Let gravity do its work

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One day, early this spring, I found myself in a hotel elevator with three other people. The cohort consisted of two theoretical physicists, one computer scientist, and what appeared to be a normal person. I pressed the elevator’s 4 button, as my husband (the computer scientist) and I were staying on the hotel’s fourth floor. The button refused to light up.

“That happened last time,” the normal person remarked. He was staying on the fourth floor, too.

The other theoretical physicist pressed the 3 button.

“Should we press the 5 button,” the normal person continued, “and let gravity do its work?”

I took a moment to realize that he was suggesting we ascend to the fifth floor and then induce the elevator to fall under gravity’s influence to the fourth. We were reaching floor three, so I exchanged a “have a good evening” with the other physicist, who left. The door shut, and we began to ascend.

“As it happens,” I remarked, “he’s an expert on gravity.” The other physicist was Herman Verlinde, a professor at Princeton.

Such is a side effect of visiting the Simons Center for Geometry and Physics. The Simons Center graces the Stony Brook University campus, which was awash in daffodils and magnolia blossoms last month. The Simons Center derives its name from hedge-fund manager Jim Simons (who passed away during the writing of this article). He achieved landmark physics and math research before earning his fortune on Wall Street as a quant. Simons supported his early loves by funding the Simons Center and other scientific initiatives. The center reminded me of the Perimeter Institute for Theoretical Physics, down to the café’s linen napkins, so I felt at home.

I was participating in the Simons Center workshop “Entanglement, thermalization, and holography.” It united researchers from quantum information and computation, black-hole physics and string theory, quantum thermodynamics and many-body physics, and nuclear physics. We were to share our fields’ approaches to problems centered on thermalization, entanglement, quantum simulation, and the like. I presented about the eigenstate thermalization hypothesis, which elucidates how many-particle quantum systems thermalize. The hypothesis fails, I argued, if a system’s dynamics conserve quantities (analogous to energy and particle number) that can’t be measured simultaneously. Herman Verlinde discussed the ER=EPR conjecture.

My PhD advisor, John Preskill, blogged about ER=EPR almost exactly eleven years ago. Read his blog post for a detailed introduction. Briefly, ER=EPR posits an equivalence between wormholes and entanglement. 

The ER stands for Einstein–Rosen, as in Einstein–Rosen bridge. Sean Carroll provided the punchiest explanation I’ve heard of Einstein–Rosen bridges. He served as the scientific advisor for the 2011 film Thor. Sean suggested that the film feature a wormhole, a connection between two black holes. The filmmakers replied that wormholes were passé. So Sean suggested that the film feature an Einstein–Rosen bridge. “What’s an Einstein–Rosen bridge?” the filmmakers asked. “A wormhole.” So Thor features an Einstein–Rosen bridge.

EPR stands for Einstein–Podolsky–Rosen. The three authors published a quantum paradox in 1935. Their EPR paper galvanized the community’s understanding of entanglement.

ER=EPR is a conjecture that entanglement is closely related to wormholes. As Herman said during his talk, “You probably need entanglement to realize a wormhole.” Or any two maximally entangled particles are connected by a wormhole. The idea crystallized in a paper by Juan Maldacena and Lenny Susskind. They drew on work by Mark Van Raamsdonk (who masterminded the workshop behind this Quantum Frontiers post) and Brian Swingle (who’s appeared in further posts).

Herman presented four pieces of evidence for the conjecture, as you can hear in the video of his talk. One piece emerges from the AdS/CFT duality, a parallel between certain space-times (called anti–de Sitter, or AdS, spaces) and quantum theories that have a certain symmetry (called conformal field theories, or CFTs). A CFT, being quantum, can contain entanglement. One entangled state is called the thermofield double. Suppose that a quantum system is in a thermofield double and you discard half the system. The remaining half looks thermal—we can attribute a temperature to it. Evidence indicates that, if a CFT has a temperature, then it parallels an AdS space that contains a black hole. So entanglement appears connected to black holes via thermality and temperature.

Despite the evidence—and despite the eleven years since John’s publication of his blog post—ER=EPR remains a conjecture. Herman remarked, “It’s more like a slogan than anything else.” His talk’s abstract contains more hedging than a suburban yard. I appreciated the conscientiousness, a college acquaintance having once observed that I spoke carefully even over sandwiches with a friend.

A “source of uneasiness” about ER=EPR, to Herman, is measurability. We can’t check whether a quantum state is entangled via any single measurement. We have to prepare many identical copies of the state, measure the copies, and process the outcome statistics. In contrast, we seem able to conclude that a space-time is connected without measuring multiple copies of the space-time. We can check that a hotel’s first floor is connected to its fourth, for instance, by riding in an elevator once.

Or by riding an elevator to the fifth floor and descending by one story. My husband, the normal person, and I took the stairs instead of falling. The hotel fixed the elevator within a day or two, but who knows when we’ll fix on the truth value of ER=EPR?

With thanks to the conference organizers for their invitation, to the Simons Center for its hospitality, to Jim Simons for his generosity, and to the normal person for inspiration.

To thermalize, or not to thermalize, that is the question.

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The Noncommuting-Charges World Tour (Part 3 of 4)

This is the third part of a four-part series covering the recent Perspective on noncommuting charges. I’ll post one part every ~5 weeks leading up to my PhD thesis defence. You can find Part 1 here and Part 2 here.

If Hamlet had been a system of noncommuting charges, his famous soliloquy may have gone like this…

To thermalize, or not to thermalize, that is the question:
Whether ’tis more natural for the system to suffer
The large entanglement of thermalizing dynamics,
Or to take arms against the ETH
And by opposing inhibit it. To die—to thermalize,
No more; and by thermalization to say we end
The dynamical symmetries and quantum scars
That complicate dynamics: ’tis a consummation
Devoutly to be wish’d. To die, to thermalize;
To thermalize, perchance to compute—ay, there’s the rub:
For in that thermalization our quantum information decoheres,
When our coherence has shuffled off this quantum coil,
Must give us pause—there’s the respect
That makes calamity of resisting thermalization.

Hamlet (the quantum steampunk edition)


In the original play, Hamlet grapples with the dilemma of whether to live or die. Noncommuting charges have a dilemma regarding whether they facilitate or impede thermalization. Among the five research opportunities highlighted in the Perspective article, resolving this debate is my favourite opportunity due to its potential implications for quantum technologies. A primary obstacle in developing scalable quantum computers is mitigating decoherence; here, thermalization plays a crucial role. If systems with noncommuting charges are shown to resist thermalization, they may contribute to quantum technologies that are more resistant to decoherence. Systems with noncommuting charges, such as spin systems and squeezed states of light, naturally occur in quantum computing models like quantum dots and optical approaches. This possibility is further supported by recent advances demonstrating that non-Abelian symmetric operations are universal for quantum computing (see references 1 and 2).

In this penultimate blog post of the series, I will review some results that argue both in favour of and against noncommuting charges hindering thermalization. This discussion includes content from Sections III, IV, and V of the Perspective article, along with a dash of some related works at the end—one I recently posted and another I recently found. The results I will review do not directly contradict one another because they arise from different setups. My final blog post will delve into the remaining parts of the Perspective article.

