If I ever mention a crazy high-school English teacher to you, I might be referring to Mr. Lukacs. One morning, before the first bell rang, I found him wandering among the lockers, wearing a white beard and a mischievous grin. (The school had pronounced the day “Dress Up as Your Favorite Writer” Day, or some such designation, but still.1) Mr. Lukacs was carrying a copy of Leaves of Grass, a book by the nineteenth-century American poet Walt Whitman, and yawping. To yawp is to cry out, and Whitman garnered acclaim for weaving such colloquialisms into his poetry. “I sound my barbaric yawp over the roofs of the world,” he wrote in Leaves of Grass—as Mr. Lukacs illustrated until the bells rang for class. And, for all I know, until the final bell.
I call Mr. Lukacs one of my crazy high-school English teachers despite never having taken any course of his.2 He served as the faculty advisor for the school’s literary magazine, on whose editorial board I served. As a freshman and sophomore, I kept my head down and scarcely came to know Mr. Lukacs. He wore small, round glasses and a bowtie. As though to ham up the idiosyncrasy, he kept a basket of bowties in his classroom. His hair had grayed, he spoke slowly, and he laughed in startling little bursts that resembled gasps.
Junior year, I served as co-editor-in-chief of the literary magazine; and, senior year, as editor-in-chief. I grew to conjecture that Mr. Lukacs spoke slowly because he was hunting for the optimal word to use next. Finding that word cost him a pause, but learning his choice enriched the listener. And Mr. Lukacs adored literature. You could hear, when he read aloud, how he invested himself in it.
I once submitted to the literary magazine a poem about string theory, inspired by a Brian Greene book.3 As you might expect, if you’ve ever read about string theory, the poem invoked music. Mr. Lukacs pretended to no expertise in science; he even had a feud with the calculus teacher.4 But he wrote that the poem made him feel like dancing.
You might fear that Mr. Lukacs too strongly echoed the protagonist of Dead Poets Society to harbor any originality. The 1989 film Dead Poets Society stars Robin Williams as an English teacher who inspires students to discover their own voices, including by yawping à la Whitman. But Mr. Lukacs leaned into the film, with a gleeful sort of exultation. He even interviewed one of the costars, who’d left acting to teach, for a job. The interview took place beside a cardboard-cutout advertisement for Dead Poets Society—a possession, I’m guessing, of Mr. Lukacs’s.
This winter, friends of Mr. Lukacs’s helped him create a Youtube video for his former students. He sounded as he had twenty years before. But he said goodbye, expecting his cancer journey to end soon. Since watching the video, I’ve been waffling between reading Goodbye, Mr. Chips—a classic novella I learned of around the time the video debuted—and avoiding it. I’m not sure what Mr. Lukacs would advise—probably to read, rather than not to read. But I like the thought of saluting a literary-magazine advisor on Quantum Frontiers. We became Facebook friends years ago; and, although I’ve rarely seen activity by him, he’s occasionally effused over some physics post of mine.
Physics brought me to the Washington, DC area, where a Whitman quote greets entrants to the Dupont Circle metro station. The DC area also houses Abraham Lincoln’s Cottage, where the president moved with his wife. They sought quietude to mourn their son Willie, who’d succumbed to an illness. Lincoln rode from the cottage to the White House every day. Whitman lived along his commute, according to a panel in the visitors’ center. I was tickled to learn that the two men used to exchange bows during that commute—one giant of politics and one giant of literature.
I wrote the text above this paragraph, as well as the text below, within a few weeks of watching the Youtube video. The transition between the two bothered me; it felt too abrupt. But I asked Mr. Lukacs via email whether he’d mind my posting the story. I never heard back. I learned why this weekend: he’d passed away on Friday. The announcement said, “please consider doing something that reminds you of George in the coming days. Read a few lines of a cherished text. Marvel at a hummingbird…” So I determined to publish the story without approval. I can think of no tribute more fitting than a personal essay published on a quantum blog that’s charted my intellectual journey of the past decade.
