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 book for the general public, 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.

Hello, my name is QUANTUM MASTER EQUATION

Posted on June 21, 2015 by Nicole Yunger Halpern

“Why does it have that name?”

I’ve asked in seminars, in lectures, in offices, and at group meetings. I’ve asked about physical conjectures, about theorems, and about mathematical properties.

“I don’t know.” Lecturers have shrugged. “It’s just a name.”

This spring, I asked about master equations. I thought of them as tools used in statistical mechanics, the study of vast numbers of particles. We can’t measure vast numbers of particles, so we can’t learn about stat-mech systems everything one might want to know. The magma beneath Santorini, for example, consists of about 10²⁴ molecules. Good luck measuring every one.

Imagine, as another example, using a quantum computer to solve a problem. We load information by initializing the computer to a certain state: We orient the computer’s particles in certain directions. We run a program, then read out the output.

Suppose the computer sits on a tabletop, exposed to the air like leftover casserole no one wants to save for tomorrow. Air molecules bounce off the computer, becoming entangled with the hardware. This entanglement, or quantum correlation, alters the computer’s state, just as flies alter a casserole.^* To understand the computer’s output—which depends on the state, which depends on the air—we must have a description of the air. But we can’t measure all those air molecules, just as we can’t measure all the molecules in Santorini’s magma.

We can package our knowledge about the computer’s state into a mathematical object, called a density operator, labeled by ρ(t). A quantum master equation describes how ρ(t) changes. I had no idea, till this spring, why we call master equations “master equations.” Had someone named “John Master” invented them? Had the inspiration for the Russell Crowe movie Master and Commander? Or the Igor who lisps, “Yeth, mathter” in adaptations of Frankenstein?

Jenia Mozgunov, a fellow student and Preskillite, proposed an answer: Using master equations, we can calculate how averages of observable properties change. Imagine describing a laser, a cavity that spews out light. A master equation reveals how the average number of photons (particles of light) in the cavity changes. We want to predict these averages because experimentalists measure them. Because master equations spawn many predictions—many equations—they merit the label “master.”

Jenia’s hypothesis appealed to me, but I wanted certainty. I wanted Truth. I opened my laptop and navigated to Facebook.

“Does anyone know,” I wrote in my status, “why master equations are called ‘master equations’?”

Ian Durham, a physicist at St. Anselm College, cited Tom Moore’s Six Ideas that Shaped Physics. Most physics problems, Ian wrote, involve “some overarching principle.” Example principles include energy conservation and invariance under discrete translations (the system looks the same after you step in some direction). A master equation encapsulates this principle.

Ian’s explanation sounded sensible. But fewer people “liked” his reply on Facebook than “liked” a quip by a college friend: Master equations deserve their name because “[t]hey didn’t complete all the requirements for the doctorate.”

My advisor, John Preskill, dug through two to three books, one set of lecture notes, one German Wikipedia page, one to two articles, and Google Scholar. He concluded that Nordsieck, Lamb, and Uhlenbeck coined “master equation.” According to a 1940 paper of theirs,^** “When the probabilities of the elementary processes are known, one can write down a continuity equation for W [a set of probabilities], from which all other equations can be derived and which we will call therefore the ‘master’ equation.”

“Are you sure you were meant to be a physicist,” I asked John, “rather than a historian?”

“Procrastination is a powerful motivator,” he replied.

Lecturers have shrugged at questions about names. Then they’ve paused, pondered, and begun, “I guess because…” Theorems and identities derive their names from symmetries, proof techniques, geometric illustrations, and applications to problems I’d thought unrelated. A name taught me about uses for master equations. Names reveal physics I wouldn’t learn without asking about names. Names aren’t just names. They’re lamps and guides.

Pity about the origin of “master equation,” though. I wish an Igor had invented them.

^*Apologies if I’ve spoiled your appetite.

^**A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I,” Physica 7, 344-60 (1940), p. 353.

Mingling stat mech with quantum info in Maryland

Posted on May 8, 2015 by Nicole Yunger Halpern

I felt like a yoyo.

