like the “power” button on a video-game console.

Fluctuation relations are equations derived in nonequilibrium statistical mechanics. They describe systems driven far from equilibrium, like a DNA strand whose ends you’ve yanked apart. Experimentalists use fluctuation theorems to infer a difficult-to-measure quantity, a difference \Delta F between free energies. Fluctuation relations imply the Second Law of Thermodynamics. The Second Law relates to the flow of time and the loss of information.

Time…loss of information…Fluctuation relations smelled like the OTOC. The two had to join together.


I spent the next four days sitting, writing, obsessed. I’d read a paper, three years earlier, that casts a fluctuation relation in terms of a correlator. I unearthed the paper and redid the proof. Could I deform the proof until the paper’s correlator became the out-of-time-ordered correlator?

Apparently. I presented my argument to my research group. John encouraged me to clarify a point: I’d defined a mathematical object A, a probability amplitude. Did A have physical significance? Could anyone measure it? I consulted measurement experts. One identified A as a quasiprobability, a quantum generalization of a probability, used to model light in quantum optics. With the experts’ assistance, I devised two schemes for measuring the quasiprobability.

The result is a fluctuation-like relation that contains the OTOC. The OTOC, the theorem reveals, is a combination of quasiprobabilities. Experimentalists can measure quasiprobabilities with weak measurements, gentle probings that barely disturb the probed system. The theorem suggests two experimental protocols for inferring the difficult-to-measure OTOC, just as fluctuation relations suggest protocols for inferring the difficult-to-measure \Delta F. Just as fluctuation relations cast \Delta F in terms of a characteristic function of a probability distribution, this relation casts F(t) in terms of a characteristic function of a (summed) quasiprobability distribution. Quasiprobabilities reflect entanglement, as the OTOC does.


Collaborators and I are extending this work theoretically and experimentally. How does the quasiprobability look? How does it behave? What mathematical properties does it have? The OTOC is motivating questions not only about our quasiprobability, but also about quasiprobability and weak measurements. We’re pushing toward measuring the OTOC quasiprobability with superconducting qubits or cold atoms.

Chaos has evolved from an enemy to a curiosity, from a god of destruction to an inspiration. I no longer play the electric-blue hedgehog. But I remain electrified.


1I hadn’t started studying physics, ok?

2Don’t ask me how the liquid’s surface tension rises enough to maintain the limbs’ shapes.

3Black holes obey quantum mechanics. Quantum systems can solve certain problems more quickly than ordinary (classical) computers. Computers make mistakes. We fix mistakes using error-correcting codes. The codes required by quantum computers differ from the codes required by ordinary computers. Systems that contain black holes, we can regard as performing quantum computations. Black-hole systems’ mistakes admit of correction via the code constructed by Beni & co. 

Hamiltonian: An American Musical (without Americana or music)

Author’s note: I intended to post this article three months ago. Other developments delayed the release. Thanks in advance for pardoning the untimeliness.

Critics are raving about it. Barak Obama gave a speech about it. It’s propelled two books onto bestseller lists. Committees have showered more awards on it than clouds have showered rain on California this past decade.

What is it? The Hamiltonian, represented by \hat{H}. It’s an operator (a mathematical object) that basically represents a system’s energy. Hamiltonians characterize systems classical and quantum, from a brick in a Broadway theater to the photons that form a spotlight. \hat{H} determines how a system evolves, or changes in time.


I lied: Obama didn’t give a speech about the Hamiltonian. He gave a speech about Hamilton. Hamilton: An American Musical spotlights 18th-century revolutionary Alexander Hamilton. Hamilton conceived the United States’s national bank. He nurtured the economy as our first Secretary of the Treasury. The year after Alexander Hamilton died, William Rowan Hamilton was born. Rowan Hamilton conceived four-dimensional numbers called quaternions. He nurtured the style of physics, Hamiltonian mechanics, used to model quantum systems today.


Hamilton has enchanted audiences and critics. Ticket sell out despite costing over $1,000. Tonys, Grammys, and Pulitzers have piled up. Lawmakers, one newspaper reported, ridicule colleagues who haven’t seen the show. One political staff member confessed that “dodging ‘Hamilton’ barbs has affected her work—so much so that she hasn’t returned certain phone calls ‘because I couldn’t handle the anxiety’ of being harangued for her continued failure to see the show.”

Musical-theater fans across the country are applauding Alexander. Hamilton forbid that William Rowan should envy him. Let’s celebrate Hamiltonians.


I’ve been pondering the Hamiltonian


It describes a chain of L sites. L ranges from 10 to 30 in most computer simulations. The cast consists of quantum particles. Each site houses one particle or none. \hat{n}_j represents the number of particles at site j. c_j represents the removal of a particle from site j, and c_j^\dag represents the adding of a particle.

The last term in \hat{H} represents the repulsion between particles that border each other. The “nn” in “E_{\rm nn}” stands for “nearest-neighbor.” The J term encodes particles’ hopping between sites. \hat{c}_j^\dag \hat{c}_{j+1} means, “A particle jumps from site j+1 to site j.”

The first term in \hat{H}, we call disorder. Imagine a landscape of random dips and hills. Imagine, for instance, crouching on the dirt and snow in Valley Forge. Boots and hooves have scuffed the ground. Zoom in; crouch lower. Imagine transplanting the row of sites into this landscape. h_j denotes the height of site j.

Say that the dips sink low and the hills rise high. The disorder traps particles like soldiers behind enemy lines. Particles have trouble hopping. We call this system many-body localized.

Imagine flattening the landscape abruptly, as by stamping on the snow. This flattening triggers a phase transition.  Phase transitions are drastic changes, as from colony to country. The flattening frees particles to hop from site to site. The particles spread out, in accordance with the Hamiltonian’s J term. The particles come to obey thermodynamics, a branch of physics that I’ve effused about.