Here’s to another giant of literature. Goodbye, Mr. Lukacs.
Image from wmata.com
1I was too boring to dress up as anyone.
2I call him one of my crazy high-school English teachers because his wife merits the epithet, too. She called herself senile, enacted the climax of Jude the Obscure with a student’s person-shaped pencil case, and occasionally imitated a chipmunk; but damn, do I know my chiasmus from my caesura because of her.
3That fact sounds hackneyed to me now. But I’m proud never to have entertained grand dreams of discovering a theory of everything.
4AKA my crazy high-school calculus teacher. My high school had loads of crazy teachers, but it also had loads of excellent teachers, and the crazy ones formed a subset of the excellent ones.
It’s the beginning of another summer, and I’m looking forward to outdoor barbecues, swimming in lakes and pools, and sharing my home-made ice cream with friends and family. One thing that I won’t encounter this summer, but I did last year, is a Canadian goose. In summer 2023, I ventured north from the University of Maryland – College Park to Waterloo, Canada, for a position at the University of Waterloo. The university houses the Institute for Quantum Computing (IQC), and the Perimeter Institute (PI) for Theoretical Physics is nearby. I spent my summer at these two institutions because I was accepted into the IQC’s Undergraduate School on Experimental Quantum Information Processing (USEQIP) and received an Undergraduate Research Award. I’ll detail my experiences in the program and the fun social activities I participated in along the way.
For my first two weeks in Waterloo, I participated in USEQIP. This program is an intense boot camp in quantum hardware. I learned about many quantum-computing platforms, including trapped ions, superconducting circuits, and nuclear magnetic resonance systems. There were interactive lab sessions where I built a low-temperature thermometer, assembled a quantum key distribution setup, and designed an experiment of the Quantum Zeno Effect using nuclear magnetic resonance systems. We also toured the IQC’s numerous research labs and their nano-fabrication clean room. I learned a lot from these two weeks, and I settled into life in goose-filled Waterloo, trying to avoid goose poop on my daily walks around campus.
I pour liquid nitrogen into a low-temperature container.
Once USEQIP ended, I began the work for my Undergraduate Research Award, joining Dr. Raymond Laflamme’s group. My job was to read Dr. Laflamme’s soon-to-be-published textbook about quantum hardware, which he co-wrote with graduate student Shayan Majidy and Dr. Chris Wilson. I read through the sections for clarity and equation errors. I also worked through the textbook’s exercises to ensure they were appropriate for the book. Additionally, I contributed figures to the book.
The most challenging part of this work was completing the exercises. I would become frustrated with the complex problems, sometimes toiling over a single problem for over three hours. My frustrations were aggravated when I asked Shayan for help, and my bitter labor was to him a simple trick I had not seen. I had to remind myself that I had been asked to test drive this textbook because I am the target audience for it. I offered an authentic undergraduate perspective on the material that would be valuable to the book’s development. Despite the challenges, I successfully completed my book review, and Shayan sent the textbook for publication at the beginning of August.
After, I moved on to another project. I worked on the quantum thermodynamics research that I conduct with Dr. Nicole Yunger Halpern. My work with Dr. Yunger Halpern concerns systems with noncommuting charges. I run numerical calculations on these systems to understand how they thermalize internally. I enjoyed working at both the IQC and the Perimeter Institute with their wonderful office views and free coffee.
Dr. Laflamme and I at the Perimeter Institute on my last day in Waterloo.
Midway through the summer, Dr. Laflamme’s former and current students celebrated his 60th birthday with a birthday conference. As one of his newest students, I had a wonderful time meeting many of his past students who’ve had exciting careers following their graduation from the group. During the birthday conference, we had six hours of talks daily, but these were not traditional research talks. The talks were on any topic the speaker wanted to share with the audience. I learned about how a senior data scientist at TD Bank uses machine learning, a museum exhibit organized by the University of Waterloo called Quantum: The Exhibition, and photonic quantum science at the Raman Research Institute. For the socializing portion, we played street hockey and enjoyed delicious sushi, sandwiches, and pastries. By coincidence, Dr. Laflamme’s birthday and mine are one day apart!