Playing Hamlet is like jury duty for actors–sooner or later, you’re getting the call (source).

Arguments for hindering thermalization

The first argument supporting the idea that noncommuting charges hinder thermalization is that they can reduce the production of thermodynamic entropy. In their study, Manzano, Parrondo, and Landi explore a collisional model involving two systems, each composed of numerous subsystems. In each “collision,” one subsystem from each system is randomly selected to “collide.” These subsystems undergo a unitary evolution during the collision and are subsequently returned to their original systems. The researchers derive a formula for the entropy production per collision within a certain regime (the linear-response regime). Notably, one term of this formula is negative if and only if the charges do not commute. Since thermodynamic entropy production is a hallmark of thermalization, this finding implies that systems with noncommuting charges may thermalize more slowly. Two other extensions support this result.

The second argument stems from an essential result in quantum computing. This result is that every algorithm you want to run on your quantum computer can be broken down into gates you run on one or two qubits (the building blocks of quantum computers). Marvian’s research reveals that this principle fails when dealing with charge-conserving unitaries. For instance, consider the charge as energy. Marvian’s results suggest that energy-preserving interactions between neighbouring qubits don’t suffice to construct all energy-preserving interactions across all qubits. The restrictions become more severe when dealing with noncommuting charges. Local interactions that preserve noncommuting charges impose stricter constraints on the system’s overall dynamics compared to commuting charges. These constraints could potentially reduce chaos, something that tends to lead to thermalization.

Adding to the evidence, we revisit the eigenstate thermalization hypothesis (ETH), which I discussed in my first post. The ETH essentially asserts that if an observable and Hamiltonian adhere to the ETH, the observable will thermalize. This means its expectation value stabilizes over time, aligning with the expectation value of the thermal state, albeit with some important corrections. Noncommuting charges cause all kinds of problems for the ETH, as detailed in these two posts by Nicole Yunger Halpern. Rather than reiterating Nicole’s succinct explanations, I’ll present the main takeaway: noncommuting charges undermine the ETH. This has led to the development of a non-Abelian version of the ETH by Murthy and collaborators. This new framework still predicts thermalization in many, but not all, cases. Under a reasonable physical assumption, the previously mentioned corrections to the ETH may be more substantial.

If this story ended here, I would have needed to reference a different Shakespearean work. Fortunately, the internal conflict inherent in noncommuting aligns well with Hamlet. Noncommuting charges appear to impede thermalization in various aspects, yet paradoxically, they also seem to promote it in others.

Arguments for promoting thermalization

Among the many factors accompanying the thermalization of quantum systems, entanglement is one of the most studied. Last year, I wrote a blog post explaining how my collaborators and I constructed analogous models that differ in whether their charges commute. One of the paper’s results was that the model with noncommuting charges had higher average entanglement entropy. As a result of that blog post, I was invited to CBC’s “Quirks & Quarks” Podcast to explain, on national radio, whether quantum entanglement can explain the extreme similarities we see in identical twins who are raised apart. Spoilers for the interview: it can’t, but wouldn’t it be grand if it could?

Following up on that work, my collaborators and I introduced noncommuting charges into monitored quantum circuits (MQCs)—quantum circuits with mid-circuit measurements. MQCs offer a practical framework for exploring how, for example, entanglement is affected by the interplay between unitary dynamics and measurements. MQCs with no charges or with commuting charges have a weakly entangled phase (“area-law” phase) when the measurements are done often enough, and a highly entangled phase (“volume-law” phase) otherwise. However, in MQCs with noncommuting charges, this weakly entangled phase never exists. In its place, there is a critical phase marked by long-range entanglement. This finding supports our earlier observation that noncommuting charges tend to increase entanglement.

I recently looked at a different angle to this thermalization puzzle. It’s well known that most quantum many-body systems thermalize; some don’t. In those that don’t, what effect do noncommuting charges have? One paper that answers this question is covered in the Perspective. Here, Potter and Vasseur study many-body localization (MBL). Imagine a chain of spins that are strongly interacting. We can add a disorder term, such as an external field whose magnitude varies across sites on this chain. If the disorder is sufficiently strong, the system “localizes.” This implies that if we measured the expectation value of some property of each qubit at some time, it would maintain that same value for a while. MBL is one type of behaviour that resists thermalization. Potter and Vasseur found that noncommuting charges destabilize MBL, thereby promoting thermalizing behaviour.

In addition to the papers discussed in our Perspective article, I want to highlight two other studies that study how systems can avoid thermalization. One mechanism is through the presence of “dynamical symmetries” (there are “spectrum-generating algebras” with a locality constraint). These are operators that act similarly to ladder operators for the Hamiltonian. For any observable that overlaps with these dynamical symmetries, the observable’s expectation value will continue to evolve over time and will not thermalize in accordance with the Eigenstate Thermalization Hypothesis (ETH). In my recent work, I demonstrate that noncommuting charges remove the non-thermalizing dynamics that emerge from dynamical symmetries.

Additionally, I came across a study by O’Dea, Burnell, Chandran, and Khemani, which proposes a method for constructing Hamiltonians that exhibit quantum scars. Quantum scars are unique eigenstates of the Hamiltonian that do not thermalize despite being surrounded by a spectrum of other eigenstates that do thermalize. Their approach involves creating a Hamiltonian with noncommuting charges and subsequently breaking the non-Abelian symmetry. When the symmetry is broken, quantum scars appear; however, if the non-Abelian symmetry were to be restored, the quantum scars vanish. These last three results suggest that noncommuting charges impede various types of non-thermalizing dynamics.

Unlike Hamlet, the narrative of noncommuting charges is still unfolding. I wish I could conclude with a dramatic finale akin to the duel between Hamlet and Laertes, Claudius’s poisoning, and the proclamation of a new heir to the Danish throne. However, that chapter is yet to be written. “To thermalize or not to thermalize?” We will just have to wait and see.

How I didn’t become a philosopher (but wound up presenting a named philosophy lecture anyway)

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Many people ask why I became a theoretical physicist. The answer runs through philosophy—which I thought, for years, I’d left behind in college.

My formal relationship with philosophy originated with Mr. Bohrer. My high school classified him as a religion teacher, but he co-opted our junior-year religion course into a philosophy course. He introduced us to Plato’s cave, metaphysics, and the pursuit of the essence beneath the skin of appearance. The essence of reality overlaps with quantum theory and relativity, which fascinated him. Not that he understood them, he’d hasten to clarify. But he passed along that fascination to me. I’d always loved dealing in abstract ideas, so the notion of studying the nature of the universe attracted me. A friend and I joked about growing up to be philosophers and—on account of not being able to find jobs—living in cardboard boxes next to each other.

After graduating from high school, I searched for more of the same in Dartmouth College’s philosophy department. I began with two prerequisites for the philosophy major: Moral Philosophy and Informal Logic. I adored those courses, but I adored all my courses.

