# I spy with my little eye…something algebraic.

Look at this picture.

Does any part of it surprise you? Look more closely.

Do you see a boy’s name?

I spell “Peter” with two e’s, but “Piotr” and “Pyotr” appear as authors’ names in papers’ headers. Finding “Petr” in a paper shouldn’t have startled me. But how often does “Gretchen” or “Amadeus” materialize in an equation?

When I was little, my reading list included Eye Spy, Where’s Waldo?, and Puzzle Castle. The books teach children to pay attention, notice details, and evaluate ambiguities.

That’s what physicists do. The first time I saw the picture above, I saw a variation on “Peter.” I was reading (when do I not?) about the intersection of quantum information and thermodynamics. The authors were discussing heat and algebra, not saints or boys who picked pecks of pickled peppers. So I looked more closely.

Each letter resolved into part of a story about a physical system. The P represents a projector. A projector is a mathematical object that narrows one’s focus to a particular space, as blinders on a horse do. The E tells us which space to focus on: a space associated with an amount E of energy, like a country associated with a GDP of $500 billion. Some of the energy E belongs to a heat reservoir. We know so because “reservoir” begins with r, and R appears in the picture. A heat reservoir is a system, like a colossal bathtub, whose temperature remains constant. The Greek letter $\tau$, pronounced “tau,” represents the reservoir’s state. The reservoir occupies an equilibrium state: The bath’s large-scale properties—its average energy, volume, etc.—remain constant. Never mind about jacuzzis. Piecing together the letters, we interpret the picture as follows: Imagine a vast, constant-temperature bathtub (R). Suppose we shut the tap long enough ago that the water in the tub has calmed ($\tau$). Suppose the tub neighbors a smaller system—say, a glass of Perrier.* Imagine measuring how much energy the bath-and-Perrier composite contains (P). Our measurement device reports the number E. Quite a story to pack into five letters. Didn’t Peter deserve a second glance? The equation’s right-hand side forms another story. I haven’t seen Peters on that side, nor Poseidons nor Gallahads. But look closely, and you will find a story. The images above appear in “Fundamental limitations for quantum and nanoscale thermodynamics,” published by Michał Horodecki and Jonathan Oppenheim in Nature Communications in 2013. *Experts: The ρS that appears in the first two images represents the smaller system. The tensor product represents the reservoir-and-smaller-system composite. # Generally speaking My high-school calculus teacher had a mustache like a walrus’s and shoulders like a rower’s. At 8:05 AM, he would demand my class’s questions about our homework. Students would yawn, and someone’s hand would drift into the air. “I have a general question,” the hand’s owner would begin. “Only private questions from you,” my teacher would snap. “You’ll be a general someday, but you’re not a colonel, or even a captain, yet.” Then his eyes would twinkle; his voice would soften; and, after the student asked the question, his answer would epitomize why I’ve chosen a life in which I use calculus more often than laundry detergent. Many times though I witnessed the “general” trap, I fell into it once. Little wonder: I relish generalization as other people relish hiking or painting or Michelin-worthy relish. When inferring general principles from examples, I abstract away details as though they’re tomato stains. My veneration of generalization led me to quantum information (QI) theory. One abstract theory can model many physical systems: electrons, superconductors, ion traps, etc. Little wonder that generalizing a QI model swallowed my summer. QI has shed light on statistical mechanics and thermodynamics, which describe energy, information, and efficiency. Models called resource theories describe small systems’ energies, information, and efficiencies. Resource theories help us calculate a quantum system’s value—what you can and can’t create from a quantum system—if you can manipulate systems in only certain ways. Suppose you can perform only operations that preserve energy. According to the Second Law of Thermodynamics, systems evolve toward equilibrium. Equilibrium amounts roughly to stasis: Averages of properties like energy remain constant. Out-of-equilibrium systems have value because you can suck energy from them to power laundry machines. How much energy can you draw, on average, from a system in a constant-temperature environment? Technically: How much “work” can you draw? We denote this average work by < W >. According to thermodynamics, < W > equals the change ∆F in the system’s Helmholtz free energy. The Helmholtz free energy is a thermodynamic property similar to the energy stored in a coiled spring. One reason to study thermodynamics? Suppose