Outside of my work, I spent almost every weekend exploring Ontario. I beheld the majesty of Niagara Falls for the first time; I visited Canada’s wine country, Niagara on the Lake; I met with friends and family in Toronto; I stargazed with the hope of seeing the aurora borealis (unfortunately, the Northern Lights did not appear). I also joined a women’s ultimate frisbee team, PPF (sorry, we can’t tell you what it stands for), during my stay in Canada. I had a blast getting to play while sharpening my skills for the collegiate ultimate frisbee season. Finally, my summer would not have been great without the friendships that I formed with my fellow USEQIP undergraduates. We shared more than just meals; we shared our hopes and dreams, and I am so lucky to have met such inspiring people.
I spent my first weekend in Canada at Niagara Falls.
Though my summer in Waterloo has come to an end now, I’ll never forget the incredible experiences I had.
This is the final part of a four-part series covering the recent Perspective on noncommuting charges. I’ve been posting one part every ~5 weeks leading up to my PhD thesis defence. You can find Part 1 here, Part 2 here, and Part 3 here.
In four months, I’ll embark on the adventure of a lifetime—fatherhood.
To prepare, I’ve been honing a quintessential father skill—storytelling. If my son inherits even a fraction of my tastes, he’ll soon develop a passion for film noir detective stories. And really, who can resist the allure of a hardboiled detective, a femme fatale, moody chiaroscuro lighting, and plot twists that leave you reeling? For the uninitiated, here’s a quick breakdown of the genre.
To sharpen my storytelling skills, I’ve decided to channel my inner noir writer and craft this final blog post—the opportunities for future work, as outlined in the Perspective—in that style.
I wouldn’t say film noir needs to be watched in black and white like how I wouldn’t say jazz needs to be listened to on vinyl. But it adds a charm that’s hard to replicate.
Theft at the Quantum Frontier
Under the dim light of a flickering bulb, private investigator Max Kelvin leaned back in his creaky chair, nursing a cigarette. The steady patter of rain against the window was interrupted by the creak of the office door. In walked trouble. Trouble with a capital T.
She was tall, moving with a confident stride that barely masked the worry lines etched into her face. Her dark hair was pulled back in a tight bun, and her eyes were as sharp as the edges of the papers she clutched in her gloved hand.
“Mr. Kelvin?” she asked, her voice a low, smoky whisper.
“That’s what the sign says,” Max replied, taking a long drag of his cigarette, the ember glowing a fiery red. “What can I do for you, Miss…?”
“Doctor,” she corrected, her tone firm, “Shayna Majidy. I need your help. Someone’s about to scoop my research.”
Max’s eyebrows arched. “Scooped? You mean someone stole your work?”
“Yes,” Shayna said, frustration seeping into her voice. “I’ve been working on noncommuting charge physics, a topic recently highlighted in a Perspective article. But someone has stolen my paper. We need to find who did it before they send it to the local rag, The Ark Hive.”
Max leaned forward, snuffing out his cigarette and grabbing his coat in one smooth motion. “Alright, Dr. Majidy, let’s see where your work might have wandered off to.”
They started their investigation with Joey “The Ant” Guzman, an experimental physicist whose lab was a tangled maze of gleaming equipment. Superconducting qubits, quantum dots, ultracold atoms, quantum optics, and optomechanics cluttered the room, each device buzzing with the hum of cutting-edge science. Joey earned his nickname due to his meticulous and industrious nature, much like an ant in its colony.
Guzman was a prime suspect, Shayna had whispered as they approached. His experiments could validate the predictions of noncommuting charges. “The first test of noncommuting-charge thermodynamics was performed with trapped ions,” she explained, her voice low and tense. “But there’s a lot more to explore—decreased entropy production rates, increased entanglement, to name a couple. There are many platforms to test these results, and Guzman knows them all. It’s a major opportunity for future work.”
