# The Grand Tour of quantum thermodynamics

Young noblemen used to undertake a “Grand Tour” during the 1600s and 1700s. Many of the tourists hailed from England, though well-to-do compatriots traveled from Scandinavia, Germany, and the United States. The men had just graduated from university—in many cases, Oxford or Cambridge. They’d studied classical history, language, and literature; and now, they’d experience what they’d read. Tourists flocked to Rome, Venice, and Florence, as well as to Paris; optional additions included Naples, Switzerland, Germany, and the Netherlands.

Tutors accompanied the tourists, guiding their charges across Europe. The tutors rounded out the young men’s education, instructing them in art, music, architecture, and continental society. I felt like those tutors, this month and last.1

I was lecturing in a quantum-thermodynamics mini course, with fellow postdoctoral scholar Matteo Lostaglio. Gabriel Landi, a professor of theoretical physics at the University of São Paolo in Brazil, organized the course. It targeted early-stage graduate students, who’d mastered the core of physics and who wished to immerse in quantum thermodynamics. But the enrollment ranged from PhD and Masters students to undergraduates, postdocs, faculty members, and industry employees.

The course toured quantum thermodynamics similarly to how young noblemen toured Europe. I imagine quantum thermodynamics as a landscape—one inked on a parchment map, with blue whorls representing the sea and with a dragon breathing fire in one corner. Quantum thermodynamics encompasses many communities whose perspectives differ and who wield different mathematical and conceptual tools. These communities translate into city-states, principalities, republics, and other settlements on the map. The class couldn’t visit every city, just as Grand Tourists couldn’t. But tourists had a leg up on us in their time budgets: A Grand Tour lasted months or years, whereas we presented nine hour-and-a-half lectures.

Grand Tourists returned home with trinkets, books, paintings, and ancient artifacts. I like to imagine that the tutors, too, acquired souvenirs. Here are four of my favorite takeaways from the course:

1) Most captivating subfield that I waded into for the course: Thermodynamic uncertainty relations. Researchers have derived these inequalities using nonequilibrium statistical mechanics, a field that encompasses molecular motors, nanorobots, and single strands of DNA. Despite the name “uncertainty relations,” classical and quantum systems obey these inequalities.

Imagine a small system interacting with big systems that have different temperatures and different concentrations of particles. Energy and particles hop between the systems, dissipating entropy ($\Sigma$) and forming currents. The currents change in time, due to the probabilistic nature of statistical mechanics.

How much does a current vary, relative to its average value, $\langle J \rangle$? We quantify this variation with the relative variance, ${\rm var}(J) / \langle J \rangle^2$. Say that you want a low-variance, predictable current. You’ll have to pay a high entropy cost: $\frac{ {\rm var} (J) }{\langle J \rangle^2 }$ $\geq \frac{2 k_{\rm B} }{\Sigma}$, wherein $k_{\rm B}$ denotes Boltzmann’s constant.

Thermodynamic uncertainty relations govern systems arbitrarily far from equilibrium. We know loads about systems at equilibrium, in which large-scale properties remain approximately constant and no net flows (such as flows of particles) enter or leave the system. We know much about systems close to equilibrium. The regime arbitrarily far from equilibrium is the Wild, Wild West of statistical mechanics. Proving anything about this regime tends to require assumptions and specific models, to say nothing of buckets of work. But thermodynamic uncertainty relations are general, governing classical and quantum systems from molecular motors to quantum dots.

2) Most unexpected question: During lecture one, I suggested readings that introduce quantum thermodynamics. The suggestions included two reviews and the article I wrote for Scientific American about quantum steampunk, my angle on quantum thermodynamics. The next day, a participant requested recommendations of steampunk novels. I’d prepared more for requests for justifications of the steps in my derivations. But I forwarded a suggestion given to me twice: The Difference Engine, by William Gibson and Bruce Sterling.

3) Most insightful observation: My fellow tutor—I mean lecturer—pointed out how quantum thermodynamics doesn’t and does diverge from classical thermodynamics. Quantum systems can’t break the second law of thermodynamics, as classical systems can’t. Quantum engines can’t operate more efficiently than Carnot’s engine. Erasing information costs work, regardless of whether the information-bearing degree of freedom is classical or quantum. So broad results about quantum thermodynamics coincide with broad results about classical thermodynamics. We can find discrepancies by focusing on specific physical systems, such as a spring that can be classical or quantum.

4) Most staggering numbers: Unlike undertaking a Grand Tour, participating in the mini course cost nothing. We invited everyone across the world to join, and 420 participants from 48 countries enrolled. I learned of the final enrollment days before the course began, scrolling through the spreadsheet of participants. Motivated as I had been to double-check my lecture notes, the number spurred my determination like steel on a horse’s flanks.

The Grand Tour gave rise to travelogues and guidebooks read by tourists across the centuries: Mark Twain has entertained readers—partially at his own expense—since 1869 in the memoir The Innocents Abroad. British characters in the 1908 novel A Room with a View diverge in their views of Baedeker’s Handbook to Northern Italy. Our course material, and videos of the lectures, remain online and available to everyone for free. You’re welcome to pack your trunk, fetch your cloak, and join the trip.

1In addition to guiding their wards, tutors kept the young men out of trouble—and one can only imagine what trouble wealthy young men indulged in the year after college. I didn’t share that responsibility.

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## About Nicole Yunger Halpern

I’m a theoretical physicist at the Joint Center for Quantum Information and Computer Science in Maryland. My research group re-envisions 19th-century thermodynamics for the 21st century, using the mathematical toolkit of quantum information theory. We then apply quantum thermodynamics as a lens through which to view the rest of science. I call this research “quantum steampunk,” after the steampunk genre of art and literature that juxtaposes Victorian settings (à la thermodynamics) with futuristic technologies (à la quantum information). For more information, check out my upcoming book Quantum Steampunk: The Physics of Yesterday’s Tomorrow. I earned my PhD at Caltech under John Preskill’s auspices; one of my life goals is to be the subject of one of his famous (if not Pullitzer-worthy) poems. Follow me on Twitter @nicoleyh11.