I was standing in a hallway at the University of Maryland. On one side stood quantum-information theorists. On the other side stood statistical-mechanics scientists.^* The groups eyed each other, like Jets and Sharks in West Side Story, except without fighting or dancing.

This March, the groups were generous enough to host me for a visit. I parked first at QuICS, the Joint Center for Quantum Information and Computer Science. Established in October 2014, QuICS had moved into renovated offices the previous month. QuICSland boasts bright colors, sprawling armchairs, and the scent of novelty. So recently had QuICS arrived that the restroom had not acquired toilet paper (as I learned later than I’d have preferred).

Photo credit: QuICS

From QuICS, I yoyo-ed to the chemistry building, where Chris Jarzynski’s group studies fluctuation relations. Fluctuation relations, introduced elsewhere on this blog, describe out-of-equilibrium systems. A system is out of equilibrium if large-scale properties of it change. Many systems operate out of equilibrium—boiling soup, combustion engines, hurricanes, and living creatures, for instance. Physicists want to describe nonequilibrium processes but have trouble: Living creatures are complicated. Hence the buzz about fluctuation relations.

My first Friday in Maryland, I presented a seminar about quantum voting for QuICS. The next Tuesday, I was to present about one-shot information theory for stat-mech enthusiasts. Each week, the stat-mech crowd invites its speaker to lunch. Chris Jarzynski recommended I invite QuICS. Hence the Jets-and-Sharks tableau.

“Have you interacted before?” I asked the hallway.

“No,” said a voice. QuICS hadn’t existed till last fall, and some QuICSers hadn’t had offices till the previous month.**

Silence.

“We’re QuICS,” volunteered Stephen Jordan, a quantum-computation theorist, “the Joint Center for Quantum Information and Computer Science.”

So began the mingling. It continued at lunch, which we shared at three circular tables we’d dragged into a chain. The mingling continued during the seminar, as QuICSers sat with chemists, materials scientists, and control theorists. The mingling continued the next day, when QuICSer Alexey Gorshkov joined my discussion with the Jarzynski group. Back and forth we yoyo-ed, between buildings and topics.

“Mingled,” said Yigit Subasi. Yigit, a postdoc of Chris’s, specialized in quantum physics as a PhD student. I’d asked how he thinks about quantum fluctuation relations. Since Chris and colleagues ignited fluctuation-relation research, theorems have proliferated like vines in a jungle. Everyone and his aunty seems to have invented a fluctuation theorem. I canvassed Marylanders for bushwhacking tips.

Imagine, said Yigit, a system whose state you know. Imagine a gas, whose temperature you’ve measured, at equilibrium in a box. Or imagine a trapped ion. Begin with a state about which you have information.

Imagine performing work on the system “violently.” Compress the gas quickly, so the particles roil. Shine light on the ion. The system will leave equilibrium. “The information,” said Yigit, “gets mingled.”

Imagine halting the compression. Imagine switching off the light. Combine your information about the initial state with assumptions and physical laws.^*** Manipulate equations in the right way, and the information might “unmingle.” You might capture properties of the violence in a fluctuation relation.

With Zhiyue Lu and Andrew Maven Smith of Chris Jarzynski’s group (left) and with QuICSers (right)

I’m grateful to have exchanged information in Maryland, to have yoyo-ed between groups. We have work to perform together. I have transformations to undergo.^**** Let the unmingling begin.

With gratitude to Alexey Gorshkov and QuICS, and to Chris Jarzynski and the University of Maryland Department of Chemistry, for their hospitality, conversation, and camaraderie.

^*Statistical mechanics is the study of systems that contain vast numbers of particles, like the air we breathe and white dwarf stars. I harp on about statistical mechanics often.

**Before QuICS’s birth, a future QuICSer had collaborated with a postdoc of Chris’s on combining quantum information with fluctuation relations.

^***Yes, physical laws are assumptions. But they’re glorified assumptions.

^****Hopefully nonviolent transformations.

Paul Dirac and poetry

Posted on April 12, 2015 by Nicole Yunger Halpern

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it’s the exact opposite!

