Toward physical realizations of thermodynamic resource theories

“This is your arch-nemesis.”

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

The question-asker pointed at a listener.

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

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

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

“My new friend,” I corrected the questioner.

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

A Q&A more successful than mine.

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

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

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

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

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

Something to get off your chest. Like the contents of a conference-proceedings paper, according to my advisor.

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

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

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

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

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

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

*So do theorists-in-training. I have.

**A definition that involves the trace distance.

Bits, Bears, and Beyond in Banff: Part Deux

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

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

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

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

Screen Shot 2015-09-12 at 3.55.17 PM

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

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

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

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

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

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

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

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

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

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

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

Renyi generalizations of the conditional quantum mutual information
Fidelity of recovery, geometric squashed entanglement, and measurement recoverability
Renyi squashed entanglement, discord, and relative entropy differences
Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps
Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution
Multipartite quantum correlations and local recoverability
Monotonicity of quantum relative entropy and recoverability
Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures
The Fidelity of Recovery is Multiplicative
Universal recovery map for approximate Markov chains
Recoverability in quantum information theory
Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
Universal recovery from a decrease of quantum relative entropy

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

Bits, bears, and beyond in Banff

Another conference about entropy. Another graveyard.

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

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


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

Suppose that, every day, exactly one bear attacks you as you hike in Banff. Every day, you have a probability p1 of facing down a black bear, a probability p2 of facing down a grizzly, and so on. These probabilities form a distribution {pi} over the set of possible events (of possible attacks). We call the type of attack that occurs on a given day a random variable. The distribution associated with each day equals the distribution associated with each other day. Hence the variables are identically distributed. The Monday distribution doesn’t affect the Tuesday distribution and so on, so the distributions are independent.

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

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

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

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

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

Entropy zoo

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

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

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

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

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


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

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


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

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

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

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


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

Holography and the MERA

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

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

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

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


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

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


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

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

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

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


“Why does it have that name?”

I’ve asked in seminars, in lectures, in offices, and at group meetings. I’ve asked about physical conjectures, about theorems, and about mathematical properties.

“I don’t know.” Lecturers have shrugged. “It’s just a name.”

This spring, I asked about master equations. I thought of them as tools used in statistical mechanics, the study of vast numbers of particles. We can’t measure vast numbers of particles, so we can’t learn about stat-mech systems everything one might want to know. The magma beneath Santorini, for example, consists of about 1024 molecules. Good luck measuring every one.

Imagine, as another example, using a quantum computer to solve a problem. We load information by initializing the computer to a certain state: We orient the computer’s particles in certain directions. We run a program, then read out the output.

Suppose the computer sits on a tabletop, exposed to the air like leftover casserole no one wants to save for tomorrow. Air molecules bounce off the computer, becoming entangled with the hardware. This entanglement, or quantum correlation, alters the computer’s state, just as flies alter a casserole.* To understand the computer’s output—which depends on the state, which depends on the air—we must have a description of the air. But we can’t measure all those air molecules, just as we can’t measure all the molecules in Santorini’s magma.

We can package our knowledge about the computer’s state into a mathematical object, called a density operator, labeled by ρ(t). A quantum master equation describes how ρ(t) changes. I had no idea, till this spring, why we call master equations “master equations.” Had someone named “John Master” invented them? Had the inspiration for the Russell Crowe movie Master and Commander? Or the Igor who lisps, “Yeth, mathter” in adaptations of Frankenstein?

Jenia Mozgunov, a fellow student and Preskillite, proposed an answer: Using master equations, we can calculate how averages of observable properties change. Imagine describing a laser, a cavity that spews out light. A master equation reveals how the average number of photons (particles of light) in the cavity changes. We want to predict these averages because experimentalists measure them. Because master equations spawn many predictions—many equations—they merit the label “master.”

Jenia’s hypothesis appealed to me, but I wanted certainty. I wanted Truth. I opened my laptop and navigated to Facebook.

“Does anyone know,” I wrote in my status, “why master equations are called ‘master equations’?”