you want to calculate more than the average extractable work. How much work will you probably extract during some particular trial? Though statistical physics offers no answer, resource theories do. One answer derived from resource theories resembles ∆F mathematically but involves one-shot information theory, which I’ve discussed elsewhere. If you average this one-shot extractable work, you recover < W > = ∆F. “Helmholtz” resource theories recapitulate statistical-physics results while offering new insights about single trials. Helmholtz resource theories sit atop a silver-tasseled pillow in my heart. Why not, I thought, spread the joy to the rest of statistical physics? Why not generalize thermodynamic resource theories? The average work <W > extractable equals ∆F if heat can leak into your system. If heat and particles can leak, <W > equals the change in your system’s grand potential. The grand potential, like the Helmholtz free energy, is a free energy that resembles the energy in a coiled spring. The grand potential characterizes Bose-Einstein condensates, low-energy quantum systems that may have applications to metrology and quantum computation. If your system responds to a magnetic field, or has mass and occupies a gravitational field, or has other properties, <W > equals the change in another free energy. A collaborator and I designed resource theories that describe heat-and-particle exchanges. In our paper “Beyond heat baths: Generalized resource theories for small-scale thermodynamics,” we propose that different thermodynamic resource theories correspond to different interactions, environments, and free energies. I detailed the proposal in “Beyond heat baths II: Framework for generalized thermodynamic resource theories.” “II” generalizes enough to satisfy my craving for patterns and universals. “II” generalizes enough to merit a hand-slap of a pun from my calculus teacher. We can test abstract theories only by applying them to specific systems. If thermodynamic resource theories describe situations as diverse as heat-and-particle exchanges, magnetic fields, and polymers, some specific system should shed light on resource theories’ accuracy. If you find such a system, let me know. Much as generalization pleases aesthetically, the detergent is in the details. # Making sci-fi teleportation sound less crazy Laser beam bending due to a change in the speed of light in water. If you ever wanted to see a sci-fi plot that expertly applied advanced physical concepts so that with a bit of imagination teleporting a human was not as unbelievable as most of the teleportation scenarios we see in the movies, keep reading. Years ago, when I was still in Russia, I was working on a back-story for a sci-fi game I was playing with friends. In the game, players were given stones (from Mars!) that could change the fundamental constants of nature: electron charge e, speed of light c and Planck constant h. I had already worked out the effects these stones would produce on the space around them (I suggest it as an exercise to the nerdy reader – once you are done thinking about it, see my answer below), so my next task was to envision a big scientific project centered around those stones, with a solid foundation on real physics and a portal to Mars as a final goal. As it turned out, some unforeseen consequences included blowing up the whole lab and scattering the stones in the nearby forest. That’s the back-story. It all worked beautifully on paper. I imagined that materials could be programmed to obey different values of fundamental constants. These Martian stones were supposed to be the first encounter humanity had with matter where such effects could be observed and studied. The effects extended to a region around the stone, with weird things happening on the boundary of that region. For one thing, energy was not conserved in the vicinity of these stones. Now having control over e, c and h, the scientists would leverage this new-found power to try to move the fine structure constant $e^2/(\hbar c)$ to what is known as the Landau pole. Such a feat would result in infinitely strong interactions between particles, so that the energetic content of space-time would jump through the roof and a black hole would form. If one was lucky, even a traversable wormhole would form, which is what the scientists were hoping for, because back on Mars these things could have formed naturally, and the lab wormhole would connect to the Martian network. If you’ve read all this and are asking yourself “What just happened?”, see all the physical concepts explained below: Continue reading # Where are you, Dr. Frank Baxter? This year marks the 50th anniversary of my first publication. In 1964, when we were eleven-year-old fifth graders, my best friend Mace Rosenstein and I launched The Pres-stein Gazette, a not-for-profit monthly. Though the first issue sold well, the second issue never