– Paul Dirac

Paul Dirac

I tacked Dirac’s quote onto the bulletin board above my desk, the summer before senior year of high school. I’d picked quotes by T.S. Elliot and Einstein, Catullus and Hatshepsut.^* In a closet, I’d found amber-, peach-, and scarlet-colored paper. I’d printed the quotes and arranged them, starting senior year with inspiration that looked like a sunrise.

Not that I knew who Paul Dirac was. Nor did I evaluate his opinion. But I’d enrolled in Advanced Placement Physics C and taken the helm of my school’s literary magazine. The confluence of two passions of mine—science and literature—in Dirac’s quote tickled me.

A fiery lecturer began to alleviate my ignorance in college. Dirac, I learned, had co-invented quantum theory. The “Dee-rac Equa-shun,” my lecturer trilled in her Italian accent, describes relativistic quantum systems—tiny particles associated with high speeds. I developed a taste for spin, a quantum phenomenon encoded in Dirac’s equation. Spin serves quantum-information scientists as two-by-fours serve carpenters: Experimentalists have tried to build quantum computers from particles that have spins. Theorists keep the idea of electron spins in a mental car trunk, to tote out when illustrating abstract ideas with examples.

The next year, I learned that Dirac had predicted the existence of antimatter. Three years later, I learned to represent antimatter mathematically. I memorized the Dirac Equation, forgot it, and re-learned it.

One summer in grad school, visiting my parents, I glanced at my bulletin board.

The sun rises beyond a window across the room from the board. Had the light faded the papers’ colors? If so, I couldn’t tell.

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it’s the exact opposite!

Do poets try to obscure ideas everyone understands? Some poets express ideas that people intuit but feel unable, lack the attention, or don’t realize one should, articulate. Reading and hearing poetry helps me grasp the ideas. Some poets express ideas in forms that others haven’t imagined.

Did Dirac not represent physics in a form that others hadn’t imagined?

The Dirac Equation

Would you have imagined that form? I didn’t imagine it until learning it. Do scientists not express ideas—about gravity, time, energy, and matter—that people feel unable, lack the attention, or don’t realize we should, articulate?

The U.S. and Canada have designated April as National Poetry Month. A hub for cousins of poets, Quantum Frontiers salutes. Carry a poem in your pocket this month. Or carry a copy of the Dirac Equation. Or tack either on a bulletin board; I doubt whether their colors will fade.

^*“Now my heart turns this way and that, as I think what the people will say. Those who see my monuments in years to come, and who shall speak of what I have done.” I expect to build no such monuments. But here’s to trying.

Quantum Frontiers salutes Terry Pratchett.

Posted on March 15, 2015 by Nicole Yunger Halpern

I blame British novels for my love of physics. Philip Pullman introduced me to elementary particles; Jasper Fforde, to the possibility that multiple worlds exist; Diana Wynne Jones, to questions about space and time.

So began the personal statement in my application to Caltech’s PhD program. I didn’t mention Sir Terry Pratchett, but he belongs in the list. Pratchett wrote over 70 books, blending science fiction with fantasy, humor, and truths about humankind. Pratchett passed away last week, having completed several novels after doctors diagnosed him with early-onset Alzheimer’s. According to the San Francisco Chronicle, Pratchett “parodie[d] everything in sight.” Everything in sight included physics.

Terry Pratchett continues to influence my trajectory through physics: This cover has a cameo in a seminar I’m presenting in Maryland this March.

Pratchett set many novels on the Discworld, a pancake of a land perched atop four elephants, which balance on the shell of a turtle that swims through space. Discworld wizards quantify magic in units called thaums. Units impressed their importance upon me in week one of my first high-school physics class. We define one meter as “the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.” Wizards define one thaum as “the amount of magic needed to create one small white pigeon or three normal-sized billiard balls.”

Wizards study the thaum in a High-Energy Magic Building reminiscent of Caltech’s Lauritsen-Downs Building. To split the thaum, the wizards built a Thaumatic Resonator. Particle physicists in our world have split atoms into constituent particles called mesons and baryons. Discworld wizards discovered that the thaum consists of resons. Mesons and baryons consist of quarks, seemingly elementary particles that we believe cannot be split. Quarks fall into six types, called flavors: up, down, charmed, strange, top (or truth), and bottom (or beauty). Resons, too, consist of quarks. The Discworld’s quarks have the flavors up, down, sideways, sex appeal, and peppermint.

