About Nicole Yunger Halpern

I recently completed a physics PhD with the buccaneers of Quantum Frontiers. Starting in Sept. 2018, I'll work as an ITAMP Postdoctoral Fellow at the Harvard-Smithsonian Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP). You'll be able to catch me at ITAMP, in Harvard's physics department, or at MIT. My research consists of what I like to call "quantum steampunk": I combine quantum information with thermodynamics, contributing to the theory of quantum thermodynamics and applying quantum thermodynamics across science. Before beginning my PhD at Caltech, I studied at Dartmouth College and the Perimeter Institute for Theoretical Physics. I like my quantum information physical, my math algebraic, and my spins rotated but not stirred.

What’s the worst that could happen?

The archaeologist Howard Carter discovered Tutankhamun’s burial site in 1922. No other Egyptian pharaoh’s tomb had survived mostly intact until the modern era. Gold and glass and faience, statues and pendants and chariots, had evaded looting. The discovery would revolutionize the world’s understanding of, and enthusiasm for, ancient Egypt.

First, the artifacts had to leave the tomb.

Tutankhamun lay in three coffins nested like matryoshka dolls. Carter describes the nesting in his book The Tomb of Tutankhamen. Lifting the middle coffin from the outer coffin raised his blood pressure:

Everything may seem to be going well until suddenly, in the crisis of the process, you hear a crack—little pieces of surface ornament fall. Your nerves are at an almost painful tension. What is happening? All available room in the narrow space is crowded by your men. What action is needed to avert a catastrophe?

In other words, “What’s the worst that could happen?”

Matryoshka dolls

Collaborators and I asked that question in a paper published last month. We had in mind less Egyptology than thermodynamics and information theory. But never mind the distinction; you’re reading Quantum Frontiers! Let’s mix the fields like flour and oil in a Biblical grain offering.

Carter’s team had trouble separating the coffins: Ancient Egyptian priests (presumably) had poured fluid atop the innermost, solid-gold coffin. The fluid had congealed into a brown gunk, gluing the gold coffin to the bottom of the middle coffin. Removing the gold coffin required work—thermodynamic work.

Work consists of “well-ordered” energy usable in tasks like levering coffins out of sarcophagi and motoring artifacts from Egypt’s Valley of the Kings toward Cairo. We can model the gunk as a spring, one end of which was fixed to the gold coffin and one end of which was fixed to the middle coffin. The work W required to stretch a spring depends on the spring’s stiffness (the gunk’s viscosity) and on the distance stretched through.

W depends also on details: How many air molecules struck the gold coffin from above, opposing the team’s effort? How quickly did Carter’s team pull? Had the gunk above Tuankhamun’s nose settled into a hump or spread out? How about the gunk above Tutankhamun’s left eye socket? Such details barely impact the work required to open a 6.15-foot-long coffin. But air molecules would strongly impact W if Tutankhamun measured a few nanometers in length. So imagine Egyptian matryoshka dolls as long as stubs of DNA.

DNA

Imagine that Carter found one million sets of these matryoshka dolls. Lifting a given set’s innermost coffin would require an amount W of work that would vary from set of coffins to set of coffins. W would satisfy fluctuation relations, equalities I’ve blogged about many times.

Fluctuation relations resemble the Second Law of Thermodynamics, which illuminates why time flows in just one direction. But fluctuation relations imply more-precise predictions about W than the Second Law does.

Some predictions concern dissipated work: Carter’s team could avoid spending much work by opening the coffin infinitesimally slowly. Speeding up would heat the gunk, roil air molecules, and more. The heating and roiling would cost extra work, called dissipated work, denoted by W_{\rm diss}.

Suppose that Carter’s team has chosen a lid-opening speed v. Consider the greatest W_{\rm diss} that the team might have to waste on any nanoscale coffin. W_{\rm diss}^{\rm worst} is proportional to each of three information-theoretic quantities, my coauthors and I proved.