Guzman looked up from his work as they entered, his expression guarded. “Can I help you?” he asked, wiping his hands on a rag.
Max stepped forward, his eyes scanning the room. “A rag? I guess you really are a quantum mechanic.” He paused for laughter, but only silence answered. “We’re investigating some missing research,” he said, his voice calm but edged with intensity. “You wouldn’t happen to know anything about noncommuting charges, would you?”
Guzman’s eyes narrowed, a flicker of suspicion crossing his face. “Almost everyone is interested in that right now,” he replied cautiously.
Shayna stepped forward, her eyes boring into Guzman’s. “So what’s stopping you from doing experimental tests? Do you have enough qubits? Long enough decoherence times?”
Guzman shifted uncomfortably but kept his silence. Max took another drag of his cigarette, the smoke curling around his thoughts. “Alright, Guzman,” he said finally. “If you think of anything that might help, you know where to find us.”
As they left the lab, Max turned to Shayna. “He’s hiding something,” he said quietly. “But whether it’s your work or how noisy and intermediate scale his hardware is, we need more to go on.”
Shayna nodded, her face set in grim determination. The rain had stopped, but the storm was just beginning.
I bless the night my mom picked up “Who Framed Roger Rabbit” at Blockbuster. That, along with the criminally underrated “Dog City,” likely ignited my love for the genre.
Their next stop was the dimly lit office of Alex “Last Piece” Lasek, a puzzle enthusiast with a sudden obsession with noncommuting charge physics. The room was a chaotic labyrinth, papers strewn haphazardly, each covered with intricate diagrams and cryptic scrawlings. The stale aroma of old coffee and ink permeated the air.
Lasek was hunched over his desk, scribbling furiously, his eyes darting across the page. He barely acknowledged their presence as they entered. “Noncommuting charges,” he muttered, his voice a gravelly whisper, “they present a fascinating puzzle. They hinder thermalization in some ways and enhance it in others.”
“Last Piece Lasek, I presume?” Max’s voice sliced through the dense silence.
Lasek blinked, finally lifting his gaze. “Yeah, that’s me,” he said, pushing his glasses up the bridge of his nose. “Who wants to know?”
“Max Kelvin, private eye,” Max replied, flicking his card onto the cluttered desk. “And this is Dr. Majidy. We’re investigating some missing research.”
Shayna stepped forward, her eyes sweeping the room like a hawk. “I’ve read your papers, Lasek,” she said, her tone a blend of admiration and suspicion. “You live for puzzles, and this one’s as tangled as they come. How do you plan to crack it?”
Lasek shrugged, leaning back in his creaky chair. “It’s a tough nut,” he admitted, a sly smile playing at his lips. “But I’m no thief, Dr. Majidy. I’m more interested in solving the puzzle than in academic glory.”
As they exited Lasek’s shadowy lair, Max turned to Shayna. “He’s a riddle wrapped in an enigma, but he doesn’t strike me as a thief.”
Shayna nodded, her expression grim. “Then we keep digging. Time’s slipping away, and we’ve got to find the missing pieces before it’s too late.”
Their third stop was the office of Billy “Brass Knuckles,” a classical physicist infamous for his no-nonsense attitude and a knack for punching holes in established theories.
Max’s skepticism was palpable as they entered the office. “He’s a classical physicist; why would he give a damn about noncommuting charges?” he asked Shayna, raising an eyebrow.
Billy, overhearing Max’s question, let out a gravelly chuckle. “It’s not as crazy as it sounds,” he said, his eyes glinting with amusement. “Sure, the noncommutation of observables is at the core of quantum quirks like uncertainty, measurement disturbances, and the Einstein-Podolsky-Rosen paradox.”
Max nodded slowly, “Go on.”