As a sophomore, I embarked upon an upper-level philosophy course: philosophy of mind. I was one of the course’s youngest students, but the professor assured me that I’d accumulated enough background information in science and philosophy classes. Yet he and the older students threw around technical terms, such as qualia, that I’d never heard of. Those terms resurfaced in the assigned reading, again without definitions. I struggled to follow the conversation.

Meanwhile, I’d been cycling through the sciences. I’d taken my high school’s highest-level physics course, senior year—AP Physics C: Mechanics and Electromagnetism. So, upon enrolling in college, I made the rounds of biology, chemistry, and computer science. I cycled back to physics at the beginning of sophomore year, taking Modern Physics I in parallel with Informal Logic. The physics professor, Miles Blencowe, told me, “I want to see physics in your major.” I did, too, I assured him. But I wanted to see most subjects in my major.

Miles, together with department chair Jay Lawrence, helped me incorporate multiple subjects into a physics-centric program. The major, called “Physics Modified,” stood halfway between the physics major and the create-your-own major offered at some American liberal-arts colleges. The program began with heaps of prerequisite courses across multiple departments. Then, I chose upper-level physics courses, a math course, two history courses, and a philosophy course. I could scarcely believe that I’d planted myself in a physics department; although I’d loved physics since my first course in it, I loved all subjects, and nobody in my family did anything close to physics. But my major would provide a well-rounded view of the subject.

From shortly after I declared my Physics Modified major. Photo from outside the National Academy of Sciences headquarters in Washington, DC.

The major’s philosophy course was an independent study on quantum theory. In one project, I dissected the “EPR paper” published by Einstein, Podolsky, and Rosen (EPR) in 1935. It introduced the paradox that now underlies our understanding of entanglement. But who reads the EPR paper in physics courses nowadays? I appreciated having the space to grapple with the original text. Still, I wanted to understand the paper more deeply; the philosophy course pushed me toward upper-level physics classes.

What I thought of as my last chance at philosophy evaporated during my senior spring. I wanted to apply to graduate programs soon, but I hadn’t decided which subject to pursue. The philosophy and history of physics remained on the table. A history-of-physics course, taught by cosmologist Marcelo Gleiser, settled the matter. I worked my rear off in that course, and I learned loads—but I already knew some of the material from physics courses. Moreover, I knew the material more deeply than the level at which the course covered it. I couldn’t stand the thought of understanding the rest of physics only at this surface level. So I resolved to burrow into physics in graduate school. 

Appropriately, Marcelo published a book with a philosopher (and an astrophysicist) this March.

Burrow I did: after a stint in condensed-matter research, I submerged up to my eyeballs in quantum field theory and differential geometry at the Perimeter Scholars International master’s program. My research there bridged quantum information theory and quantum foundations. I appreciated the balance of fundamental thinking and possible applications to quantum-information-processing technologies. The rigorous mathematical style (lemma-theorem-corollary-lemma-theorem-corollary) appealed to my penchant for abstract thinking. Eating lunch with the Perimeter Institute’s quantum-foundations group, I felt at home.

Craving more research at the intersection of quantum thermodynamics and information theory, I enrolled at Caltech for my PhD. As I’d scarcely believed that I’d committed myself to my college’s physics department, I could scarcely believe that I was enrolling in a tech school. I was such a child of the liberal arts! But the liberal arts include the sciences, and I ended up wrapping Caltech’s hardcore vibe around myself like a favorite denim jacket.

Caltech kindled interests in condensed matter; atomic, molecular, and optical physics; and even high-energy physics. Theorists at Caltech thought not only abstractly, but also about physical platforms; so I started to, as well. I began collaborating with experimentalists as a postdoc, and I’m now working with as many labs as I can interface with at once. I’ve collaborated on experiments performed with superconducting qubits, photons, trapped ions, and jammed grains. Developing an abstract idea, then nursing it from mathematics to reality, satisfies me. I’m even trying to redirect quantum thermodynamics from foundational insights to practical applications.

At the University of Toronto in 2022, with my experimental collaborator Batuhan Yılmaz—and a real optics table!

So I did a double-take upon receiving an invitation to present a named lecture at the University of Pittsburgh Center for Philosophy of Science. Even I, despite not being a philosopher, had heard of the cache of Pitt’s philosophy-of-science program. Why on Earth had I received the invitation? I felt the same incredulity as when I’d handed my heart to Dartmouth’s physics department and then to a tech school. But now, instead of laughing at the image of myself as a physicist, I couldn’t see past it.

Why had I received that invitation? I did a triple-take. At Perimeter, I’d begun undertaking research on resource theories—simple, information-theoretic models for situations in which constraints restrict the operations one can perform. Hardly anyone worked on resource theories then, although they form a popular field now. Philosophers like them, and I’ve worked with multiple classes of resource theories by now.

More recently, I’ve worked with contextuality, a feature that distinguishes quantum theory from classical theories. And I’ve even coauthored papers about closed timelike curves (CTCs), hypothetical worldlines that travel backward in time. CTCs are consistent with general relativity, but we don’t know whether they exist in reality. Regardless, one can simulate CTCs, using entanglement. Collaborators and I applied CTC simulations to metrology—to protocols for measuring quantities precisely. So we kept a foot in practicality and a foot in foundations.

Perhaps the idea of presenting a named lecture on the philosophy of science wasn’t hopelessly bonkers. All right, then. I’d present it.

Presenting at the Center for Philosophy of Science

This March, I presented an ALS Lecture (an Annual Lecture Series Lecture, redundantly) entitled “Field notes on the second law of quantum thermodynamics from a quantum physicist.” Scientists formulated the second law the early 1800s. It helps us understand why time appears to flow in only one direction. I described three enhancements of that understanding, which have grown from quantum thermodynamics and nonequilibrium statistical mechanics: resource-theory results, fluctuation theorems, and thermodynamic applications of entanglement. I also enjoyed talking with Center faculty and graduate students during the afternoon and evening. Then—being a child of the liberal arts—I stayed in Pittsburgh for half the following Saturday to visit the Carnegie Museum of Art.

With a copy of a statue of the goddess Sekhmet. She lives in the Carnegie Museum of Natural History, which shares a building with the art museum, from which I detoured to see the natural-history museum’s ancient-Egypt area (as Quantum Frontiers regulars won’t be surprised to hear).

Don’t get me wrong: I’m a physicist, not a philosopher. I don’t have the training to undertake philosophy, and I have enough work to do in pursuit of my physics goals. But my high-school self would approve—that self is still me.

My experimental adventures in quantum thermodynamics

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Imagine a billiard ball bouncing around on a pool table. High-school level physics enables us to predict its motion until the end of time using simple equations for energy and momentum conservation, as long as you know the initial conditions – how fast the ball is moving at launch, and in which direction.

What if you add a second ball? This makes things more complicated, but predicting the future state of this system would still be possible based on the same principles. What about if you had a thousand balls, or a million? Technically, you could still apply the same equations, but the problem would not be tractable in any practical sense.