Ian Durham, a physicist at St. Anselm College, cited Tom Moore’s Six Ideas that Shaped Physics. Most physics problems, Ian wrote, involve “some overarching principle.” Example principles include energy conservation and invariance under discrete translations (the system looks the same after you step in some direction). A master equation encapsulates this principle.

Ian’s explanation sounded sensible. But fewer people “liked” his reply on Facebook than “liked” a quip by a college friend: Master equations deserve their name because “[t]hey didn’t complete all the requirements for the doctorate.”

My advisor, John Preskill, dug through two to three books, one set of lecture notes, one German Wikipedia page, one to two articles, and Google Scholar. He concluded that Nordsieck, Lamb, and Uhlenbeck coined “master equation.” According to a 1940 paper of theirs,** “When the probabilities of the elementary processes are known, one can write down a continuity equation for W [a set of probabilities], from which all other equations can be derived and which we will call therefore the ‘master’ equation.”

“Are you sure you were meant to be a physicist,” I asked John, “rather than a historian?”

“Procrastination is a powerful motivator,” he replied.

Lecturers have shrugged at questions about names. Then they’ve paused, pondered, and begun, “I guess because…” Theorems and identities derive their names from symmetries, proof techniques, geometric illustrations, and applications to problems I’d thought unrelated. A name taught me about uses for master equations. Names reveal physics I wouldn’t learn without asking about names. Names aren’t just names. They’re lamps and guides.

Pity about the origin of “master equation,” though. I wish an Igor had invented them.

*Apologies if I’ve spoiled your appetite.

**A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I,” Physica 7, 344-60 (1940), p. 353.

Mingling stat mech with quantum info in Maryland

I felt like a yoyo.

I was standing in a hallway at the University of Maryland. On one side stood quantum-information theorists. On the other side stood statistical-mechanics scientists.* The groups eyed each other, like Jets and Sharks in West Side Story, except without fighting or dancing.

This March, the groups were generous enough to host me for a visit. I parked first at QuICS, the Joint Center for Quantum Information and Computer Science. Established in October 2014, QuICS had moved into renovated offices the previous month. QuICSland boasts bright colors, sprawling armchairs, and the scent of novelty. So recently had QuICS arrived that the restroom had not acquired toilet paper (as I learned later than I’d have preferred).

Interaction space

Photo credit: QuICS

From QuICS, I yoyo-ed to the chemistry building, where Chris Jarzynski’s group studies fluctuation relations. Fluctuation relations, introduced elsewhere on this blog, describe out-of-equilibrium systems. A system is out of equilibrium if large-scale properties of it change. Many systems operate out of equilibrium—boiling soup, combustion engines, hurricanes, and living creatures, for instance. Physicists want to describe nonequilibrium processes but have trouble: Living creatures are complicated. Hence the buzz about fluctuation relations.

My first Friday in Maryland, I presented a seminar about quantum voting for QuICS. The next Tuesday, I was to present about one-shot information theory for stat-mech enthusiasts. Each week, the stat-mech crowd invites its speaker to lunch. Chris Jarzynski recommended I invite QuICS. Hence the Jets-and-Sharks tableau.

“Have you interacted before?” I asked the hallway.

“No,” said a voice. QuICS hadn’t existed till last fall, and some QuICSers hadn’t had offices till the previous month.**


“We’re QuICS,” volunteered Stephen Jordan, a quantum-computation theorist, “the Joint Center for Quantum Information and Computer Science.”

So began the mingling. It continued at lunch, which we shared at three circular tables we’d dragged into a chain. The mingling continued during the seminar, as QuICSers sat with chemists, materials scientists, and control theorists. The mingling continued the next day, when QuICSer Alexey Gorshkov joined my discussion with the Jarzynski group. Back and forth we yoyo-ed, between buildings and topics.

