Tag Archives: quantum thermodynamics

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.

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.