The Hamiltonian encodes repulsion, hopping, localization, thermalization, and more behaviors. A richer biography you’ll not find amongst the Founding Fathers.


As Hamiltonians constrain particles, politics constrain humans. A play has primed politicians to smile upon the name “Hamilton.” Physicists study Hamiltonians and petition politicians for funding. Would politicians fund us more if we emphasized the Hamiltonians in our science?

Gold star for whoever composes the most rousing lyrics about many-body localization. Or, rather, fifty white stars.

The weak shall inherit the quasiprobability.

Justin Dressel’s office could understudy for the archetype of a physicist’s office. A long, rectangular table resembles a lab bench. Atop the table perches a tesla coil. A larger tesla coil perches on Justin’s desk. Rubik’s cubes and other puzzles surround a computer and papers. In front of the desk hangs a whiteboard.

A puzzle filled the whiteboard in August. Justin had written a model for a measurement of a quasiprobability. I introduced quasiprobabilities here last Halloween. Quasiprobabilities are to probabilities as ebooks are to books: Ebooks resemble books but can respond to touchscreen interactions through sounds and animation. Quasiprobabilities resemble probabilities but behave in ways that probabilities don’t.


A tesla coil of Justin Dressel’s


Let p denote the probability that any given physicist keeps a tesla coil in his or her office. p ranges between zero and one. Quasiprobabilities can dip below zero. They can assume nonreal values, dependent on the imaginary number i = \sqrt{-1}. Probabilities describe nonquantum phenomena, like tesla-coil collectors,1 and quantum phenomena, like photons. Quasiprobabilities appear nonclassical.2,3

We can infer the tesla-coil probability by observing many physicists’ offices:

\text{Prob(any given physicist keeps a tesla coil in his/her office)}  =  \frac{ \text{\# physicists who keep tesla coils in their offices} }{ \text{\# physicists} } \, . We can infer quasiprobabilities from weak measurements, Justin explained. You can measure the number of tesla coils in an office by shining light on the office, correlating the light’s state with the tesla-coil number, and capturing the light on photographic paper. The correlation needn’t affect the tesla coils. Observing a quantum state changes the state, by the Uncertainty Principle heralded by Heisenberg.

We could observe a quantum system weakly. We’d correlate our measurement device (the analogue of light) with the quantum state (the analogue of the tesla-coil number) unreliably. Imagining shining a dull light on an office for a brief duration. Shadows would obscure our photo. We’d have trouble inferring the number of tesla coils. But the dull, brief light burst would affect the office less than a strong, long burst would.

Justin explained how to infer a quasiprobability from weak measurements. He’d explained on account of an action that others might regard as weak: I’d asked for help.


Chaos had seized my attention a few weeks earlier. Chaos is a branch of math and physics that involves phenomena we can’t predict, like weather. I had forayed into quantum chaos for reasons I’ll explain in later posts. I was studying a function F(t) that can flag chaos in cold atoms, black holes, and superconductors.

I’d derived a theorem about F(t). The theorem involved a UFO of a mathematical object: a probability amplitude that resembled a probability but could assume nonreal values. I presented the theorem to my research group, which was kind enough to provide feedback.

“Is this amplitude physical?” John Preskill asked. “Can you measure it?”

“I don’t know,” I admitted. “I can tell a story about what it signifies.”

“If you could measure it,” he said, “I might be more excited.”

You needn’t study chaos to predict that private clouds drizzled on me that evening. I was grateful to receive feedback from thinkers I respected, to learn of a weakness in my argument. Still, scientific works are creative works. Creative works carry fragments of their creators. A weakness in my argument felt like a weakness in me. So I took the step that some might regard as weak—by seeking help.



Some problems, one should solve alone. If you wake me at 3 AM and demand that I solve the Schrödinger equation that governs a particle in a box, I should be able to comply (if you comply with my demand for justification for the need to solve the Schrödinger equation at 3 AM).One should struggle far into problems before seeking help.

Some scientists extend this principle into a ban on assistance. Some students avoid asking questions for fear of revealing that they don’t understand. Some boast about passing exams and finishing homework without the need to attend office hours. I call their attitude “scientific machismo.”

I’ve all but lived in office hours. I’ve interrupted lectures with questions every few minutes. I didn’t know if I could measure that probability amplitude. But I knew three people who might know. Twenty-five minutes after I emailed them, Justin replied: “The short answer is yes!”


I visited Justin the following week, at Chapman University’s Institute for Quantum Studies. I sat at his bench-like table, eyeing the nearest tesla coil, as he explained. Justin had recognized my probability amplitude from studies of the Kirkwood-Dirac quasiprobability. Experimentalists infer the Kirkwood-Dirac quasiprobability from weak measurements. We could borrow these experimentalists’ techniques, Justin showed, to measure my probability amplitude.

The borrowing grew into a measurement protocol. The theorem grew into a paper. I plunged into quasiprobabilities and weak measurements, following Justin’s advice. John grew more excited.

The meek might inherit the Earth. But the weak shall measure the quasiprobability.

With gratitude to Justin for sharing his expertise and time; and to Justin, Matt Leifer, and Chapman University’s Institute for Quantum Studies for their hospitality.

Chapman’s community was gracious enough to tolerate a seminar from me about thermal states of quantum systems. You can watch the seminar here.

1Tesla-coil collectors consists of atoms described by quantum theory. But we can describe tesla-coil collectors without quantum theory.

2Readers foreign to quantum theory can interpret “nonclassical” roughly as “quantum.”

3Debate has raged about whether quasiprobabilities govern classical phenomena.

4I should be able also to recite the solutions from memory.