“Mingled,” said Yigit Subasi. Yigit, a postdoc of Chris’s, specialized in quantum physics as a PhD student. I’d asked how he thinks about quantum fluctuation relations. Since Chris and colleagues ignited fluctuation-relation research, theorems have proliferated like vines in a jungle. Everyone and his aunty seems to have invented a fluctuation theorem. I canvassed Marylanders for bushwhacking tips.

Imagine, said Yigit, a system whose state you know. Imagine a gas, whose temperature you’ve measured, at equilibrium in a box. Or imagine a trapped ion. Begin with a state about which you have information.

Imagine performing work on the system “violently.” Compress the gas quickly, so the particles roil. Shine light on the ion. The system will leave equilibrium. “The information,” said Yigit, “gets mingled.”

Imagine halting the compression. Imagine switching off the light. Combine your information about the initial state with assumptions and physical laws.*** Manipulate equations in the right way, and the information might “unmingle.” You might capture properties of the violence in a fluctuation relation.

2 photos - cut

With Zhiyue Lu and Andrew Maven Smith of Chris Jarzynski’s group (left) and with QuICSers (right)

I’m grateful to have exchanged information in Maryland, to have yoyo-ed between groups. We have work to perform together. I have transformations to undergo.**** Let the unmingling begin.

With gratitude to Alexey Gorshkov and QuICS, and to Chris Jarzynski and the University of Maryland Department of Chemistry, for their hospitality, conversation, and camaraderie.

*Statistical mechanics is the study of systems that contain vast numbers of particles, like the air we breathe and white dwarf stars. I harp on about statistical mechanics often.

**Before QuICS’s birth, a future QuICSer had collaborated with a postdoc of Chris’s on combining quantum information with fluctuation relations.

***Yes, physical laws are assumptions. But they’re glorified assumptions.

****Hopefully nonviolent transformations.

Quantum gravity from quantum error-correcting codes?

The lessons we learned from the Ryu-Takayanagi formula, the firewall paradox and the ER=EPR conjecture have convinced us that quantum information theory can become a powerful tool to sharpen our understanding of various problems in high-energy physics. But, many of the concepts utilized so far rely on entanglement entropy and its generalizations, quantities developed by Von Neumann more than 60 years ago. We live in the 21st century. Why don’t we use more modern concepts, such as the theory of quantum error-correcting codes?

In a recent paper with Daniel Harlow, Fernando Pastawski and John Preskill, we have proposed a toy model of the AdS/CFT correspondence based on quantum error-correcting codes. Fernando has already written how this research project started after a fateful visit by Daniel to Caltech and John’s remarkable prediction in 1999. In this post, I hope to write an introduction which may serve as a reader’s guide to our paper, explaining why I’m so fascinated by the beauty of the toy model.

This is certainly a challenging task because I need to make it accessible to everyone while explaining real physics behind the paper. My personal philosophy is that a toy model must be as simple as possible while capturing key properties of the system of interest. In this post, I will try to extract some key features of the AdS/CFT correspondence and construct a toy model which captures these features. This post may be a bit technical compared to other recent posts, but anyway, let me give it a try…

Bulk locality paradox and quantum error-correction

The AdS/CFT correspondence says that there is some kind of correspondence between quantum gravity on (d+1)-dimensional asymptotically-AdS space and d-dimensional conformal field theory on its boundary. But how are they related?

The AdS-Rindler reconstruction tells us how to “reconstruct” a bulk operator from boundary operators. Consider a bulk operator \phi and a boundary region A on a hyperbolic space (in other words, a negatively-curved plane). On a fixed time-slice, the causal wedge of A is a bulk region enclosed by the geodesic line of A (a curve with a minimal length). The AdS-Rindler reconstruction says that \phi can be represented by some integral of local boundary operators supported on A if and only if \phi is contained inside the causal wedge of A. Of course, there are multiple regions A,B,C,… whose causal wedges contain \phi, and the reconstruction should work for any such region.