appeared. Front page of the inaugural issue of the Pres-stein Gazette. Faded but still legible, it was produced using a mimeograph machine, a low-cost printing press which was popular in the pre-Xerox era. One of my contributions to the inaugural issue was a feature article on solar energy, which concluded that fossil fuel “isn’t of such terrific abundance and it cannot support the world for very long. We must come up with some other source of energy. Solar energy is that source … when developed solar energy will be a cheap powerful “fuel” serving the entire world generously forever.” This statement holds up reasonably well 50 years later. You might wonder how an eleven-year-old in 1964 would know something like that. I can explain … In the 1950s and early 1960s, AT&T and the Bell Telephone System produced nine films about science, which were broadcast on prime-time network television and attracted a substantial audience. After broadcast, the films were distributed to schools as 16 mm prints and frequently shown to students for many years afterward. I don’t remember seeing any of the films on TV, but I eventually saw all nine in school. It was always a treat to watch one of the “Bell Telephone Movies” instead of hearing another boring lecture. For educational films, the production values were uncommonly high. Remarkably, the first four were all written and directed by the legendary Frank Capra (a Caltech alum), in consultation with a scientific advisory board provided by Bell Labs. Those four (Our Mr. Sun, Hemo the Magnificent, The Strange Case of the Cosmic Rays, and Unchained Goddess, originally broadcast in 1956-58) are the ones I remember most vividly. DVDs of these films exist, but I have not watched any of them since I was a kid. The star of the first eight films was Dr. Frank Baxter, who played Dr. Research, the science expert. Baxter was actually an English professor at USC who had previous television experience as the popular host of a show about Shakespeare, but he made a convincing and pleasingly avuncular scientist. (The ninth film, Restless Sea, was produced by Disney, and Walt Disney himself served as host.) The other lead role was Mr. Writer, a skeptical and likeable Everyman who learned from Dr. Research’s clear explanations and sometimes translated them into vernacular. The first film, Our Mr. Sun, debuted in 1956 (broadcast in color, a rarity at that time) and was seen by 24 million prime-time viewers. Mr. Writer was Eddie Albert, a well-known screen actor who later achieved greater fame as the lead on the 1960s TV situation comedy Green Acres. Lionel Barrymore appeared in a supporting role. Dr. Frank Baxter and Eddie Albert in Our Mr. Sun. (Source: Wikipedia) Our Mr. Sun must have been the primary (unacknowledged) source for my article in the Pres-stein Gazette. Though I learned from Wikipedia that Capra insisted (to the chagrin of some of his scientific advisers) on injecting some religious themes into the film, I don’t remember that aspect at all. The scientific content was remarkably sophisticated for a film that could be readily enjoyed by elementary school students, and I remember (or think I do) clever animations teaching me about the carbon cycle in stellar nuclear furnaces and photosynthesis as the ultimate source of all food sustaining life on earth. But I was especially struck by Dr. Baxter’s dire warning that, as the earth’s population grows, our planet will face shortages of food and fuel. On a more upbeat note he suggested that advanced technologies for harnessing the power of the sun would be the key to our survival, which inspired the optimistic conclusion of my article. A lavishly produced prime-time show about science was a real novelty in 1956, many years before NOVA or the Discovery Channel. I wonder how many readers remember seeing the Dr. Frank Baxter movies when you were kids, either on TV or in school. Or was there another show that inspired you like Our Mr. Sun inspired me? I hope some of you will describe your experiences in the comments. And I also wonder what resource could have a comparable impact on an eleven-year-old in today’s very different media environment. The obvious comparison is with Neil deGrasse Tyson’s revival of Cosmos, which aired on Fox in 2014. The premiere episode of Cosmos drew 8.5 million viewers on the night it was broadcast, but that is a poor measure of impact nowadays. Each episode has been rebroadcast many times, not just in the US and Canada but internationally as well, and the whole series is now available in DVD and Blu-ray. Will lots of kids in the coming years own it and watch it? Is Cosmos likely to be shown in classrooms as well? Science is accessible to the curious through many other avenues today, particularly on YouTube. One can watch TED talks, or Minute Physics, or Veritasium, or Khan Academy, or Lenny Susskind’s lectures, not to mention our own IQIM videos on PHD Comics. And