Reading about the Discworld since high school, I’ve wanted to grasp Pratchett’s allusions. I’ve wanted to do more than laugh at them. In Pyramids, Pratchett describes “ideas that would make even a quantum mechanic give in and hand back his toolbox.” Pratchett’s ideas have given me a hankering for that toolbox. Pratchett nudged me toward training as a quantum mechanic.

Pratchett hasn’t only piqued my curiosity about his allusions. He’s piqued my desire to create as he did, to do physics as he wrote. While reading or writing, we build worlds in our imaginations. We visualize settings; we grow acquainted with characters; we sense a plot’s consistency or the consistency of a system of magic. We build worlds in our imaginations also when doing and studying physics and math. The Standard Model is a system that encapsulates the consistency of our knowledge about particles. We tell stories about electrons’ behaviors in magnetic fields. Theorems’ proofs have logical structures like plots’. Pratchett and other authors trained me to build worlds in my imagination. Little wonder I’m training to build worlds as a physicist.

Around the time I graduated from college, Diana Wynne Jones passed away. So did Brian Jacques (another British novelist) and Madeleine L’Engle. L’Engle wasn’t British, but I forgave her because her Time Quartet introduced me to dimensions beyond three. As I completed one stage of intellectual growth, creators who’d led me there left.

Terry Pratchett has joined Jones, Jacques, and L’Engle. I will probably create nothing as valuable as his Discworld, let alone a character in the Standard Model toward which the Discworld steered me.

But, because of Terry Pratchett, I have to try.

Always look on the bright side…of CPTP maps.

Posted on February 28, 2015 by Nicole Yunger Halpern

Once upon a time, I worked with a postdoc who shaped my views of mathematical physics, research, and life. Each week, I’d email him a PDF of the calculations and insights I’d accrued. He’d respond along the lines of, “Thanks so much for your notes. They look great! I think they’re mostly correct; there are just a few details that might need fixing.” My postdoc would point out the “details” over espresso, at a café table by a window. “Are you familiar with…?” he’d begin, and pull out of his back pocket some bit of math I’d never heard of. My calculations appeared to crumble like biscotti.

Some of the math involved CPTP maps. “CPTP” stands for a phrase little more enlightening than the acronym: “completely positive trace-preserving”. CPTP maps represent processes undergone by quantum systems. Imagine preparing some system—an electron, a photon, a superconductor, etc.—in a state I’ll call “ $\rho$ “. Imagine turning on a magnetic field, or coupling one electron to another, or letting the superconductor sit untouched. A CPTP map, labeled as $\mathcal{E}$ , represents every such evolution.

“Trace-preserving” means the following: Imagine that, instead of switching on the magnetic field, you measured some property of $\rho$ . If your measurement device (your photodetector, spectrometer, etc.) worked perfectly, you’d read out one of several possible numbers. Let $p_i$ denote the probability that you read out the $i^{\rm{th}}$ possible number. Because your device outputs some number, the probabilities sum to one: $\sum_i p_i = 1$ . We say that $\rho$ “has trace one.” But you don’t measure $\rho$ ; you switch on the magnetic field. $\rho$ undergoes the process $\mathcal{E}$ , becoming a quantum state $\mathcal{E(\rho)}$ . Imagine that, after the process ended, you measured a property of $\mathcal{E(\rho)}$ . If your measurement device worked perfectly, you’d read out one of several possible numbers. Let $q_a$ denote the probability that you read out the $a^{\rm{th}}$ possible number. The probabilities sum to one: $\sum_a q_a =1$ . $\mathcal{E(\rho)}$ “has trace one”, so the map $\mathcal{E}$ is “trace preserving”.