Billiard balls bouncing around on a pool table are a good analogy for a many-body system like a gas of molecules. Image credit

Thermodynamics lets us make precise predictions about averaged (over all the particles) properties of complicated, many-body systems, like millions of billiard balls or atoms bouncing around, without needing to know the gory details. We can make these predictions by introducing the notion of probabilities. Even though the system is deterministic – we can in principle calculate the exact motion of every ball – there are so many balls in this system, that the properties of the whole will be very close to the average properties of the balls. If you throw a six-sided die, the result is in principle deterministic and predictable, based on the way you throw it, but it’s in practice completely random to you – it could be 1 through 6, equally likely. But you know that if you cast a thousand dice, the average will be close to 3.5 – the average of all possibilities. Statistical physics enables us to calculate a probability distribution over the energies of the balls, which tells us everything about the average properties of the system. And because of entropy – the tendency for the system to go from ordered to disordered configurations, even if the probability distribution of the initial system is far from the one statistical physics predicts, after the system is allowed to bounce around and settle, this final distribution will be extremely close to a generic distribution that depends on average properties only. We call this the thermal distribution, and the process of the system mixing and settling to one of the most likely configurations – thermalization.

For a practical example – instead of billiard balls, consider a gas of air molecules bouncing around. The average energy of this gas is proportional to its temperature, which we can calculate from the probability distribution of energies. Being able to predict the temperature of a gas is useful for practical things like weather forecasting, cooling your home efficiently, or building an engine. The important properties of the initial state we needed to know – energy and number of particles – are conserved during the evolution, and we call them “thermodynamic charges”. They don’t actually need to be electric charges, although it is a good example of something that’s conserved.

Let’s cross from the classical world – balls bouncing around – to the quantum one, which deals with elementary particles that can be entangled, or in a superposition. What changes when we introduce this complexity? Do systems even thermalize in the quantum world? Because of the above differences, we cannot in principle be sure that the mixing and settling of the system will happen just like in the classical cases of balls or gas molecules colliding.

A visualization of a complex pattern called a quantum scar that can develop in quantum systems. Image credit

It turns out that we can predict the thermal state of a quantum system using very similar principles and equations that let us do this in the classical case. Well, with one exception – what if we cannot simultaneously measure our critical quantities – the charges?

One of the quirks of quantum mechanics is that observing the state of the system can change it. Before the observation, the system might be in a quantum superposition of many states. After the observation, a definite classical value will be recorded on our instrument – we say that the system has collapsed to this state, and thus changed its state. There are certain observables that are mutually incompatible – we cannot know their values simultaneously, because observing one definite value collapses the system to a state in which the other observable is in a superposition. We call these observables noncommuting, because the order of observation matters – unlike in multiplication of numbers, which is a commuting operation you’re familiar with. 2 * 3 = 6, and also 3 * 2 = 6 – the order of multiplication doesn’t matter.

Electron spin is a common example that entails noncommutation. In a simplified picture, we can think of spin as an axis of rotation of our electron in 3D space. Note that the electron doesn’t actually rotate in space, but it is a useful analogy – the property is “spin” for a reason. We can measure the spin along the x-,y-, or z-axis of a 3D coordinate system and obtain a definite positive or negative value, but this observation will result in a complete loss of information about spin in the other two perpendicular directions.

An illustration of electron spin. We can imagine it as an axis in 3D space that points in a particular direction. Image from Wikimedia Commons.

If we investigate a system that conserves the three spin components independently, we will be in a situation where the three conserved charges do not commute. We call them “non-Abelian” charges, because they enjoy a non-Abelian, that is, noncommuting, algebra. Will such a system thermalize, and if so, to what kind of final state?

This is precisely what we set out to investigate. Noncommutation of charges breaks usual derivations of the thermal state, but researchers have managed to show that with non-Abelian charges, a subtly different non-Abelian thermal state (NATS) should emerge. Myself and Nicole Yunger Halpern at the Joint Center for Quantum Information and Computer Science (QuICS) at the University of Maryland have collaborated with Amir Kalev from the Information Sciences Institute (ISI) at the University of Southern California, and experimentalists from the University of Innsbruck (Florian Kranzl, Manoj Joshi, Rainer Blatt and Christian Roos) to observe thermalization in a non-Abelian system – and we’ve recently published this work in PRX Quantum .

The experimentalists used a device that can trap ions with electric fields, as well as manipulate and read out their states using lasers. Only select energy levels of these ions are used, which effectively makes them behave like electrons. The laser field can couple the ions in a way that approximates the Heisenberg Hamiltonian – an interaction that conserves the three total spin components individually. We thus construct the quantum system we want to study – multiple particles coupled with interactions that conserve noncommuting charges.

We conceptually divide the ions into a system of interest and an environment. The system of interest, which consists of two particles, is what we want to measure and compare to theoretical predictions. Meanwhile, the other ions act as the effective environment for our pair of ions – the environment ions interact with the pair in a way that simulates a large bath exchanging heat and spin.

Photo of our University of Maryland group. From left to right: Twesh Upadhyaya, Billy Braasch, Shayan Majidy, Nicole Yunger Halpern, Aleks Lasek, Jose Antonio Guzman, Anthony Munson.

If we start this total system in some initial state, and let it evolve under our engineered interaction for a long enough time, we can then measure the final state of the system of interest. To make the NATS distinguishable from the usual thermal state, I designed an initial state that is easy to prepare, and has the ions pointing in directions that result in high charge averages and relatively low temperature. High charge averages make the noncommuting nature of the charges more pronounced, and low temperature makes the state easy to distinguish from the thermal background. However, we also show that our experiment works for a variety of more-arbitrary states.

We let the system evolve from this initial state for as long as possible given experimental limitations, which was 15 ms. The experimentalists then used quantum state tomography to reconstruct the state of the system of interest. Quantum state tomography makes multiple measurements over many experimental runs to approximate the average quantum state of the system measured. We then check how close the measured state is to the NATS. We have found that it’s about as close as one can expect in this experiment!

And we know this because we have also implemented a different coupling scheme, one that doesn’t have non-Abelian charges. The expected thermal state in the latter case was reached within a distance that’s a little smaller than our non-Abelian case. This tells us that the NATS is almost reached in our experiment, and so it is a good, and the best known, thermal state for the non-Abelian system – we have compared it to competitor thermal states.

Working with the experimentalists directly has been a new experience for me. While I was focused on the theory and analyzing the tomography results they obtained, they needed to figure out practical ways to realize what we asked of them. I feel like each group has learned a lot about the tasks of the other. I have become well acquainted with the trapped ion experiment and its capabilities and limitation. Overall, it has been great collaborating with the Austrian group.

Our result is exciting, as it’s the first experimental observation within the field of non-Abelian thermodynamics! This result was observed in a realistic, non-fine-tuned system that experiences non-negligible errors due to noise. So the system does thermalize after all. We have also demonstrated that the trapped ion experiment of our Austrian friends can be used to simulate interesting many-body quantum systems. With different settings and programming, other types of couplings can be simulated in different types of experiments.