there are many other options. Maybe too many? But do kids watch this stuff? If not, what online sources inspire them? Do they get as excited as I did when I watched Dr. Frank Baxter at age 11? I don’t know. What do you think? # Macroscopic quantum teleportation: the story of my chair In the summer of 2000, a miracle occurred: The National Science Foundation decided to fund a new Institute for Quantum Information at Caltech with a 5 million dollar award from their Information Technology Research program. I was to be the founding director of the IQI. Jeff Kimble explained to me why we should propose establishing the IQI. He knew I had used my slice of our shared DARPA grant to bring Alexei Kitaev to Caltech as a visiting professor, which had been wonderful. Recalling how much we had both benefited from Kitaev’s visit, Jeff remarked emphatically that “This stuff’s not free.” He had a point. To have more fun we’d need more money. Jeff took the lead in recruiting a large team of Caltech theorists and experimentalists to join the proposal we submitted, but the NSF was primarily interested in supporting the theory of quantum computation rather than the experimental part of the proposal. That was how I wound up in charge, though I continued to rely on Jeff’s advice and support. This was a new experience for me and I worried a lot about how directing an institute would change my life. But I had one worry above all: space. We envisioned a thriving institute brimming over with talented and enthusiastic young scientists and visitors drawn from the physics, computer science, and engineering communities. But how could we carve out a place on the Caltech campus where they could work and interact? To my surprise and delight, Jeff and I soon discovered that someone else at Caltech shared our excitement over the potential of IQI — Richard Murray, who was then the Chair of Caltech’s Division of Engineering and Applied Science. Richard arranged for the IQI to occupy office space in Steele Laboratory and some space we could configure as we pleased in Jorgensen Laboratory. The hub of the IQI became the lounge in Jorgensen, which we used for our seminar receptions, group meetings, and innumerable informal discussions, until our move to the beautiful Annenberg Center when it opened in 2009. I sketched a rough plan for the Jorgensen layout, including furniture for the lounge. The furniture, I was told, was “NIC”. Though I was too embarrassed to ask, I eventually inferred this meant “Not in Contract” — I would need to go furniture shopping, one of my many burgeoning responsibilities as Director. By this time, Ann Harvey was in place as IQI administrator, a huge relief. But furniture was something I thought I knew about, because I had designed and furnished a common area for the particle theory group a couple of years earlier. As we had done on that previous occasion, my wife Roberta and I went to Krause’s Sofa Factory to order a custom-made couch, love seat, and lounge chair, in a grayish green leather which we thought would blend well with the carpeting. Directing an institute is not as simple as it sounds, though. Before the furniture was delivered, Krause’s declared bankruptcy! We had paid in full, but I had some anxious moments wondering whether there would be a place to sit down in the IQI lounge. In the end, after some delay, our furniture was delivered in time for the grand opening of the new space in September 2001. A happy ending, but not really the end of the story. Before the move to Annenberg in 2009, I ordered furniture to fill our (much smaller) studio space, which became the new IQI common area. The Jorgensen furniture was retired, and everything was new! It was nice … But every once in a while I felt a twinge of sadness. I missed my old leather chair, from which I had pontificated at eight years worth of group meetings. That chair and I had been through a lot together, and I couldn’t help but feel that my chair’s career had been cut short before its time. I don’t recall mentioning these feelings to anyone, but someone must have sensed by regrets. Because one day not long after the move another miracle occurred … my chair was baaack! Sitting in it again felt … good. For five years now I’ve been pontificating from my old chair in our new studio, just like I used to. No one told me how my chair had been returned to me, and I knew better than to ask. My chair today. Like me, a bit worn but still far from retirement. Eventually the truth comes out. At my 60th birthday celebration last year, Stephanie Wehner and Darrick Chang admitted to being the perpetrators, and revealed the whole amazing story in their article on “Macroscopic Quantum Teleportation” in a special issue of Nature Relocations. Their breakthrough article was enhanced by Stephanie’s extraordinary artwork, which you really have to see to believe. So if your curiosity is piqued, please follow this link to find out