For experts: Each information-theoretic quantity is an order-infinity Rényi divergence D_\infty ( X || Y). The Rényi divergences generalize the relative entropy D ( X || Y ). D quantifies how efficiently one can distinguish between probability distributions, or quantum states, X and Y on average. The average is over many runs of a guessing game.

Imagine the worst possible run, which offers the lowest odds of guessing correctly. D_\infty quantifies your likelihood of winning. We related W_{\rm diss}^{\rm worst} to a D_\infty between two statistical-mechanical phase-space distributions (when we described classical systems), to a D_\infty between two quantum states (when we described quantum systems), and to a D_\infty between two probability distributions over work quantities W (when we described systems quantum and classical).

Book-paper

The worst case marks an extreme. How do the extremes consistent with physical law look? As though they’ve escaped from a mythologist’s daydream.

In an archaeologist’s worst case, arriving at home in the evening could lead to the following conversation:

“How was your day, honey?”

“The worst possible.”

“What happened?”

“I accidentally eviscerated a 3.5-thousand-year-old artifact—the most valuable, best-preserved, most information-rich, most lavishly wrought ancient Egyptian coffin that existed yesterday.”

Suppose that the archaeologist lived with a physicist. My group (guided by a high-energy physicist) realized that the conversation could continue as follows:

“And how was your day?”

“Also the worst possible.”

“What happened?”

“I created a black hole.”

General relativity and high-energy physics have begun breeding with quantum information and thermodynamics. The offspring bear extremes like few other systems imaginable. I wonder what our results would have to say about those offspring.

Oops

National Geographic reprinted Carter’s The Tomb of Tutankhamen in its “Adventure Classics” series. The series title fits Tomb as a mummy’s bandages fit the mummy. Carter’s narrative stretches from Egypt’s New Kingdom (of 3.5 thousand years ago) through the five-year hunt for the tomb (almost fruitless until the final season), to a water boy’s discovery of steps into the tomb, to the unsealing of the burial chamber, to the confrontation of Tutankhamun’s mummy.

Carter’s book guided me better than any audio guide could have at the California Science Center. The center is hosting the exhibition “King Tut: Treasures of the Golden Pharaoh.” After debuting in Los Angeles, the exhibition will tour the world. The tour showcases 150 artifacts from Tutankhamun’s tomb.

Those artifacts drove me to my desk—to my physics—as soon as I returned home from the museum. Tutankhamun’s tomb, Carter argues in his book, ranks amongst the 20th century’s most important scientific discoveries. I’d seen a smidgeon of the magnificence that Carter’s team— with perseverance, ingenuity, meticulousness, and buckets of sweat shed in Egypt’s heat—had discovered. I don’t expect to discover anything a tenth as magnificent. But how can a young scientist resist trying?

People say, “Prepare for the worst. Hope for the best.” I prefer “Calculate the worst. Hope and strive for a Tutankhamun.”

Outside exhibition

Postscript: Carter’s team failed to unglue the gold coffin by just “stretching” the gunky “spring.” The team resorted to heat, a thermodynamic quantity alternative to work: The team flipped the middle coffin upside-down above a heat lamp. The lamp raised the temperature to 932°F, melting the goo. The melting, with more work, caused the gold coffin to plop out of the middle coffin.

A finger painting for John Preskill

I’d completed about half my candidacy exam.

Four Caltech faculty members sat in front of me, in a bare seminar room. I stood beside a projector screen, explaining research I’d undertaken. The candidacy exam functions as a milepost in year three of our PhD program. The committee confirms that the student has accomplished research and should continue.

I was explaining a quantum-thermodynamics problem. I reviewed the problem’s classical doppelgänger and a strategy for solving the doppelgänger. Could you apply the classical strategy in the quantum problem? Up to a point. Beyond it, you’d need

Beatles

“Does anyone here like the Beatles?” I asked the committee. Three professors had never participated in an exam committee before. The question from the examinee appeared to startle them.

One committee member had participated in cartloads of committees. He recovered first, raising a hand.

The committee member—John Preskill—then began singing the Beatles song.