“However,” Billy continued, leaning forward, “classical mechanics also deals with quantities that don’t commute, like rotations around different axes. So, how unique is noncommuting-charge thermodynamics to the quantum realm? What parts of this new physics can we find in classical systems?”
Shayna crossed her arms, a devious smile playing on her lips. “Wouldn’t you like to know?”
“Wouldn’t we all?” Billy retorted, his grin mirroring hers. “But I’m about to retire. I’m not the one sneaking around your work.”
Max studied Billy for a moment longer, then nodded. “Alright, Brass Knuckles. Thanks for your time.”
As they stepped out of the shadowy office and into the damp night air, Shayna turned to Max. “Another dead end?”
Max nodded and lit a cigarette, the smoke curling into the misty air. “Seems so. But the clock’s ticking, and we can’t afford to stop now.”
If you want contemporary takes on the genre, Sin City (2005), Memento (2000), and L.A. Confidential (1997) each deliver in their own distinct ways.
Their fourth suspect, Tony “Munchies” Munsoni, was a specialist in chaos theory and thermodynamics, with an insatiable appetite for both science and snacks.
“Another non-quantum physicist?” Max muttered to Shayna, raising an eyebrow.
Shayna nodded, a glint of excitement in her eyes. “The most thrilling discoveries often happen at the crossroads of different fields.”
Dr. Munson looked up from his desk as they entered, setting aside his bag of chips with a wry smile. “I’ve read the Perspective article,” he said, getting straight to the point. “I agree—every chaotic or thermodynamic phenomenon deserves another look under the lens of noncommuting charges.”
Max leaned against the doorframe, studying Munsoni closely.
“We’ve seen how they shake up the Eigenstate Thermalization Hypothesis, monitored quantum circuits, fluctuation relations, and Page curves,” Munson continued, his eyes alight with intellectual fervour. “There’s so much more to uncover. Think about their impact on diffusion coefficients, transport relations, thermalization times, out-of-time-ordered correlators, operator spreading, and quantum-complexity growth.”
Shayna leaned in, clearly intrigued. “Which avenue do you think holds the most promise?”
Munsoni’s enthusiasm dimmed slightly, his expression turning regretful. “I’d love to dive into this, but I’m swamped with other projects right now. Give me a few months, and then you can start grilling me.”
Max glanced at Shayna, then back at Munsoni. “Alright, Munchies. If you hear anything or stumble upon any unusual findings, keep us in the loop.”
As they stepped back into the dimly lit hallway, Max turned to Shayna. “I saw his calendar; he’s telling the truth. His schedule is too packed to be stealing your work.”
Shayna’s shoulders slumped slightly. “Maybe. But we’re not done yet. The clock’s ticking, and we’ve got to keep moving.”
Finally, they turned to a pair of researchers dabbling in the peripheries of quantum thermodynamics. One was Twitch Uppity, an expert on non-Abelian gauge theories. The other, Jada LeShock, specialized in hydrodynamics and heavy-ion collisions.
Max leaned against the doorframe, his voice casual but probing. “What exactly are non-Abelian gauge theories?” he asked (setting up the exposition for the Quantum Frontiers reader’s benefit).
Uppity looked up, his eyes showing the weary patience of someone who had explained this concept countless times. “Imagine different particles interacting, like magnets and electric charges,” he began, his voice steady. “We describe the rules for these interactions using mathematical objects called ‘fields.’ These rules are called field theories. Electromagnetism is one example. Gauge theories are a class of field theories where the laws of physics are invariant under certain local transformations. This means that a gauge theory includes more degrees of freedom than the physical system it represents. We can choose a ‘gauge’ to eliminate the extra degrees of freedom, making the math simpler.”
Max nodded slowly, his eyes fixed on Uppity. “Go on.”
“These transformations form what is called a gauge group,” Uppity continued, taking a sip of his coffee. “Electromagnetism is described by the gauge group U(1). Other interactions are described by more complex gauge groups. For instance, quantum chromodynamics, or QCD, uses an SU(3) symmetry and describes the strong force between particles in an atom. QCD is a non-Abelian gauge theory because its gauge group is noncommutative. This leads to many intriguing effects.”
