Author Archives: Shayan Majidy

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 ~6 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.

Noncommuting charges are much like Batman

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The Noncommuting-Charges World Tour Part 2 of 4

This is the second part in a four part series covering the recent Perspective on noncommuting charges. I’ll be posting one part every 6 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.

“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. ↩︎

Identical twins and quantum entanglement

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“If I had a nickel for every unsolicited and very personal health question I’ve gotten at parties, I’d have paid off my medical school loans by now,” my doctor friend complained. As a physicist, I can somewhat relate. I occasionally find myself nodding along politely to people’s eccentric theories about the universe. A gentleman once explained to me how twin telepathy (the phenomenon where, for example, one twin feels the other’s pain despite being in separate countries) comes from twins’ brains being entangled in the womb. Entanglement is a nonclassical correlation that can exist between spatially separated systems. If two objects are entangled, it’s possible to know everything about both of them together but nothing about either one. Entangling two particles (let alone full brains) over tens of kilometres (let alone full countries) is incredibly challenging. “Using twins to study entanglement, that’ll be the day,” I thought. Well, my last paper did something like that. 

In theory, a twin study consists of two people that are as identical as possible in every way except for one. What that allows you to do is isolate the effect of that one thing on something else. Aleksander Lasek (postdoc at QuICS), David Huse (professor of physics at Princeton), Nicole Yunger Halpern (NIST physicist and Quantum Frontiers blogger), and I were interested in isolating the effects of quantities’ noncommutation (explained below) on entanglement. To do so, we first built a pair of twins and then compared them. 

Consider a well-insulated thermos filled with soup. The heat and the number of “soup particles” inside the thermos are conserved. So the energy and the number of “soup particles” are conserved quantities. In classical physics, conserved quantities commute. This means that we can simultaneously measure the amount of each conserved quantity in our system, like the energy and number of soup particles. However, in quantum mechanics, this needn’t be true. Measuring one property of a quantum system can change another measurement’s outcome.

Conserved quantities’ noncommutation in thermodynamics has led to some interesting results. For example, it’s been shown that conserved quantities’ noncommutation can decrease the rate of entropy production. For the purposes of this post, entropy production is something that limits engine efficiency—how well engines can convert fuel to useful work. For example, if your car engine had zero entropy production (which is impossible), it would convert 100% of the energy in your car’s fuel into work that moved your car along the road. Current car engines can convert about 30% of this energy, so it’s no wonder that people are excited about the prospective application of decreasing entropy production. Other results (like this one and that one) have connected noncommutation to potentially hindering thermalization—the phenomenon where systems interact until they have similar properties, like when a cup of coffee cools. Thermalization limits memory storage and battery lifetimes. Thus, learning how to resist thermalization could also potentially lead to better technologies, such as longer-lasting batteries. 

One can measure the amount of entanglement within a system, and as quantum particles thermalize, they entangle. Given the above results about thermalization, we might expect that noncommutation would decrease entanglement. Testing this expectation is where the twins come in.

Say we built a pair of twins that were identical in every way except for one. Nancy, the noncommuting twin, has some features that don’t commute, say, her hair colour and height. This means that if we measure her height, we’ll have no idea what her hair colour is. For Connor, the commuting twin, his hair colour and height commute, so we can determine them both simultaneously. Which twin has more entanglement? It turns out it’s Nancy.

Disclaimer: This paragraph is written for an expert audience. Our actual models consist of 1D chains of pairs of qubits. Each model has three conserved quantities (“charges”), which are sums over local charges on the sites. In the noncommuting model, the three local charges are tensor products of Pauli matrices with the identity (XI, YI, ZI). In the commuting model, the three local charges are tensor products of the Pauli matrices with themselves (XX, YY, ZZ). The paper explains in what sense these models are similar. We compared these models numerically and analytically in different settings suggested by conventional and quantum thermodynamics. In every comparison, the noncommuting model had more entanglement on average.

Our result thus suggests that noncommutation increases entanglement. So does charges’ noncommutation promote or hinder thermalization? Frankly, I’m not sure. But I’d bet the answer won’t be in the next eccentric theory I hear at a party.

Mo’ heights mo’ challenges – Climbing mount grad school

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My wife’s love of mountain hiking and my interest in quantum thermodynamics collided in Telluride, Colorado.

We spent ten days in Telluride, where I spoke at the Information Engines at the Frontiers of Nanoscale Thermodynamics workshop. Telluride is a gorgeous city surrounded by mountains and scenic trails. My wife (Hasti) and I were looking for a leisurely activity one morning. We chose hiking Bear Creek Trail, a 5.1-mile hike with a 1092-foot elevation. This would have been a reasonable choice… back home.

Telluride’s elevation is 8,750 feet (ten times that of our hometown’s). This meant there was nothing leisurely about the hike. Ill-prepared, I dragged myself up the mountain in worn runners and tight jeans. My gasps for breath reminded me how new heights (a literal one in this case) could bring unexpected challenges – A lesson I’ve encountered many times as a graduate student. 