In the middle of my candidacy exam.

The moment remains one of the highlights of my career.

FP3

Throughout my PhD career, I’ve reported to John. I’ve emailed an update every week and requested a meeting about once a month. I sketch the work that’s firing me, relate my plans, and request feedback.

Much of the feedback, I’ve discerned over the years, condenses into aphorisms buried in our conversations. I doubt whether John has noticed his aphorisms. But they’ve etched themselves in me, and I hope they remain there.

“Think big.” What would impact science? Don’t buff a teapot if you could be silversmithing.

Education serves as “money in the bank.” Invest in yourself, and draw on the interest throughout your career.

“Stay broad.” (A stretching outward of both arms accompanies this aphorism.) Embrace connections with diverse fields. Breadth affords opportunities to think big.

“Keep it simple,” but “do something technical.” A teapot cluttered with filigree, spouts, and eighteen layers of gold leaf doesn’t merit a spot at the table. A Paul Revere does.

“Do what’s best for Nicole.” I don’t know how many requests to speak, to participate on committees, to explain portions of his lecture notes, to meet, to contribute to reports, and more John receives per week. The requests I receive must look, in comparison, like a mouse to a mammoth. But John exhorts me to to guard my time for research—perhaps, partially, because he gives so much time, including to students.

“Move on.” If you discover an opportunity, study background information for a few months, seize the opportunity, wrap up the project, and seek the next window.

John has never requested my updates, but he’s grown used to them. I’ve grown used to how meetings end. Having brought him questions, I invite him to ask questions of me.

“Are you having fun?” he says.

FP2

I tell the Beatles story when presenting that quantum-thermodynamics problem in seminars.

“I have to digress,” I say when the “Help!” image appears. “I presented this slide at a talk at Caltech, where John Preskill was in the audience. Some of you know John.” People nod. “He’s a…mature gentleman.”

I borrowed the term from the apparel industry. “Mature gentleman” means “at a distinguished stage by which one deserves to have celebrated a birthday of his with a symposium.”

Many physicists lack fluency in apparel-industry lingo. My audience members take “mature” at face value.

Some audience members grin. Some titter. Some tilt their heads from side to side, as though thinking, “Eh…”

John has impact. He’s logged boatloads of technical achievements. He has the scientific muscle of a scientific rhinoceros.

And John has fun. He doesn’t mind my posting an article about audience members giggling about him.

Friends ask me whether professors continue doing science after meriting birthday symposia, winning Nobel Prizes, and joining the National Academy of Sciences. I point to the number of papers with which John has, with coauthors, electrified physics over the past 20 years. Has coauthored because science is fun. It merits singing about during candidacy exams. Satisfying as passing the exam felt two years ago, I feel more honored when John teases me about my enthusiasm for science.

FP5

A year ago, I ate lunch with an alumnus who’d just graduated from our group. Students, he reported, have a tradition of gifting John a piece of art upon graduating. I relayed the report to another recent alumnus.

“Really?” the second alumnus said. “Maybe someone gave John a piece of art and then John invented the tradition.”

Regardless of its origin, the tradition appealed to me. John has encouraged me to blog as he’s encouraged me to do theoretical physics. Writing functions as art. And writing resembles theoretical physics: Each requires little more than a pencil, paper, and thought. Each requires creativity, aesthetics, diligence, and style. Each consists of ideas, of abstractions; each lacks substance but can outlive its creator. Let this article serve as a finger painting for John Preskill.

Thanks for five fun years.

Committee

With my PhD-thesis committee, after my thesis defense. Photo credit to Nick Hutzler, who cracked the joke that accounts for everyone’s laughing. (Left to right: Xie Chen, Fernando Brandão, John Preskill, Nicole Yunger Halpern, Manuel Endres.)

Catching up with the quantum-thermo crowd

You have four hours to tour Oxford University.