The Rindler-wedge reconstruction

That a bulk operator in the causal wedge can be reconstructed by local boundary operators, however, leads to a rather perplexing paradox in the AdS/CFT correspondence. Consider a bulk operator \phi at the center of a hyperbolic space, and split the boundary into three pieces, A, B, C. Then the geodesic line for the union of BC encloses the bulk operator, that is, \phi is contained inside the causal wedge of BC. So, \phi can be represented by local boundary operators supported on BC. But the same argument applies to AB and CA, implying that the bulk operator \phi corresponds to local boundary operators which are supported inside AB, BC and CA simultaneously. It would seem then that the bulk operator \phi must correspond to an identity operator times a complex phase. In fact, similar arguments apply to any bulk operators, and thus, all the bulk operators must correspond to identity operators on the boundary. Then, the AdS/CFT correspondence seems so boring…


The bulk operator at the center is contained inside causal wedges of BC, AB, AC. Does this mean that the bulk operator corresponds to an identity operator on the boundary?

Almheiri, Dong and Harlow have recently proposed an intriguing way of reconciling this paradox with the AdS/CFT correspondence. They proposed that the AdS/CFT correspondence can be viewed as a quantum error-correcting code. Their idea is as follows. Instead of \phi corresponding to a single boundary operator, \phi may correspond to different operators in different regions, say O_{AB}, O_{BC}, O_{CA} living in AB, BC, CA respectively. Even though O_{AB}, O_{BC}, O_{CA} are different boundary operators, they may be equivalent inside a certain low energy subspace on the boundary.

This situation resembles the so-called quantum secret-sharing code. The quantum information at the center of the bulk cannot be accessed from any single party A, B or C because \phi does not have representation on A, B, or C. It can be accessed only if multiple parties cooperate and perform joint measurements. It seems that a quantum secret is shared among three parties, and the AdS/CFT correspondence somehow realizes the three-party quantum secret-sharing code!

Entanglement wedge reconstruction?

Recently, causal wedge reconstruction has been further generalized to the notion of entanglement wedge reconstruction. Imagine we split the boundary into four pieces A,B,C,D such that A,C are larger than B,D. Then the geodesic lines for A and C do not form the geodesic line for the union of A and C because we can draw shorter arcs by connecting endpoints of A and C, which form the global geodesic line. The entanglement wedge of AC is a bulk region enclosed by this global geodesic line of AC. And the entanglement wedge reconstruction predicts that \phi can be represented as an integral of local boundary operators on AC if and only if \phi is inside the entanglement wedge of AC [1].


Causal wedge vs entanglement wedge.

Building a minimal toy model; the five-qubit code

Okay, now let’s try to construct a toy model which admits causal and entanglement wedge reconstructions of bulk operators. Because I want a simple toy model, I take a rather bold assumption that the bulk consists of a single qubit while the boundary consists of five qubits, denoted by A, B, C, D, E.


Reconstruction of a bulk operator in the “minimal” model.

What does causal wedge reconstruction teach us in this minimal setup of five and one qubits? First, we split the boundary system into two pieces, ABC and DE and observe that the bulk operator \phi is contained inside the causal wedge of ABC. From the rotational symmetries, we know that the bulk operator \phi must have representations on ABC, BCD, CDE, DEA, EAB. Next, we split the boundary system into four pieces, AB, C, D and E, and observe that the bulk operator \phi is contained inside the entanglement wedge of AB and D. So, the bulk operator \phi must have representations on ABD, BCE, CDA, DEB, EAC. In summary, we have the following:

  • The bulk operator must have representations on R if and only if R contains three or more qubits.

This is the property I want my toy model to possess.