The experiment also opened avenues for future work. The distance to the NATS was greater than the analogous distance to the Abelian system. This suggests that thermalization is inhibited by the noncommutation of charges, but more evidence is needed to justify this claim. In fact, our other recent paper in Physical Review B suggests the opposite!

As noncommutation is one of the core features that distinguishes classical and quantum physics, it is of great interest to unravel the fine differences non-Abelian charges can cause. But we also hope that this research can have practical uses. If thermalization is disrupted by noncommutation of charges, engineered systems featuring them could possibly be used to build quantum memory that is more robust, or maybe even reduce noise in quantum computers. We continue to explore noncommutation, looking for interesting effects that we can pin on it. I am currently working on verifying the workings of a hypothesis that explains when and why quantum systems thermalize internally.

Noncommuting charges are much like Batman

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The Noncommuting-Charges World Tour Part 2 of 4

This is the second part of a four-part series covering the recent Perspective on noncommuting charges. I’ll post one part every ~5 weeks leading up to my PhD thesis defence. You can find part 1 here.

Understanding a character’s origins enriches their narrative and motivates their actions. Take Batman as an example: without knowing his backstory, he appears merely as a billionaire who might achieve more by donating his wealth rather than masquerading as a bat to combat crime. However, with the context of his tragic past, Batman transforms into a symbol designed to instill fear in the hearts of criminals. Another example involves noncommuting charges. Without understanding their origins, the question “What happens when charges don’t commute?” might appear contrived or simply devised to occupy quantum information theorists and thermodynamicists. However, understanding the context of their emergence, we find that numerous established results unravel, for various reasons, in the face of noncommuting charges. In this light, noncommuting charges are much like Batman; their backstory adds to their intrigue and clarifies their motivation. Admittedly, noncommuting charges come with fewer costumes, outside the occasional steampunk top hat my advisor Nicole Yunger Halpern might sport.

Growing up, television was my constant companion. Of all the shows I’d get lost in, ‘Batman: The Animated Series’ stands the test of time. I highly recommend giving it a watch.

In the early works I’m about to discuss, a common thread emerges: the initial breakdown of some well-understood derivations and the effort to establish a new derivation that accommodates noncommuting charges. These findings will illuminate, yet not fully capture, the multitude of results predicated on the assumption that charges commute. Removing this assumption is akin to pulling a piece from a Jenga tower, triggering a cascade of other results. Critics might argue, “If you’re merely rederiving known results, this field seems uninteresting.” However, the reality is far more compelling. As researchers diligently worked to reconstruct this theoretical framework, they have continually uncovered ways in which noncommuting charges might pave the way for new physics. That said, the exploration of these novel phenomena will be the subject of my next post, where we delve into the emerging physics. So, I invite you to stay tuned. Back to the history…

E.T. Jaynes’s 1957 formalization of the maximum entropy principle has a blink-and-you’ll-miss-it reference to noncommuting charges. Consider a quantum system, similar to the box discussed in Part 1, where our understanding of the system’s state is limited to the expectation values of certain observables. Our aim is to deduce a probability distribution for the system’s potential pure states that accurately reflects our knowledge without making unjustified assumptions. According to the maximum entropy principle, this objective is met by maximizing the entropy of the distribution, which serve as a measure of uncertainty. This resulting state is known as the generalized Gibbs ensemble. Jaynes noted that this reasoning, based on information theory for the generalized Gibbs ensemble, remains valid even when our knowledge is restricted to the expectation values of noncommuting charges. However, later scholars have highlighted that physically substantiating the generalized Gibbs ensemble becomes significantly more challenging when the charges do not commute. Due to this and other reasons, when the system’s charges do not commute, the generalized Gibbs ensemble is specifically referred to as the non-Abelian thermal state (NATS).

For approximately 60 years, discussions about noncommuting charges remain dormant, outside a few mentions here and there. This changed when two studies highlighted how noncommuting charges break commonplace thermodynamics derivations. The first of these, conducted by Matteo Lostaglio as part of his 2014 thesis, challenged expectations about a system’s free energy—a measure of the system’s capacity for performing work. Interestingly, one can define a free energy for each charge within a system. Imagine a scenario where a system with commuting charges comes into contact with an environment that also has commuting charges. We then evolve the system such that the total charges in both the system and the environment are conserved. This evolution alters the system’s information content and its correlation with the environment. This change in information content depends on a sum of terms. Each term depends on the average change in one of the environment’s charges and the change in the system’s free energy for that same charge. However, this neat distinction of terms according to each charge breaks down when the system and environment exchange noncommuting charges. In such cases, the terms cannot be cleanly attributed to individual charges, and the conventional derivation falters.

The second work delved into resource theories, a topic discussed at length in Quantum Frontiers blog posts. In short, resource theories are frameworks used to quantify how effectively an agent can perform a task subject to some constraints. For example, consider all allowed evolutions (those conserving energy and other charges) one can perform on a closed system. From these evolutions, what system can you not extract any work from? The answer is systems in thermal equilibrium. The method used to determine the thermal state’s structure also fails when the system includes noncommuting charges. Building on this result, three groups (one, two, and three) presented physically motivated derivations of the form of the thermal state for systems with noncommuting charges using resource-theory-related arguments. Ultimately, the form of the NATS was recovered in each work.

Just as re-examining Batman’s origin story unveils a deeper, more compelling reason behind his crusade against crime, diving into the history and implications of noncommuting charges reveals their untapped potential for new physics. Behind every mask—or theory—there can lie an untold story. Earlier, I hinted at how reevaluating results with noncommuting charges opens the door to new physics. A specific example, initially veiled in Part 1, involves the violation of the Onsager coefficients’ derivation by noncommuting charges. By recalculating these coefficients for systems with noncommuting charges, we discover that their noncommutation can decrease entropy production. In Part 3, we’ll delve into other new physics that stems from charges’ noncommutation, exploring how noncommuting charges, akin to Batman, can really pack a punch.

The quantum gold rush

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Even if you don’t recognize the name, you probably recognize the saguaro cactus. It’s the archetype of the cactus, a column from which protrude arms bent at right angles like elbows. As my husband pointed out, the cactus emoji is a saguaro: 🌵. In Tucson, Arizona, even the airport has a saguaro crop sufficient for staging a Western short film. I didn’t have a film to shoot, but the garden set the stage for another adventure: the ITAMP winter school on quantum thermodynamics.

Tucson airport

ITAMP is the Institute for Theoretical Atomic, Molecular, and Optical Physics (the Optical is silent). Harvard University and the Smithsonian Institute share ITAMP, where I worked as a postdoc. ITAMP hosted the first quantum-thermodynamics conference to take place on US soil, in 2017. Also, ITAMP hosts a winter school in Arizona every February. (If you lived in the Boston area, you might want to escape to the southwest then, too.) The winter school’s topic varies from year to year. 

How about a winter school on quantum thermodynamics? ITAMP’s director, Hossein Sadeghpour, asked me when I visited Cambridge, Massachusetts last spring.

Let’s do it, I said. 

