About Nicole Yunger Halpern

I'm pursuing a physics PhD with the buccaneers of Quantum Frontiers. Before moving to Caltech, I studied at Dartmouth College and the Perimeter Institute for Theoretical Physics. I apply quantum-information tools to thermodynamics and statistical mechanics (the study of heat, work, information, and time), particularly at small scales. I like my quantum information physical, my math algebraic, and my spins rotated but not stirred.

Quantum Frontiers salutes Terry Pratchett.

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.

http://www.lookoutmountainbookstore.com/

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.

http://www.google.com/imgres?imgurl=http://www.stmoroky.com/reviews/films/brian.jpg&imgrefurl=http://www.stmoroky.com/reviews/films/brian.htm&h=290&w=250&tbnid=ZVAVf4ZX_zMISM:&zoom=1&docid=D5VZDH_bepBS0M&hl=en&ei=3bLfVIiYOoSpogS6toLYCQ&tbm=isch&ved=0CFEQMygYMBg

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 H and that E of rho is defined on H'. “Channel is positive” means Positive

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 SigmaAB defined on a Hilbert space HAB. Channel acts on the system’s part of state. Let I denote the identity operation that maps every possible environment state to itself. Suppose that Channel changes the system’s state while I preserves the environment’s state. The system-and-environment composite ends up in the state Channel SigmaAB. This state is positive, so we call Channel “completely positive”:Completely pos

Democrat plus Republican over the square-root of two

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

TeXVoteClip

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-20th 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.

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.

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.

Drysdale photo 2

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.

Look at this picture.

Peter 1

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

Peter 2

Now? Try crossing your eyes.

Peter 3

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.

http://www.sell-buy-machines.com/2013/02/why-prefer-second-hand-equipment-over-new.html

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

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.”

P1040564

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.”

P1040552

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 trapsquantum 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.