What will you visit? The Ashmolean Museum, home to da Vinci drawings, samurai armor, and Egyptian mummies? The Bodleian, one of Europe’s oldest libraries? Turf Tavern, where former president Bill Clinton reportedly “didn’t inhale” marijuana?

Felix Binder showed us a cemetery.

Of course he showed us a cemetery. We were at a thermodynamics conference.

The Fifth Quantum Thermodynamics Conference took place in the City of Dreaming Spires.Participants enthused about energy, information, engines, and the flow of time. About 160 scientists attended—roughly 60 more than attended the first conference, co-organizer Janet Anders estimated.

IMG_1177

Weak measurements and quasiprobability distributions were trending. The news delighted me, Quantum Frontiers regulars won’t be surprised to hear.

Measurements disturb quantum systems, as early-20th-century physicist Werner Heisenberg intuited. Measure a system’s position strongly, and you forfeit your ability to predict the outcomes of future momentum measurements. Weak measurements don’t disturb the system much. In exchange, weak measurements provide little information about the system. But you can recoup information by performing a weak measurement in each of many trials, then processing the outcomes.

Strong measurements lead to probability distributions: Imagine preparing a particle in some quantum state, then measuring its position strongly, in each of many trials. From the outcomes, you can infer a probability distribution \{ p(x) \}, wherein p(x) denotes the probability that the next trial will yield position x.

Weak measurements lead analogously to quasiprobability distributions. Quasiprobabilities resemble probabilities but can misbehave: Probabilities are real numbers no less than zero. Quasiprobabilities can dip below zero and can assume nonreal values.

Do not disturb 2

What relevance have weak measurements and quasiprobabilities to quantum thermodynamics? Thermodynamics involves work and heat. Work is energy harnessed to perform useful tasks, like propelling a train from London to Oxford. Heat is energy that jiggles systems randomly.

Quantum properties obscure the line between work and heat. (Here’s an illustration for experts: Consider an isolated quantum, such as a spin chain. Let H(t) denote the Hamiltonian that evolves with the time t \in [0, t_f]. Consider preparing the system in an energy eigenstate | E_i(0) \rangle. This state has zero diagonal entropy: Measuring the energy yields E_i(0) deterministically. Considering tuning H(t), as by changing a magnetic field. This change constitutes work, we learn in electrodynamics class. But if H(t) changes quickly, the state can acquire weight on multiple energy eigenstates. The diagonal entropy rises. The system’s energetics have gained an unreliability characteristic of heat absorption. But the system has remained isolated from any heat bath. Work mimics heat.)

Quantum thermodynamicists have defined work in terms of a two-point measurement scheme: Initialize the quantum system, such as by letting heat flow between the system and a giant, fixed-temperature heat reservoir until the system equilibrates. Measure the system’s energy strongly, and call the outcome E_i. Isolate the system from the reservoir. Tune the Hamiltonian, performing the quantum equivalent of propelling the London train up a hill. Measure the energy, and call the outcome E_f.

Any change \Delta E in a system’s energy comes from heat Q and/or from work W, by the First Law of Thermodynamics, \Delta E = Q + W.  Our system hasn’t exchanged energy with any heat reservoir between the measurements. So the energy change consists of work: E_f - E_i =: W.

Train

Imagine performing this protocol in each of many trials. Different trials will require different amounts W of work. Upon recording the amounts, you can infer a distribution \{ p(W) \}. p(W) denotes the probability that the next trial will require an amount W of work.

Measuring the system’s energy disturbs the system, squashing some of its quantum properties. (The measurement eliminates coherences, relative to the energy eigenbasis, from the state.) Quantum properties star in quantum thermodynamics. So the two-point measurement scheme doesn’t satisfy everyone.

Enter weak measurements. They can provide information about the system’s energy without disturbing the system much. Work probability distributions \{ p(W) \} give way to quasiprobability distributions \{ \tilde{p}(W) \}.

So propose Solinas and Gasparinetti, in these papers. Other quantum thermodynamicists apply weak measurements and quasiprobabilities differently.2 I proposed applying them to characterize chaos, and the scrambling of quantum information in many-body systems, at the conference.3 Feel free to add your favorite applications to the “comments” section.