Lecturers came from near and far. Kanu Sinha, of the University of Arizona, spoke about how electric charges fluctuate in the quantum vacuum. Fluctuations feature also in extensions of the second law of thermodynamics, which helps explain why time flows in only one direction. Gabriel Landi, from the University of Rochester, lectured about these fluctuation relations. ITAMP Postdoctoral Fellow Ceren Dag explained why many-particle quantum systems register time’s arrow. Ferdinand Schmidt-Kaler described the many-particle quantum systems—the trapped ions—in his lab at the University of Mainz.

Ronnie Kosloff, of Hebrew University in Jerusalem, lectured about quantum engines. Nelly Ng, an Assistant Professor at Nanyang Technological University, has featured on Quantum Frontiers at least three times. She described resource theories—information-theoretic models—for thermodynamics. Information and energy both serve as resources in thermodynamics and computation, I explained in my lectures.

The 2024 ITAMP winter school

The winter school took place at the conference center adjacent to Biosphere 2. Biosphere 2 is an enclosure that contains several miniature climate zones, including a coastal fog desert, a rainforest, and an ocean. You might have heard of Biosphere 2 due to two experiments staged there during the 1990s: in each experiment, a group of people was sealed in the enclosure. The experimentalists harvested their own food and weren’t supposed to receive any matter from outside. The first experiment lasted for two years. The group, though, ran out of oxygen, which a support crew pumped in. Research at Biosphere 2 contributes to our understanding of ecosystems and space colonization.

Fascinating as the landscape inside Biosphere 2 is, so is the landscape outside. The winter school included an afternoon hike, and my husband and I explored the territory around the enclosure.

Did you see any snakes? my best friend asked after I returned home.

No, I said. But we were chased by a vicious beast. 

On our first afternoon, my husband and I followed an overgrown path away from the biosphere to an almost deserted-looking cluster of buildings. We eventually encountered what looked like a warehouse from which noises were emanating. Outside hung a sign with which I resonated.

Scientists, I thought. Indeed, a researcher emerged from the warehouse and described his work to us. His group was preparing to seal off a building where they were simulating a Martian environment. He also warned us about the territory we were about to enter, especially the creature that roosted there. We were too curious to retreat, though, so we set off into a ghost town.

At least, that’s what the other winter-school participants called the area, later in the week—a ghost town. My husband and I had already surveyed the administrative offices, conference center, and other buildings used by biosphere personnel today. Personnel in the 1980s used a different set of buildings. I don’t know why one site gave way to the other. But the old buildings survive—as what passes for ancient ruins to many Americans. 

Weeds have grown up in the cracks in an old parking lot’s tarmac. A sign outside one door says, “Classroom”; below it is a sign that must not have been correct in decades: “Class in progress.” Through the glass doors of the old visitors’ center, we glimpsed cushioned benches and what appeared to be a diorama exhibit; outside, feathers and bird droppings covered the ground. I searched for a tumbleweed emoji, to illustrate the atmosphere, but found only a tumbler one: 🥃.

After exploring, my husband and I rested in the shade of an empty building, drank some of the water we’d brought, and turned around. We began retracing our steps past the defunct visitors’ center. Suddenly, a monstrous Presence loomed on our right. 

I can’t tell you how large it was; I only glimpsed it before turning and firmly not running away. But the Presence loomed. And it confirmed what I’d guessed upon finding the feathers and droppings earlier: the old visitors’ center now served as the Lair of the Beast.

The Mars researcher had warned us about the aggressive male turkey who ruled the ghost town. The turkey, the researcher had said, hated men—especially men wearing blue. My husband, naturally, was wearing a blue shirt. You might be able to outrun him, the researcher added pensively.

My husband zipped up his black jacket over the blue shirt. I advised him to walk confidently and not too quickly. Hikes in bear country, as well as summers at Busch Gardens Zoo Camp, gave me the impression that we mustn’t run; the turkey would probably chase us, get riled up, and excite himself to violence. So we walked, and the monstrous turkey escorted us. For surprisingly and frighteningly many minutes. 

The turkey kept scolding us in monosyllabic squawks, which sounded increasingly close to the back of my head. I didn’t turn around to look, but he sounded inches away. I occasionally responded in the soothing voice I was taught to use on horses. But my husband and I marched increasingly quickly.

We left the old visitors’ center, curved around, and climbed most of a hill before ceasing to threaten the turkey—or before he ceased to threaten us. He squawked a final warning and fell back. My husband and I found ourselves amid the guest houses of workshops past, shaky but unmolested. Not that the turkey wreaks much violence, according to the Mars researcher: at most, he beats his wings against people and scratches up their cars (especially blue ones). But we were relieved to return to civilization.

Afternoon hike at Catalina State Park, a drive away from Biosphere 2. (Yes, that’s a KITP hat.)

The ITAMP winter school reminded me of Roughing It, a Mark Twain book I finished this year. Twain chronicled the adventures he’d experienced out West during the 1860s. The Gold Rush, he wrote, attracted the top young men of all nations. The quantum-technologies gold rush has been attracting the top young people of all nations, and the winter school evidenced their eagerness. Yet the winter school also evidenced how many women have risen to the top: 10 of the 24 registrants were women, as were four of the seven lecturers.1 

The winter-school participants in the shuttle I rode from the Tucson airport to Biosphere 2

We’ll see to what extent the quantum-technologies gold rush plays out like Mark Twain’s. Ours at least involves a ghost town and ferocious southwestern critters.

1For reference, when I applied to graduate programs, I was told that approximately 20% of physics PhD students nationwide were women. The percentage of women drops as one progresses up the academic chain to postdocs and then to faculty members. And primarily PhD students and postdocs registered for the winter school.

A classical foreshadow of John Preskill’s Bell Prize

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Editor’s Note: This post was co-authored by Hsin-Yuan Huang (Robert) and Richard Kueng.

John Preskill, Richard P. Feynman Professor of Theoretical Physics at Caltech, has been named the 2024 John Stewart Bell Prize recipient. The prize honors John’s contributions in “the developments at the interface of efficient learning and processing of quantum information in quantum computation, and following upon long standing intellectual leadership in near-term quantum computing.” The committee cited John’s seminal work defining the concept of the NISQ (noisy intermediate-scale quantum) era, our joint work “Predicting Many Properties of a Quantum System from Very Few Measurements” proposing the classical shadow formalism, along with subsequent research that builds on classical shadows to develop new machine learning algorithms for processing information in the quantum world.

We are truly honored that our joint work on classical shadows played a role in John winning this prize. But as the citation implies, this is also a much-deserved “lifetime achievement” award. For the past two and a half decades, first at IQI and now at IQIM, John has cultivated a wonderful, world-class research environment at Caltech that celebrates intellectual freedom, while fostering collaborations between diverse groups of physicists, computer scientists, chemists, and mathematicians. John has said that his job is to shield young researchers from bureaucratic issues, teaching duties and the like, so that we can focus on what we love doing best. This extraordinary generosity of spirit has been responsible for seeding the world with some of the bests minds in the field of quantum information science and technology.