more. Why, you may wonder, am I reminiscing today about the story of my chair? Well, is an excuse really necessary? But if you must know, it may be because, after two renewals and 14 years of operation, I submitted the IQI Final Report to the NSF this week. Don’t worry — the Report is not really Final, because the IQI has become part of an even grander vision, the IQIM (which has given birth to this blog among other good things). Like my chair, the IQI is not quite what it was, yet it lives on. The nostalgic feelings aroused by filing the Final Report led me to reread the wonderful volume my colleagues put together for my birthday celebration, which recounts not only the unforgettable exploits of Stephanie and Darrick, but many other stories and testimonials that deeply touched me. Browsing through that book today, one thing that struck me is the ways we sometimes have impact on others without even being aware of it. For example, Aram Harrow, Debbie Leung, Joe Renes and Stephanie all remember lectures I gave when they were undergraduate students (before I knew them), which might have influenced their later research careers. Knowing this will make it a little harder to say no the next time I’m invited to give a talk. Yaoyun Shi has vivid memories of the time I wore my gorilla mask to the IQI seminar on Halloween, which inspired him to dress up as “a butcher threatening to cut off the ears of my students with a bloody machete if they were not listening,” thus boosting his teaching evaluations. And Alexios Polychronakos, upon hearing that I had left particle theory to pursue quantum computing, felt it “was a bit like watching your father move to Las Vegas and marry a young dancer after you leave for college,” while at the same time he appreciated “that such reinventions are within the spectrum of possibilities for physicists who still have a pulse.” I’m proud of what the IQI(M) has accomplished, but we’re just getting started. After 14 years, I still have a pulse, and my chair has plenty of wear left. Together we look forward to many more years of pontification. # How Physics for Poets can boost your Physics GRE score As summer fades to autumn, prospective grad students are eyeing applications. Professors are suggesting courses, and seniors are preparing for Graduate Record Exams (GREs). American physics programs (and a few programs abroad) require or encourage applicants to take the Physics GRE. If you’ve sighted physics grad school in your crosshairs, consider adding Physics for Poets (PFP) to your course list. Many colleges develop this light-on-the-math tour of physics history for non-majors. But Physics for Poets can boost your Physics GRE score while reinforcing the knowledge gained from your physics major. My senior spring in college, PFP manifested as “PHYS 001/002: Understanding the Universe: From Atoms to the Big Bang.” The tallness of this order failed to daunt Marcelo Gleiser, a cosmologist whose lectures swayed with a rhythm like a Foucault pendulum. From creation myths and Greek philosophers, we proceeded via Copernicus and Kepler to Newton and the Enlightenment, Maxwell and electromagnetism, the Industrial Revolution and thermodynamics, Einstein’s relativity, and WWII and the first quantum revolution, ending with particle physics and cosmology. The course provided a history credit I needed. It offered a breather from problem sets, sandwiching my biophysics and quantum-computation lectures. Pragmatism aside, PHYS 2 showcased the grandness of the physicist’s legacy—of my legacy. PHYS 2 sharpened my determination to do that legacy justice. To do it justice, most of us must pass tests. Here’s how PFP can help. (1) A Foucault pendulum, invented during the 19th century to demonstrate that the Earth rotates. (2) An excuse to review noninertial reference frames. Reviewing basic physics can improve GRE scores. If thermodynamics has faded from memory, good luck calculating a Carnot engine’s efficiency. Several guides (and I) recommend reviewing notes and textbooks, working practice problems, simulating exams, and discussing material with peers. Taking PFP, you will review for the GRE. A list of the topics on the Physics GRE appears here. Of the 14 mechanics topics, eight surfaced in PHYS 2. Taking PFP will force you to review GRE topics that lack of time (or that procrastination) might force you to skip. Earning credit for GRE prep, you won’t have to shoehorn that prep into your schedule at the expense of research. The Physics GRE covers basic physics developed during the past 500 years; so do many PFP courses. The GRE, you might protest, involves more math than PFP. But GRE questions probe less deeply, and PFP can galvanize more reviews of math, than you might expect. According to Stanford’s Society of Physics Students, “Each [Physics GRE] question shouldn’t require much more than a minute’s worth of thought and computation.” Expect to use the Wave Equation