Now that we understand trace preservation, we can understand positivity. The probabilities $p_i$ are positive (actually, nonnegative) because they lie between zero and one. Since the $p_i$ characterize a crucial aspect of $\rho$ , we call $\rho$ “positive” (though we should call $\rho$ “nonnegative”). $\mathcal{E}$ turns the positive $\rho$ into the positive $\mathcal{E(\rho)}$ . Since $\mathcal{E}$ maps positive objects to positive objects, we call $\mathcal{E}$ “positive”. $\mathcal{E}$ also satisfies a stronger condition, so we call such maps “completely positive.”^**

So I called my postdoc. “It’s almost right,” he’d repeat, nudging aside his espresso and pulling out a pencil. We’d patch the holes in my calculations. We might rewrite my conclusions, strengthen my assumptions, or prove another lemma. Always, we salvaged cargo. Always, I learned.

I no longer email weekly updates to a postdoc. But I apply what I learned at that café table, about entanglement and monotones and complete positivity. “It’s almost right,” I tell myself when a hole yawns in my calculations and a week’s work appears to fly out the window. “I have to fix a few details.”

Am I certain? No. But I remain positive.

^*Experts: “Trace-preserving” means $\rm{Tr}(\rho) =1 \Rightarrow \rm{Tr}(\mathcal{E}(\rho)) = 1$ .

^**Experts: Suppose that ρ is defined on a Hilbert space and that is defined on . “ is positive” means

To understand what “completely positive” means, imagine that our quantum system interacts with an environment. For example, suppose the system consists of photons in a box. If the box leaks, the photons interact with the electromagnetic field outside the box. Suppose the system-and-environment composite begins in a state defined on a Hilbert space . acts on the system’s part of state. Let denote the identity operation that maps every possible environment state to itself. Suppose that changes the system’s state while preserves the environment’s state. The system-and-environment composite ends up in the state . This state is positive, so we call “completely positive”:

Democrat plus Republican over the square-root of two

Posted on January 25, 2015 by Nicole Yunger Halpern

I wish I could superpose votes on Election Day.

However much I agree with Candidate A about social issues, I dislike his running mate. I lean toward Candidate B’s economic plans and C’s science-funding record, but nobody’s foreign policy impresses me. Must I settle on one candidate? May I not vote

Now you can—at least in theory. Caltech postdoc Ning Bao and I concocted quantum elections in which voters can superpose, entangle, and create probabilistic mixtures of votes.

Previous quantum-voting work has focused on privacy and cryptography. Ning and I channeled quantum game theory. Quantum game theorists ask what happens if players in classical games, such as the Prisoner’s Dilemma, could superpose strategies and share entanglement. Quantization can change the landscape of possible outcomes.

The Prisoner’s Dilemma, for example, concerns two thugs whom the police have arrested and have isolated in separate cells. Each prisoner must decide whether to rat out the other. How much time each serves depends on who, if anyone, confesses. Since neither prisoner knows the other’s decision, each should rat to minimize his or her jail time. But both would serve less time if neither confessed. The prisoners can escape this dilemma using quantum resources.

Introducing superpositions and entanglement into games helps us understand the power of quantum mechanics. Elections involve gameplay; pundits have been feeding off Hilary Clinton’s for months. So superpositions and entanglement merit introduction into elections.

How can you model elections with quantum systems? Though multiple options exist, Ning and I followed two principles: (1) A general quantum process—a preparation procedure, an evolution, and a measurement—should model a quantum election. (2) Quantum elections should remain as true as possible to classical.

Given our quantum voting system, one can violate a quantum analogue of Arrow’s Impossibility Theorem. Arrow’s Theorem, developed by the Nobel-winning economist Kenneth Arrow during the mid-20^th century, is a no-go theorem about elections: If a constitution has three innocuous-seeming properties, it’s a dictatorship. Ning and I translated the theorem as faithfully as we knew how into our quantum voting scheme. The result, dubbed the Quantum Arrow Conjecture, rang false.

http://xkcd.com/661/

Superposing (and probabilistically mixing) votes entices me for a reason that science does: I feel ignorant. I read articles and interview political junkies about national defense; but I miss out on evidence and subtleties. I read quantum-physics books and work through papers; but I miss out on known mathematical tools and physical interpretations. Not to mention tools and interpretations that humans haven’t discovered.

Science involves identifying (and diminishing) what humanity doesn’t know. Science frees me to acknowledge my ignorance. I can’t throw all my weight behind Candidate A’s defense policy because I haven’t weighed all the arguments about defense, because I don’t know all the arguments. Believing that I do would violate my job description. How could I not vote for elections that accommodate superpositions?