My wife and I atop Bear Creek trail

I completely squandered my undergrad. One story sums it up best. I was studying for my third-year statistical mechanics final when I realized I could pass the course without writing the final. So, I didn’t write the final. After four years of similar negligence, I somehow graduated, certain I’d left academics forever. Two years later, I rediscovered my love for physics and grieved about wasting my undergraduate. I decided to try again and apply for grad school. 

After knocking on his door and pleading my case, Raymond Laflamme (one of the co-founders of the field of quantum computing) decided to overlook my past and take a chance on me. I would work at the Institute for Quantum Computing (IQC), supervised by Raymond. My first day at IQC felt surreal. I had become an efficient student and felt ready for the IQC. But, like the Bear Creek trail, a new height would bring a new challenge. Ultimately, grad school isn’t about getting good grades; it’s about researching. Raymond (Ray) gave me my first research project, and I was dumbfounded about where to start and too insecure to ask for help.

With the guidance of Ray and Jonathan Halliwell (professor at Imperial College London and guitarist-extraordinaire), I published my first paper and accepted a Ph.D. offer from Ray. After publishing my second paper, I thought it would be smooth sailing through my Ph.D.  Alas, I was again mistaken. It’s also not enough to solve problems others give you; you need to develop some problems independently. So, I tried. I spent the first 8-months of my Ph.D. pursuing a problem I came up with, and It was a complete dud. It turns out the problems also need to be worth solving. For those keeping track, this is challenge number three.

I have now finished the second year of my Ph.D. During that time, Nicole Yunger Halpern (NIST physicist and Quantum Frontiers blogger) introduced me to the field of quantum thermodynamics. We’ve published a paper together (related blog post and Vanier interview) and have a second on the way. Despite that, I’m still grappling with that last challenge. I have no problem finding research questions that would be fun to solve. However, I’m still not sure which ones are worth solving. But, like the other challenges, I’m hopeful I’ll figure it out.

While this lesson was inspiring, the city of Telluride inspired me the most. Telluride is at a local minimum elevation, surrounded by mountains. Meaning there is virtually nowhere to go but up. I’m hoping the same is true for me.

Building a Koi pond with Lie algebras

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When I was growing up, one of my favourite places was the shabby all-you-can-eat buffet near our house. We’d walk in, my mom would approach the hostess to explain that, despite my being abnormally large for my age, I qualified for kids-eat-free, and I would peel away to stare at the Koi pond. The display of different fish rolling over one another was bewitching. Ten-year-old me would have been giddy to build my own Koi pond, and now I finally have. However, I built one using Lie algebras.

The different fish swimming in the Koi pond are, in many ways, like charges being exchanged between subsystems. A “charge” is any globally conserved quantity. Examples of charges include energy, particles, electric charge, or angular momentum. Consider a system consisting of a cup of coffee in your office. The coffee will dynamically exchange charges with your office in the form of heat energy. Still, the total energy of the coffee and office is conserved (assuming your office walls are really well insulated). In this example, we had one type of charge (heat energy) and two subsystems (coffee and office). Consider now a closed system consisting of many subsystems and many different types of charges. The closed system is like the finite Koi pond with different charges like the different fish species. The charges can move around locally, but the total number of charges is globally fixed, like how the fish swim around but can’t escape the pond. Also, the presence of one type of charge can alter another’s movement, just as a big fish might block a little one’s path. 

Unfortunately, the Koi pond analogy reaches its limit when we move to quantum charges. Classically, charges commute. This means that we can simultaneously determine the amount of each charge in our system at each given moment. In quantum mechanics, this isn’t necessarily true. In other words, classically, I can count the number of glossy fish and matt fish. But, in quantum mechanics, I can’t.

So why does this matter? Subsystems exchanging charges are prevalent in thermodynamics. Quantum thermodynamics extends thermodynamics to include small systems and quantum effects. Noncommutation underlies many important quantum phenomena. Hence, studying the exchange of noncommuting charges is pivotal in understanding quantum thermodynamics. Consequently, noncommuting charges have emerged as a rapidly growing subfield of quantum thermodynamics. Many interesting results have been discovered from no longer assuming that charges commute (such as these). Until recently, most of these discoveries have been theoretical. Bridging these discoveries to experimental reality requires Hamiltonians (functions that tell you how your system evolves in time) that move charges locally but conserve them globally. Last year it was unknown whether these Hamiltonians exist, what they look like generally, how to build them, and for what charges you could find them.

Nicole Yunger Halpern (NIST physicist, my co-advisor, and Quantum Frontiers blogger) and I developed a prescription for building Koi ponds for noncommuting charges. Our prescription allows you to systematically build Hamiltonians that overtly move noncommuting charges between subsystems while conserving the charges globally. These Hamiltonians are built using Lie algebras, abstract mathematical tools that can describe many physical quantities (including everything in the standard model of particle physics and space-time metric). Our results were recently published in npj QI. We hope that our prescription will bolster the efforts to bridge the results of noncommuting charges to experimental reality.

In the end, a little group theory was all I needed for my Koi pond. Maybe I’ll build a treehouse next with calculus or a remote control car with combinatorics.