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.

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.