Dinner

All the quantum ladies: The conference’s female participants gathered for dinner one conference night.

Wednesday afforded an afternoon for touring. Participants congregated at the college of conference co-organizer Felix Binder.3 His tour evoked, for me, the ghosts of thermo conferences past: One conference, at the University of Cambridge, had brought me to the grave of thermodynamicist Arthur Eddington. Another conference, about entropies in information theory, had convened near Canada’s Banff Cemetery. Felix’s tour began with St. Edmund Hall’s cemetery. Thermodynamics highlights equilibrium, a state in which large-scale properties—like temperature and pressure—remain constant. Some things never change.

IMG_1192

 

With thanks to Felix, Janet, and the other coordinators for organizing the conference.

1Oxford derives its nickname from an elegy by Matthew Arnold. Happy National Poetry Month!

2https://arxiv.org/abs/1508.00438https://arxiv.org/abs/1610.04285,
https://arxiv.org/abs/1607.02404,
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.070601,
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.040602

3Michele Campisi joined me in introducing out-of-time-ordered correlators (OTOCs) into the quantum-thermo conference: He, with coauthor John Goold, combined OTOCs with the two-point measurement scheme.

3Oxford University contains 38 colleges, the epicenters of undergraduates’ social, dining, and housing experiences. Graduate students and postdoctoral scholars affiliate with colleges, and senior fellows—faculty members—govern the colleges.

Rock-paper-scissors, granite-clock-idea

I have a soft spot for lamassu. Ten-foot-tall statues of these winged bull-men guarded the entrances to ancient Assyrian palaces. Show me lamassu, or apkallu—human-shaped winged deities—or other reliefs from the Neo-Assyrian capital of Nineveh, and you’ll have trouble showing me the door.

Assyrian art fills a gallery in London’s British Museum. Lamassu flank the gallery’s entrance. Carvings fill the interior: depictions of soldiers attacking, captives trudging, and kings hunting lions. The artwork’s vastness, its endurance, and the contact with a three-thousand-year-old civilization floor me. I tore myself away as the museum closed one Sunday night.

Lamassu

I visited the British Museum the night before visiting Jonathan Oppenheim’s research group at University College London (UCL). Jonathan combines quantum information theory with thermodynamics. He and others co-invented thermodynamic resource theories (TRTs), which Quantum Frontiers regulars will know of. TRTs are quantum-information-theoretic models for systems that exchange energy with their environments.

Energy is conjugate to time: Hamiltonians, mathematical objects that represent energy, represent also translations through time. We measure time with clocks. Little wonder that one can study quantum clocks using a model for energy exchanges.

Mischa Woods, Ralph Silva, and Jonathan used a resource theory to design an autonomous quantum clock. “Autonomous” means that the clock contains all the parts it needs to operate, needs no periodic winding-up, etc. When might we want an autonomous clock? When building quantum devices that operate independently of classical engineers. Or when performing a quantum computation: Computers must perform logical gates at specific times.

Clock

Wolfgang Pauli and others studied quantum clocks, the authors recall. How, Pauli asked, would an ideal clock look? Its Hamiltonian, \hat{H}_{\rm C}, would have eigenstates | E \rangle. The labels E denote possible amounts of energy.

The Hamiltonian would be conjugate to a “time operator” \hat{t}. Let | \theta \rangle denote an eigenstate of \hat{t}. This “time state” would equal an even superposition over the | E \rangle’s. The clock would occupy the state | \theta \rangle at time t_\theta.

Imagine measuring the clock, to learn the time, or controlling another system with the clock. The interaction would disturb the clock, changing the clock’s state. The disturbance wouldn’t mar the clock’s timekeeping, if the clock were ideal. What would enable an ideal clock to withstand the disturbances? The ability to have any amount of energy: E must stretch from - \infty to \infty. Such clocks can’t exist.