A cartoon depiction of John Preskill (Middle), Hsin-Yuan Huang (Left), and Richard Kueng (Right). [Credit: Chi-Yun Cheng]

It is in this environment that the two of us (Robert and Richard) met and first developed the rudimentary form of classical shadows — inspired by Scott Aaronson’s idea of shadow tomography. While the initial form of classical shadows is mathematically appealing and was appreciated by the theorists (it was a short plenary talk at the premier quantum information theory conference), it was deemed too abstract to be of practical use. As a result, when we submitted the initial version of classical shadows for publication, the paper was rejected. John not only recognized the conceptual beauty of our initial idea, but also pointed us towards a direction that blossomed into the classical shadows we know today. Applications range from enabling scientists to more efficiently understand engineered quantum devices, speeding up various near-term quantum algorithms, to teaching machines to learn and predict the behavior of quantum systems.

Congratulations John! Thank you for bringing this community together to do extraordinarily fun research and for guiding us throughout the journey.

The rain in Portugal

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My husband taught me how to pronounce the name of the city where I’d be presenting a talk late last July: Aveiro, Portugal. Having studied Spanish, I pronounced the name as Ah-VEH-roh, with a v partway to a hard b. But my husband had studied Portuguese, so he recommended Ah-VAI-roo. 

His accuracy impressed me when I heard the name pronounced by the organizer of the conference I was participating in—Theory of Quantum Computation, or TQC. Lídia del Rio grew up in Portugal and studied at the University of Aveiro, so I bow to her in matters of Portuguese pronunciation. I bow to her also for organizing one of the world’s largest annual quantum-computation conferences (with substantial help—fellow quantum physicist Nuriya Nurgalieva shared the burden). But Lídia cofounded Quantum, a journal that’s risen from a Gedankenexperiment to a go-to venue in six years. So she gives the impression of being able to manage anything.

Aveiro architecture

Watching Lídia open TQC gave me pause. I met her in 2013, the summer before beginning my PhD at Caltech. She was pursuing her PhD at ETH Zürich, which I was visiting. Lídia took me dancing at an Argentine-tango studio one evening. Now, she’d invited me to speak at an international conference that she was coordinating.

Lídia and me in Zürich as PhD students
Lídia opening TQC

Not only Lídia gave me pause; so did the three other invited speakers. Every one of them, I’d met when each of us was a grad student or a postdoc. 

Richard Küng described classical shadows, a technique for extracting information about quantum states via measurements. Suppose we wish to infer about diverse properties of a quantum state \rho (about diverse observables’ expectation values). We have to measure many copies of \rho—some number n of copies. The community expected n to grow exponentially with the system’s size—for instance, with the number of qubits in a quantum computer’s register. We can get away with far fewer, Richard and collaborators showed, by randomizing our measurements. 

Richard postdocked at Caltech while I was a grad student there. Two properties of his stand out in my memory: his describing, during group meetings, the math he’d been exploring and the Austrian accent in which he described that math.

Did this restaurant’s owners realize that quantum physicists were descending on their city? I have no idea.

Also while I was a grad student, Daniel Stilck França visited Caltech. Daniel’s TQC talk conveyed skepticism about whether near-term quantum computers can beat classical computers in optimization problems. Near-term quantum computers are NISQ (noisy, intermediate-scale quantum) devices. Daniel studied how noise (particularly, local depolarizing noise) propagates through NISQ circuits. Imagine a quantum computer suffering from a 1% noise error. The quantum computer loses its advantage over classical competitors after 10 layers of gates, Daniel concluded. Nor does he expect error mitigation—a bandaid en route to the sutures of quantum error correction—to help much.

I’d coauthored a paper with the fourth invited speaker, Adam Bene Watts. He was a PhD student at MIT, and I was a postdoc. At the time, he resembled the 20th-century entanglement guru John Bell. Adam still resembles Bell, but he’s moved to Canada.

Adam speaking at TQC
From a 2021 Quantum Frontiers post of mine. I was tickled to see that TQC’s organizers used the photo from my 2021 post as Adam’s speaker photo.

Adam distinguished what we can compute using simple quantum circuits but not using simple classical ones. His results fall under the heading of complexity theory, about which one can rarely prove anything. Complexity theorists cling to their jobs by assuming conjectures widely expected to be true. Atop the assumptions, or conditions, they construct “conditional” proofs. Adam proved unconditional claims in complexity theory, thanks to the simplicity of the circuits he compared.

In my estimation, the talks conveyed cautious optimism: according to Adam, we can prove modest claims unconditionally in complexity theory. According to Richard, we can spare ourselves trials while measuring certain properties of quantum systems. Even Daniel’s talk inspired more optimism than he intended: a few years ago, the community couldn’t predict how noisy short-depth quantum circuits could perform. So his defeatism, rooted in evidence, marks an advance.

Aveiro nurtures optimism, I expect most visitors would agree. Sunshine drenches the city, and the canals sparkle—literally sparkle, as though devised by Elsa at a higher temperature than usual. Fresh fruit seems to wend its way into every meal.1 Art nouveau flowers scale the architecture, and fanciful designs pattern the tiled sidewalks.

What’s more, quantum information theorists of my generation were making good. Three riveted me in their talks, and another co-orchestrated one of the world’s largest quantum-computation gatherings. To think that she’d taken me dancing years before ascending to the global stage.

My husband and I made do, during our visit, by cobbling together our Spanish, his Portuguese, and occasional English. Could I hold a conversation with the Portuguese I gleaned? As adroitly as a NISQ circuit could beat a classical computer. But perhaps we’ll return to Portugal, and experimentalists are doubling down on quantum error correction. I remain cautiously optimistic.

1As do eggs, I was intrigued to discover. Enjoyed a hardboiled egg at breakfast? Have a fried egg on your hamburger at lunch. And another on your steak at dinner. And candied egg yolks for dessert.

This article takes its title from a book by former US Poet Laureate Billy Collins. The title alludes to a song in the musical My Fair Lady, “The Rain in Spain.” The song has grown so famous that I don’t think twice upon hearing the name. “The rain in Portugal” did lead me to think twice—and so did TQC.

With thanks to Lídia and Nuriya for their hospitality. You can submit to TQC2024 here.

Discoveries at the Dibner

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This past summer, our quantum thermodynamics research group had the wonderful opportunity to visit the Dibner Rare Book Library in D.C. Located in a small corner of the Smithsonian National Museum of American History, tucked away behind flashier exhibits, the Dibner is home to thousands of rare books and manuscripts, some dating back many centuries.

Our advisor, Nicole Yunger Halpern, has a special connection to the Dibner, having interned there as an undergrad. She’s remained in contact with the head librarian, Lilla Vekerdy. For our visit, the two of them curated a large spread of scientific work related to thermodynamics, physics, and mathematics. The tomes ranged from a 1500s print of Euclid’s Elements to originals of Einstein’s manuscripts with hand-written notes in the margin.

The print of Euclid’s Elements was one of the standout exhibits. It featured a number of foldout nets of 3D solids, which had been cut and glued into the book by hand. Several hundred copies of this print are believed to have been made, each of them containing painstakingly crafted paper models. At the time, this technique was an innovation, resulting from printers’ explorations of the then-young art of large-scale book publication.