and Maxwell’s Equations, not to derive the former from the latter. Some PFP instructors require students to memorize formulae needed on the GRE. PFP can verify whether your memory has interchanged the exponents in Kepler’s Third Law, or dropped the negative sign from the kinetic-energy term in Schrödinger’s Equation. Even formulae excluded from PFP exams appear in PFP classes. In a PHYS 2 Powerpoint, Marcelo included Planck’s blackbody formula. Though he never asked me to regurgitate it, his review benefited me. Given such a Powerpoint, you can re-memorize the formula. Derive the Stefan-Boltzmann and Wien Displacement Laws. Do you remember how a blackbody’s energy density varies with frequency in the low-energy limit? PFP can catalyze your review of math used on the GRE. Physics for Poets: It can work for applicants to physics grad programs. While recapitulating basic physics, PFP can introduce “specialized” GRE topics. Examples include particle physics, astrophysics, and nuclear physics. Covered in advanced classes, these subjects might have evaded mention in your courses. Signing up for biophysics, I had to drop particle theory. PHYS 2 helped compensate for the drop. I learned enough about neutrinos and quarks to answer GRE questions about them. In addition to improving your score, surveying advanced topics in PFP can enhance your understanding of physics seminars and conversations. The tour can help you identify which research you should undertake. If you’ve tried condensed-matter and atmospheric research without finding your niche, tasting cosmology in PFP might point toward your next project. Sampling advanced topics in PFP, you can not only prepare for the GRE, but also enrich your research. I am not encouraging you to replace advanced physics courses with PFP. I encourage you to complement advanced courses with PFP. If particle physics suits your schedule and intrigues you, enjoy. If you need to fulfill a history or social-sciences distribution requirement, check whether PFP can count. Consider PFP if you’ve committed to a thesis and three problem-set courses, you haven’t signed up for the minimum number of courses required by your college, and more problem sets would strangle you. Sleep deprivation improves neither exam scores nor understanding. Not that I sailed through PHYS 2 without working. I worked my rear off—fortunately for my science. Switching mindsets—pausing frustrating calculations to study Kepler—can refresh us. Stretch your calculational toolkit in advanced courses, and reinforce that toolkit with PFP. In addition to reviewing basic physics and surveying specialized topics, you can seek study help from experts in PFP. When your questions about GRE topics overlap with PFP material, ask your instructor and TA. They’ll probably enjoy answering: Imagine teaching physics with little math, with one hand tied behind your back. Some students take your class not because they want to, but because they need science credits. Wouldn’t you enjoy directing a student who cares? While seeking answers, you can get to know your professor or TA. You can learn about his or her research. Maybe PFP will lead you to join that research. PFP not only may connect you to experts able to answer questions as no study guide can. PFP offers opportunities to enhance a course as few non-physics students can and to develop relationships with role models. Instructors’ expertise has benefited science students throughout much of history. Those relationships illustrate the benefits that PFP extends beyond GREs. As mentioned earlier, surveying advanced topics can diversify the research conversations you understand. The survey can direct you toward research you’ll pursue. Further exposure to research can follow from discussions with instructors. Intermissions from problem sets can promote efficiency. Other benefits of PFP include enhancement of explanatory skills and a bird’s-eye view of the scientific process to which you’re pledging several years. What a privilege we enjoy, PFP shows. We physicists explore questions asked for millennia. We wear mantles donned by Faraday, Bernoulli, and Pauli. When integrals pour out our ears and experiments break down, PFP can remind us why we bother. And that we’re in fine company. Physics for Poets can improve your Physics GRE score and reinforce your physics major. Surveying basic physics, PFP will force you to review GRE topics. PFP may introduce specialized GRE topics absent from most physics majors. Opportunities abound to re-memorize equations and to complement lectures with math. Questioning instructors, you can deepen your understanding as with no study guide. Beyond boosting your GRE score, PFP can broaden your research repertoire, energize your calculations, improve your explanatory skills, and inspire. Good luck with the academic year, and see you in grad school! # Reading the sub(linear) text Physicists are not known for