Though Ning and I identified applications of superpositions and entanglement, more quantum strategies might await discovery. Monogamy of entanglement, discussed elsewhere on this blog, might limit the influence voters exert on each other. Also, we quantized ordinal voting systems (in which each voter ranks candidates, as in “A above C above B”). The quantization of cardinal voting (in which each voter grades the candidates, as in “5 points to A, 3 points to C, 2 points to B”) or another voting scheme might yield more insights.

If you have such insights, drop us a line. Ideally before the presidential smack-down of 2016.

Actions do change the world.

Posted on December 28, 2014 by Nicole Yunger Halpern

I heard it in a college lecture about Haskell.

Haskell is a programming language akin to Latin: Learning either language expands your vocabulary and technical skills. But programmers use Haskell as often as slam poets compose dactylic hexameter.^*

My professor could have understudied for the archetypal wise man: He had snowy hair, a beard, and glasses that begged to be called “spectacles.” Pointing at the code he’d projected onto a screen, he was lecturing about input/output, or I/O. The user inputs a request, and the program outputs a response.

That autumn was consuming me. Computer-science and physics courses had filled my plate. Atop the plate, I had thunked the soup tureen known as “XKCD Comes to Dartmouth”: I was coordinating a visit by Randall Munroe, creator of the science webcomic xkcd, to my college. The visit was to include a cake shaped like the Internet, a robotic velociraptor, and playpen balls. The output I’d promised felt offputting.

My professor knew. We sat in his office for hours each week, dispatching my questions about recursion and monads and complexity. His input shaped the functional-programming skills I apply today, and my input shaped his lecture notes. He promised to attend my event.

Most objects coded in Haskell, my professor reminded us in that lecture, remain static. The Haskell world never changes. Even the objects called “variables” behave like constants. Exceptions, my professor said, crop up in I/O. Users can affect a program through objects called “actions.”

“Actions,” he said, “do change the world.”

I caught my breath. What a zinger, I thought. What a moral. What a lesson for the ages—and from the mouths of computer scientists.

My professor had no idea. He swanned on about I/O without changing tone.

That December, I gave my professor a plaque that read, “Actions do change the world.” Around his slogan, I’d littered illustrations of fractals, Pascal’s triangle, and other subjects we’d studied. My professor’s actions had changed my world for the better.

You can change the world by writing code that revolutionizes physics or finance. You can change the world by writing code that limps through tasks but improves your understanding. You can change the world by helping someone understand code. You can change the world by understanding someone.

A new year is beginning. Which actions will you take?

^*Not that I recommend studying Haskell to boost your verbal SAT score.

I spy with my little eye…something algebraic.

Posted on November 17, 2014 by Nicole Yunger Halpern

Look at this picture.

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

Now? Try crossing your eyes.

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 ρ_S that appears in the first two images represents the smaller system. The tensor product represents the reservoir-and-smaller-system composite.

Generally speaking

Posted on October 26, 2014 by Nicole Yunger Halpern

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.

http://www.telegraph.co.uk/property/propertyadvice/jeffhowell/8013593/Home-improvements-Slime-does-come-out-in-the-wash.html

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.

The experimentalist next door

Posted on September 16, 2014 by Nicole Yunger Halpern

At 9:10 AM, the lab next door was blasting “Born to Be Wild.”

I was at Oxford, moonlighting as a visiting researcher during fall 2013. My hosts included quantum theorists in Townsend Laboratory, a craggy great-uncle of a building. Poke your head out of the theory office, and Experiment would flood your vision. Our neighbors included laser wielders, ion trappers, atom freezers, and yellow signs that warned, “DANGER OF DEATH.”

Down the corridor in Townsend Laboratory.

Hardly the neighborhood Mr. Rogers had in mind.

The lab that shared a wall with our office blasted music. To clear my head of calculations and of Steppenwolf, I would roam the halls. Some of the halls, that is. Other halls had hazmat warnings instead of welcome mats. I ran into “RADIATION,” “FIRE HAZARD,” “STRONG MAGNETIC FIELDS,” “HIGH VOLTAGE,” and “KEEP THIS TOILET NEAT AND TIDY.” Repelled from half a dozen doors, I would retreat to the office. Kelly Clarkson would be cooing through the wall.