“I see the noncommuting part,” Max stated, trying to keep up. “But, what’s the connection to noncommuting charges in quantum thermodynamics?”
“That’s the golden question,” Shayna interjected, excitement in her voice. “In QCD, particle physics uses non-Abelian groups, so it may exhibit phenomena related to noncommuting charges in thermodynamics.”
“May is the keyword,” Uppity replied. “In QCD, the symmetry is local, unlike the global symmetries described in the Perspective. An open question is how much noncommuting-charge quantum thermodynamics applies to non-Abelian gauge theories.”
Max turned his gaze to Jada. “How about you? What are hydrodynamics and heavy-ion collisions?” he asked, setting up more exposition.
Jada dropped her pencil and raised her head. “Hydrodynamics is the study of fluid motion and the forces acting on them,” she began. “We focus on large-scale properties, assuming that even if the fluid isn’t in equilibrium as a whole, small regions within it are. Hydrodynamics can explain systems in condensed matter and stages of heavy-ion collisions—collisions between large atomic nuclei at high speeds.”
“Where does the non-Abelian part come in?” Max asked, his curiosity piqued.
“Hydrodynamics researchers have identified specific effects caused by non-Abelian symmetries,” Jada answered. “These include non-Abelian contributions to conductivity, effects on entropy currents, and shortening neutralization times in heavy-ion collisions.”
“Are you looking for more effects due to non-Abelian symmetries?” Shayna asked, her interest clear. “A long-standing question is how heavy-ion collisions thermalize. Maybe the non-Abelian ETH would help explain this?”
Jada nodded, a faint smile playing on her lips. “That’s the hope. But as with all cutting-edge research, the answers are elusive.”
Max glanced at Shayna, his eyes thoughtful. “Let’s wrap this up. We’ve got some thinking to do.”
After hearing from each researcher, Max and Shayna found themselves back at the office. The dim light of the flickering bulb cast long shadows on the walls. Max poured himself a drink. He offered one to Shayna, who declined, her eyes darting around the room, betraying her nerves.
“So,” Max said, leaning back in his chair, the creak of the wood echoing in the silence. “Everyone seems to be minding their own business. Well…” Max paused, taking a slow sip of his drink, “almost everyone.”
Shayna’s eyes widened, a flicker of panic crossing her face. “I’m not sure who you’re referring to,” she said, her voice wavering slightly. “Did you figure out who stole my work?” She took a seat, her discomfort apparent.
Max stood up and began circling Shayna’s chair like a predator stalking its prey. His eyes were sharp, scrutinizing her every move. “I couldn’t help but notice all the questions you were asking and your eyes peeking onto their desks.”
Shayna sighed, her confident façade cracking under the pressure. “You’re good, Max. Too good… No one stole my work.” Shayna looked down, her voice barely above a whisper. “I read that Perspective article. It mentioned all these promising research avenues. I wanted to see what others were working on so I could get a jump on them.”
Max shook his head, a wry smile playing on his lips. “You tried to scoop the scoopers, huh?”
Shayna nodded, looking somewhat sheepish. “I guess I got a bit carried away.”
Max chuckled, pouring himself another drink. “Science is a tough game, Dr. Majidy. Just make sure next time you play fair.”
As Shayna left the office, Max watched the rain continue to fall outside. His thoughts lingered on the strange case, a world where the race for discovery was cutthroat and unforgiving. But even in the darkest corners of competition, integrity was a prize worth keeping…
That concludes my four-part series on our recent Perspective article. I hope you had as much fun reading them as I did writing them.
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 thisQuantum Frontiers post) and Brian Swingle (who’s appeared in furtherposts).
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.
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 hereand 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. Twoother 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 twoposts 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.
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.
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.
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.
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 Frontiersat leastthreetimes. 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.
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.
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.
Quantum Frontiers 