Approximations to them can. Mischa, Ralph, and Jonathan designed a finite-size clock, then characterized how accurately the clock mimics the ideal. (Experts: The clock corresponds to a Hilbert space of finite dimensionality d. The clock begins in a Gaussian state that peaks at one time state | \theta \rangle. The finite-width Gaussian offers more stability than a clock state.)

Disturbances degrade our ability to distinguish instants by measuring the clock. Imagine gazing at a kitchen clock through blurry lenses: You couldn’t distinguish 6:00 from 5:59 or 6:01. Disturbances also hinder the clock’s ability to implement processes, such as gates in a computation, at desired instants.

Mischa & co. quantified these degradations. The errors made by the clock, they found, decay inverse-exponentially with the clock’s size: Grow the clock a little, and the errors shrink a lot.

Reliefs

Time has degraded the lamassu, but only a little. You can distinguish feathers in their wings and strands in their beards. People portray such artifacts as having “withstood the flow of time,” or “evaded,” or “resisted.” Such portrayals have never appealed to me. I prefer to think of the lamassu as surviving not because they clash with time, but because they harmonize with it. The prospect of harmonizing with time—whatever that means—has enticed me throughout my life. The prospect partially underlies my research into time—perhaps childishly, foolishly—I recognize if I remove my blurry lenses before gazing in the mirror.

The creation of lasting works, like lamassu, has enticed me throughout my life. I’ve scrapbooked, archived, and recorded, and tended memories as though they were Great-Grandma’s cookbook. Ancient civilizations began alluring me at age six, partially due to artifacts’ longevity. No wonder I study the second law of thermodynamics.

Yet doing theoretical physics makes no sense from another perspective. The ancient Egyptians sculpted granite, when they could afford it. Gudea, king of the ancient city-state of Lagash, immortalized himself in diorite. I fashion ideas, which lack substance. Imagine playing, rather than rock-paper-scissors, granite-diorite-idea. The idea wouldn’t stand a chance.

Would it? Because an idea lacks substance, it can manifest in many forms. Plato’s cave allegory has manifested as a story, as classroom lectures, on handwritten pages, on word processors and websites, in cartloads of novels, in the film The Matrix, in one of the four most memorable advertisements I received from colleges as a high-school junior, and elsewhere. Plato’s allegory has survived since about the fourth century BCE. King Ashurbanipal’s lion-hunt reliefs have survived for only about 200 years longer.

The lion-hunt reliefs—and lamassu—exude a grandness, a majesty that’s attracted me as their longevity has. The nature of time and the perfect clock have as much grandness. Leaving the British Museum’s Assyrian gallery at 6 PM one Sunday, I couldn’t have asked for a more fitting location, 24 hours later, than in a theoretical-physics conversation.

 

With thanks to Jonathan, to Álvaro Martín-Alhambra, and to Mischa for their hospitality at UCL; to Ada Cohen for the “Art history of ancient Egypt and the ancient Near East” course for which I’d been hankering for years; to my brother, for transmitting the ancient-civilizations bug; and to my parents, who fed the infection with museum visits.

Click here for a follow-up to the quantum-clock paper.

The Ground Space of Babel

Librarians are committing suicide.

So relates the narrator of the short story “The Library of Babel.” The Argentine magical realist Jorge Luis Borges wrote the story in 1941.

Librarians are committing suicide partially because they can’t find the books they seek. The librarians are born in, and curate, a library called “infinite” by the narrator. The library consists of hexagonal cells, of staircases, of air shafts, and of closets for answering nature’s call. The narrator has never heard of anyone’s finding an edge of the library. Each hexagon houses 20 shelves, each of which houses 32 books, each of which contains 410 pages, each of which contains 40 lines, each of which consists of about 80 symbols. Every symbol comes from a set of 25: 22 letters, the period, the comma, and the space.