finesse. “Even if it cost us our funding,” I’ve heard a physicist declare, “we’d tell you what we think.” Little wonder I irked the porter who directed me toward central Cambridge. The University of Cambridge consists of colleges as the US consists of states. Each college has a porter’s lodge, where visitors check in and students beg for help after locking their keys in their rooms. And where physicists ask for directions. Last March, I ducked inside a porter’s lodge that bustled with deliveries. The woman behind the high wooden desk volunteered to help me, but I asked too many questions. By my fifth, her pointing at a map had devolved to jabbing. Read the subtext, I told myself. Leave. Or so I would have told myself, if not for that afternoon. That afternoon, I’d visited Cambridge’s CMS, which merits every letter in “Centre for Mathematical Sciences.” Home to Isaac Newton’s intellectual offspring, the CMS consists of eight soaring, glass-walled, blue-topped pavilions. Their majesty walloped me as I turned off the road toward the gatehouse. So did the congratulatory letter from Queen Elizabeth II that decorated the route to the restroom. I visited Nilanjana Datta, an affiliated lecturer of Cambridge’s Faculty of Mathematics, and her student, Felix Leditzky. Nilanjana and Felix specialize in entropies and one-shot information theory. Entropies quantify uncertainties and efficiencies. Imagine compressing many copies of a message into the smallest possible number of bits (units of memory). How few bits can you use per copy? That number, we call the optimal compression rate. It shrinks as the number of copies compressed grows. As the number of copies approaches infinity, that compression rate drops toward a number called the message’s Shannon entropy. If the message is quantum, the compression rate approaches the von Neumann entropy. Good luck squeezing infinitely many copies of a message onto a hard drive. How efficiently can we compress fewer copies? According to one-shot information theory, the answer involves entropies other than Shannon’s and von Neumann’s. In addition to describing data compression, entropies describe the charging of batteriesthe concentration of entanglementthe encrypting of messages, and other information-processing tasks. Speaking of compressing messages: Suppose one-shot information theory posted status updates on Facebook. Suppose that that panel on your Facebook page’s right-hand side showed news weightier than celebrity marriages. The news feed might read, “TRENDING: One-shot information theory: Second-order asymptotics.” Second-order asymptotics, I learned at the CMS, concerns how the optimal compression rate decays as the number of copies compressed grows. Imagine compressing a billion copies of a quantum message ρ. The number of bits needed about equals a billion times the von Neumann entropy HvN(ρ). Since a billion is less than infinity, 1,000,000,000 HvN(ρ) bits won’t suffice. Can we estimate the compression rate more precisely? The question reminds me of gas stations’ hidden pennies. The last time I passed a station’s billboard, some number like$3.65 caught my eye. Each gallon cost about $3.65, just as each copy of ρ costs about HvN(ρ) bits. But a 9/10, writ small, followed the$3.65. If I’d budgeted $3.65 per gallon, I couldn’t have filled my tank. If you budget HvN(ρ) bits per copy of ρ, you can’t compress all your copies. Suppose some station’s owner hatches a plan to promote business. If you buy one gallon, you pay$3.654. The more you purchase, the more the final digit drops from four. By cataloguing receipts, you calculate how a tank’s cost varies with the number of gallons, n. The cost equals $3.65 × n to a first approximation. To a second approximation, the cost might equal$3.65 × n + an, wherein a represents some number of cents. Compute a, and you’ll have computed the gas’s second-order asymptotics.

Nilanjana and Felix computed a’s associated with data compression and other quantum tasks. Second-order asymptotics met information theory when Strassen combined them in nonquantum problems. These problems developed under attention from Hayashi, Han, Polyanski, Poor, Verdu, and others. Tomamichel and Hayashi, as well as Li, introduced quantumness.

In the total-cost expression, \$3.65 × n depends on n directly, or “linearly.” The second term depends on √n. As the number of gallons grows, so does √n, but √n grows more slowly than n. The second term is called “sublinear.”

Which is the word that rose to mind in the porter’s lodge. I told myself, Read the sublinear text.

Little wonder I irked the porter. At least—thanks to quantum information, my mistake, and facial expressions’ contagiousness—she smiled.

With thanks to Nilanjana Datta and Felix Leditzky for explanations and references; to Nilanjana, Felix, and Cambridge’s Centre for Mathematical Sciences for their hospitality; and to porters everywhere for providing directions.