Another interesting exhibit was rough notes on ideal gases written by Planck, one of the fathers of quantum mechanics. Ideal gases are the prototypical model in statistical mechanics, capturing to high accuracy the behaviour of real gases within certain temperatures and pressures. The notes contained comparisons between Boltzmann, Ehrenfest, and Planck’s own calculations for classical and quantum ideal gases. Though the prose was in German, some results were instantly recognizable, such as the plot of the specific heat of a classical ideal gas, showing the stepwise jump as degrees of freedom freeze out. 

Looking through these great physicists’ rough notes, scratched-out ideas, and personal correspondences was a unique experience, helping humanize them and place their work in historical context. Understanding the history of science doesn’t just need to be for historians, it can be useful for scientists themselves! Seeing how scientists persevered through unknowns, grappling with doubts and incomplete knowledge to generate new ideas, is inspiring. But when one only reads the final, polished result in a modern textbook, it can be difficult to appreciate this process of discovery. Another reason to study the historical development of scientific results is that core concepts have a way of arising time and again across science. Recognizing how these ideas have arisen in the past is insightful. Examining the creative processes of great scientists before us helps develop our own intuition and skillset.

Thanks to our advisor for this field trip – and make sure to check out the Dibner next time you’re in DC! 

“Once Upon a Time”…with a twist

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The Noncommuting-Charges World Tour (Part 1 of 4)

This is the first part in a four part series covering the recent Perspectives article on noncommuting charges. I’ll be posting one part every ~6 weeks leading up to my PhD thesis defence.

Thermodynamics problems have surprisingly many similarities with fairy tales. For example, most of them begin with a familiar opening. In thermodynamics, the phrase “Consider an isolated box of particles” serves a similar purpose to “Once upon a time” in fairy tales—both serve as a gateway to their respective worlds. Additionally, both have been around for a long time. Thermodynamics emerged in the Victorian era to help us understand steam engines, while Beauty and the Beast and Rumpelstiltskin, for example, originated about 4000 years ago. Moreover, each conclude with important lessons. In thermodynamics, we learn hard truths such as the futility of defying the second law, while fairy tales often impart morals like the risks of accepting apples from strangers. The parallels go on; both feature archetypal characters—such as wise old men and fairy godmothers versus ideal gases and perfect insulators—and simplified models of complex ideas, like portraying clear moral dichotomies in narratives versus assuming non-interacting particles in scientific models.1

Of all the ways thermodynamic problems are like fairytale, one is most relevant to me: both have experienced modern reimagining. Sometimes, all you need is a little twist to liven things up. In thermodynamics, noncommuting conserved quantities, or charges, have added a twist.

Unfortunately, my favourite fairy tale, ‘The Hunchback of Notre-Dame,’ does not start with the classic opening line ‘Once upon a time.’ For a story that begins with this traditional phrase, ‘Cinderella’ is a great choice.

First, let me recap some of my favourite thermodynamic stories before I highlight the role that the noncommuting-charge twist plays. The first is the inevitability of the thermal state. For example, this means that, at most times, the state of most sufficiently small subsystem within the box will be close to a specific form (the thermal state).

The second is an apparent paradox that arises in quantum thermodynamics: How do the reversible processes inherent in quantum dynamics lead to irreversible phenomena such as thermalization? If you’ve been keeping up with Nicole Yunger Halpern‘s (my PhD co-advisor and fellow fan of fairytale) recent posts on the eigenstate thermalization hypothesis (ETH) (part 1 and part 2) you already know the answer. The expectation value of a quantum observable is often comprised of a sum of basis states with various phases. As time passes, these phases tend to experience destructive interference, leading to a stable expectation value over a longer period. This stable value tends to align with that of a thermal state’s. Thus, despite the apparent paradox, stationary dynamics in quantum systems are commonplace.

The third story is about how concentrations of one quantity can cause flows in another. Imagine a box of charged particles that’s initially outside of equilibrium such that there exists gradients in particle concentration and temperature across the box. The temperature gradient will cause a flow of heat (Fourier’s law) and charged particles (Seebeck effect) and the particle-concentration gradient will cause the same—a flow of particles (Fick’s law) and heat (Peltier effect). These movements are encompassed within Onsager’s theory of transport dynamics…if the gradients are very small. If you’re reading this post on your computer, the Peltier effect is likely at work for you right now by cooling your computer.

What do various derivations of the thermal state’s forms, the eigenstate thermalization hypothesis (ETH), and the Onsager coefficients have in common? Each concept is founded on the assumption that the system we’re studying contains charges that commute with each other (e.g. particle number, energy, and electric charge). It’s only recently that physicists have acknowledged that this assumption was even present.

This is important to note because not all charges commute. In fact, the noncommutation of charges leads to fundamental quantum phenomena, such as the Einstein–Podolsky–Rosen (EPR) paradox, uncertainty relations, and disturbances during measurement. This raises an intriguing question. How would the above mentioned stories change if we introduce the following twist?

“Consider an isolated box with charges that do not commute with one another.” 

This question is at the core of a burgeoning subfield that intersects quantum information, thermodynamics, and many-body physics. I had the pleasure of co-authoring a recent perspective article in Nature Reviews Physics that centres on this topic. Collaborating with me in this endeavour were three members of Nicole’s group: the avid mountain climber, Billy Braasch; the powerlifter, Aleksander Lasek; and Twesh Upadhyaya, known for his prowess in street basketball. Completing our authorship team were Nicole herself and Amir Kalev.

To give you a touchstone, let me present a simple example of a system with noncommuting charges. Imagine a chain of qubits, where each qubit interacts with its nearest and next-nearest neighbours, such as in the image below.

The figure is courtesy of the talented team at Nature. Two qubits form the system S of interest, and the rest form the environment E. A qubit’s three spin components, σa=x,y,z, form the local noncommuting charges. The dynamics locally transport and globally conserve the charges.

In this interaction, the qubits exchange quanta of spin angular momentum, forming what is known as a Heisenberg spin chain. This chain is characterized by three charges which are the total spin components in the x, y, and z directions, which I’ll refer to as Qx, Qy, and Qz, respectively. The Hamiltonian H conserves these charges, satisfying [H, Qa] = 0 for each a, and these three charges are non-commuting, [Qa, Qb] ≠ 0, for any pair a, b ∈ {x,y,z} where a≠b. It’s noteworthy that Hamiltonians can be constructed to transport various other kinds of noncommuting charges. I have discussed the procedure to do so in more detail here (to summarize that post: it essentially involves constructing a Koi pond).

This is the first in a series of blog posts where I will highlight key elements discussed in the perspective article. Motivated by requests from peers for a streamlined introduction to the subject, I’ve designed this series specifically for a target audience: graduate students in physics. Additionally, I’m gearing up to defending my PhD thesis on noncommuting-charge physics next semester and these blog posts will double as a fun way to prepare for that.

  1. This opening text was taken from the draft of my thesis. ↩︎