The library, a sage posited, contains every combination of the 25 symbols that satisfy the 410-40-and-80-ish requirement. His compatriots rejoiced:

All men felt themselves to be the masters of an intact and secret treasure. There was no personal or world problem whose eloquent solution did not exist in some hexagon. [ . . . ] a great deal was said about the Vindications: books of apology and prophecy which vindicated for all time the acts of every man in the universe and retained prodigious arcana for his future. Thousands of the greedy abandoned their sweet native hexagons and rushed up the stairways, urged on by the vain intention of finding their Vindication.

Probability punctured their joy: “the possibility of a man’s finding his Vindication, or some treacherous variation thereof, can be computed as zero.”

Many-body quantum physicists can empathize with Borges’s librarian.

A handful of us will huddle over a table or cluster in front of a chalkboard.

“Has anyone found this Hamiltonian’s ground space?” someone will ask.1

Library of Babel

A Hamiltonian is an observable, a measurable property. Consider a quantum system S, such as a set of particles hopping between atoms. We denote the system’s Hamiltonian by H. H determines how the system’s state changes in time. A musical about H swept Broadway last year.

A quantum system’s energy, E, might assume any of many possible values. H encodes the possible values. The least possible value, E0, we call the ground-state energy.

Under what condition does S have an amount E0 of energy? S must occupy a ground state. Consider Olympic snowboarder Shaun White in a half-pipe. He has kinetic energy, or energy of motion, when sliding along the pipe. He gains gravitational energy upon leaving the ground. He has little energy when sitting still on the snow. A quantum analog of that sitting constitutes a ground state.2

Consider, for example, electrons in a magnetic field. Each electron has a property called spin, illustrated with an arrow. The arrow’s direction represents the spin’s state. The system occupies a ground state when every arrow points in the same direction as the magnetic field.

Shaun White has as much energy, sitting on the ground in the half-pipe’s center, as he has sitting at the bottom of an edge of the half-pipe. Similarly, a quantum system might have multiple ground states. These states form the ground space.

“Has anyone found this Hamiltonian’s ground space?”

Olympic crashes

“Find” means, here,“identify the form of.” We want to derive a mathematical expression for the quantum analog of “sitting still, at the bottom of the half-pipe.”

“Find” often means “locate.” How do we locate an object such as a library? By identifying its spatial coordinates. We specify coordinates relative to directions, such as north, east, and up. We specify coordinates also when “finding” ground states.

Libraries occupy the physical space we live in. Ground states occupy an abstract mathematical space, a Hilbert space. The Hilbert space consists of the (pure) quantum states accessible to the system—loosely speaking, how the spins can orient themselves.

Libraries occupy a three-dimensional space. An N-spin system corresponds to a 2N-dimensional Hilbert space. Finding a ground state amounts to identifying 2N coordinates. The problem’s size grows exponentially with the number of particles.

An exponential quantifies also the size of the librarian’s problem. Imagine trying to locate some book in the Library of Babel. How many books should you expect to have to check? How many books does the library hold? Would you have more hope of finding the book, wandering the Library of Babel, or finding a ground state, wandering the Hilbert space? (Please take this question with a grain of whimsy, not as instructions for calculating ground states.)

A book’s first symbol has one of 25 possible values. So does the second symbol. The pair of symbols has one of 25 \times 25 = 25^2 possible values. A trio has one of 25^3 possible values, and so on.

How many symbols does a book contain? About \frac{ 410 \text{ pages} }{ 1 \text{ book} }  \:  \frac{ 40 \text{ lines} }{ 1 \text{ page} }  \:  \frac{ 80 \text{ characters} }{ 1 \text{ line} }  \approx  10^6 \, , or a million. The number of books grows exponentially with the number of symbols per book: The library contains about 25^{ 10^6 } books. You contain only about 10^{24} atoms. No wonder librarians are committing suicide.