What kinds of physical systems have such a property? Luckily, we quantum information theorists know the answer; the five-qubit code. The five-qubit code, proposed here and here, has an ability to encode one logical qubit into five-qubit entangled states and corrects any single qubit error. We can view the five-qubit code as a quantum encoding isometry from one-qubit states to five-qubit states:

\alpha | 0 \rangle + \beta | 1 \rangle \rightarrow \alpha | \tilde{0} \rangle + \beta | \tilde{1} \rangle

where | \tilde{0} \rangle and | \tilde{1} \rangle are the basis for a logical qubit. In quantum coding theory, logical Pauli operators \bar{X} and \bar{Z} are Pauli operators which act like Pauli X (bit flip) and Z (phase flip) on a logical qubit spanned by | \tilde{0} \rangle and | \tilde{1} \rangle. In the five-qubit code, for any set of qubits R with volume 3, some representations of logical Pauli X and Z operators, \bar{X}_{R} and \bar{Z}_{R}, can be found on R. While \bar{X}_{R} and \bar{X}_{R'} are different operators for R \not= R', they act exactly in the same manner on the codeword subspace spanned by | \tilde{0} \rangle and | \tilde{1} \rangle. This is exactly the property I was looking for.

Holographic quantum error-correcting codes

We just found possibly the smallest toy model of the AdS/CFT correspondence, the five-qubit code! The remaining task is to construct a larger model. For this goal, we view the encoding isometry of the five-qubit code as a six-leg tensor. The holographic quantum code is a network of such six-leg tensors covering a hyperbolic space where each tensor has one open leg. These open legs on the bulk are interpreted as logical input legs of a quantum error-correcting code while open legs on the boundary are identified as outputs where quantum information is encoded. Then the entire tensor network can be viewed as an encoding isometry.

The six-leg tensor has some nice properties. Imagine we inject some Pauli operator into one of six legs in the tensor. Then, for any given choice of three legs, there always exists a Pauli operator acting on them which counteracts the effect of the injection. An example is shown below:


In other words, if an operator is injected from one tensor leg, one can “push” it into other three tensor legs.

Finally, let’s demonstrate causal wedge reconstruction of bulk logical operators. Pick an arbitrary open tensor leg in the bulk and inject some Pauli operator into it. We can “push” it into three tensor legs, which are then injected into neighboring tensors. By repeatedly pushing operators to the boundary in the network, we eventually have some representation of the operator living on a piece of boundary region A. And the bulk operator is contained inside the causal wedge of A. (Here, the length of the curve can be defined as the number of tensor legs cut by the curve). You can also push operators into the boundary by choosing different tensor legs which lead to different representations of a logical operator. You can even have a rather exotic representation which is supported non-locally over two disjoint pieces of the boundary, realizing entanglement wedge reconstruction.


Causal wedge and entanglement wedge reconstruction.

What’s next?

This post is already pretty long and I need to wrap it up…

Shor’s quantum factoring algorithm is a revolutionary invention which opened a whole new research avenue of quantum information science. It is often forgotten, but the first quantum error-correcting code is another important invention by Peter Shor (and independently by Andrew Steane) which enabled a proof that the quantum computation can be performed fault-tolerantly. The theory of quantum error-correcting codes has found interesting applications in studies of condensed matter physics, such as topological phases of matter. Perhaps then, quantum coding theory will also find applications in high energy physics.

Indeed, many interesting open problems are awaiting us. Is entanglement wedge reconstruction a generic feature of tensor networks? How do we describe black holes by quantum error-correcting codes? Can we build a fast scrambler by tensor networks? Is entanglement a wormhole (or maybe a perfect tensor)? Can we resolve the firewall paradox by holographic quantum codes? Can the physics of quantum gravity be described by tensor networks? Or can the theory of quantum gravity provide us with novel constructions of quantum codes?

I feel that now is the time for quantum information scientists to jump into the research of black holes. We don’t know if we will be burned by a firewall or not … , but it is worth trying.

1. Whether entanglement wedge reconstruction is possible in the AdS/CFT correspondence or not still remains controversial. In the spirit of the Ryu-Takayanagi formula which relates entanglement entropy to the length of a global geodesic line, entanglement wedge reconstruction seems natural. But that a bulk operator can be reconstructed from boundary operators on two separate pieces A and C non-locally sounds rather exotic. In our paper, we constructed a toy model of tensor networks which allows both causal and entanglement wedge reconstruction in many cases. For details, see our paper.