“We can hear them,” a theorist observed about the experimentalists, “but they can’t hear us.”

Dangers lurked even in the bathroom.

Experiment should test, disprove, and motivate theories; and theory should galvanize and (according to some thinkers) explain experiments. But some theorists stray from experiment like North America from Pangaea.

The theoretical physics I’ve enjoyed is abstract. I rarely address platforms, particular physical systems in which theory might incarnate. Quantum-information platforms include electrons in magnetic fields, photons (particles of light), ion traps, quantum dots, and nuclei such as the ones that image internal organs in MRI machines.

Instead of addressing electrons and photons, I address mathematics and abstract physical concepts. Each of these concepts can incarnate in different forms in different platforms. Examples of such concepts include preparation procedures, evolutions, measurements, and memories. One preparation procedure defined by one piece of math can result from a constant magnetic field in one platform and from a laser in another. Abstractness has power, enabling one idea to describe diverse systems.

I’ve enjoyed wandering the hills and sampling the vistas of Theory Land. Yet the experimentalist next door cranked up the radio of reality in my mind. “We can hear them,” a theorist said. In Townsend Laboratory, I began listening. My Oxford collaborators and I interwove two theoretical frameworks that describe heat transferred and work performed on small scales. One framework, one-shot statistical mechanics, has guest-starred on this blog.

The other framework consists of fluctuation relations, which describe deviations from average behaviors by small physical systems. A quantum particle on one side of a wall has a tiny probability of tunneling through without boring any hole. Since the probability is tiny, the average particle doesn’t tunnel (during any reasonably short amount of time). When analyzing macroscopic systems—say, the roughly 10²⁴ atoms that form your left thumbnail—we assume that every particle behaves like the average particle. We can’t when analyzing minuscule systems such as one short strand of DNA. Deviations from average behaviors appear in experimental data about small systems as they do not appear in data about large systems. Fluctuation relations help us understand those deviations.

My colleagues and I addressed “information,” “systems,” and “interactions.” We deployed abstract ideas, referencing platforms only when motivating our work. Then a collaborator challenged me to listen through the wall.

Experimentalists have tested fluctuation relations. Why not check whether their data supports our theory? At my friend’s urging, I contacted experimentalists who’d shown that DNA obeys a fluctuation relation. The experimentalists had unzipped and re-zipped single DNA molecules using optical tweezers, which resemble ordinary tweezers but involve lasers. Whenever the experimentalists pulled the DNA, they measured the force they applied. They concluded that their platform obeyed an abstract fluctuation theorem. The experimentalists generously shared their data, which supported our results.

http://www.europhysicsnews.org/articles/epn/abs/2010/02/epn20102p27/epn20102p27.html

Experimentalists unzipped and rezipped DNA to test fluctuation relations. This depiction of the set-up comes from this article.

My colleagues and I didn’t propose experiments. We didn’t explain why platforms had behaved in unexpected ways. We checked calculations with recycled data. But we ventured outside Theory Land. We learned that one-shot theory models systems modeled also by fluctuation relations, which govern experiments. This link from one-shot theory to experiment, like the forbidden corridors in Townsend Laboratory, invite exploration.

In Townsend, I didn’t suffer the electric shocks or the explosions advertised on the doors (though the hot water in the bathroom nearly burned me). I turned out not to need those shocks. Blasting rock music at 9:10 AM can wake even a theorist up to reality.

Quantum Frontiers

A blog by the Institute for Quantum Information and Matter @ Caltech

Author Archives: Nicole Yunger Halpern

About Nicole Yunger Halpern

Hello, my name is QUANTUM MASTER EQUATION

Mingling stat mech with quantum info in Maryland

Paul Dirac and poetry

Quantum Frontiers salutes Terry Pratchett.

Always look on the bright side…of CPTP maps.

Democrat plus Republican over the square-root of two

Actions do change the world.

I spy with my little eye…something algebraic.

Generally speaking

The experimentalist next door

About Nicole Yunger Halpern

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