Do quantum physicists deserve more hope? Physicists want to find ground states of chemical systems. Example systems are discussed here and here. The second paper refers to 65 electrons distributed across 57 orbitals (spatial regions). How large a Hilbert space does this system have? Each electron has a spin that, loosely speaking, can point upward or downward (that corresponds to a two-dimensional Hilbert space). One might expect each electron to correspond to a Hilbert space of dimensionality (57 \text{ orbitals}) \frac{ 2 \text{ spin states} }{ 1 \text{ orbital} } = 114. The 65 electrons would correspond to a Hilbert space \mathcal{H}_{\rm tot} of dimensionality 114^{65}.

But no two electrons can occupy the same one-electron state, due to Pauli’s exclusion principle. Hence \mathcal{H}_{\rm tot} has dimensionality {114 \choose 65} (“114 choose 65″), the number of ways in which you can select 65 states from a set of 114 states.

{114 \choose 65} equals approximately 10^{34}. Mathematica (a fancy calculator) can print a one followed by 34 zeroes. Mathematica refuses to print the number 25^{ 10^6 } of Babel’s books. Pity the librarians more than the physicists.

Catalogue

Pity us less when we have quantum computers (QCs). They could find ground states far more quickly than today’s supercomputers. But building QCs is taking about as long as Borges’s narrator wandered the library, searching for “the catalogue of catalogues.”

What would Borges and his librarians make of QCs? QCs will be able to search unstructured databases quickly, via Grover’s algorithm. Babel’s library lacks structure. Grover’s algorithm outperforms classical algorithms just when fed large databases. 25^{ 10^6 } books constitute a large database. Researchers seek a “killer app” for QCs. Maybe Babel’s librarians could vindicate quantum computing and quantum computing could rescue the librarians. If taken with a grain of magical realism.

 

1Such questions remind me of an Uncle Alfred who’s misplaced his glasses. I half-expect an Auntie Muriel to march up to us physicists. She, sensible in plaid, will cross her arms.

“Where did you last see your ground space?” she’ll ask. “Did you put it on your dresser before going to bed last night? Did you use it at breakfast, to read the newspaper?”

We’ll bow our heads and shuffle off to double-check the kitchen.

2More accurately, a ground state parallels Shaun White’s lying on the ground, stone-cold.

What makes extraordinary science extraordinary?

My article for this month appears on Sean Carroll’s blog, Preposterous UniverseSean is a theoretical physicist who practices cosmology at Caltech. He interfaces with philosophy, which tinges the question I confront: What distinguishes extraordinary science from good science? The topic seemed an opportunity to take Sean up on an invitation to guest-post on Preposterous Universe. Head there for my article. Thanks to Sean for hosting!

Big Dipper

Panza’s paradox

I finished reading a translation of Don Quixote this past spring. Miguel de Cervantes wrote the novel during the 1600s. The hero, a Spanish gentleman, believes the tales of chivalry’s golden days. He determines to outdo his heroes as a knight. Don Quixote enlists a peasant, Sancho Panza, to serve as his squire. Bony Don Quixote quotes classical texts; tubby Sancho Panza can’t sign his name. The pair roams the countryside, seeking adventures.

Don Quixote might have sold more copies than any other novel in history. Historians have dubbed Don Quixote “the first modern novel”; “quixotic” appears in English dictionaries; and artists and writers still spin off the story. Don Quixote reverberates throughout the 500 years that have followed it.

Cervantes, I discovered, had grasped a paradox that mathematicians had exposed last century.

xkcd.jpg

Artists continue to spin off Don Quixote.

Don Quixote will vanquish so many villains, the pair expects, rulers will shower gifts on him. Someone will bequeath him a kingdom or an empire. Don Quixote promises to transfer part of his land to Sancho. Sancho expects to govern an ínsula, or island.

Sancho’s expectation amuses a duke and duchess. They pretend to grant Sancho an ínsula as a joke. How would such a simpleton rule? they wonder. They send servants and villagers to Sancho with fabricated problems. Sancho arbitrates the actors’ cases. Grossman translates one case as follows:

the first [case] was an engima presented to him by a stranger, who said:
“Señor, a very large river divided a lord’s lands into two parts [ . . . ] a bridge crossed this river, and at the end of it was a gallows and a kind of tribunal hall in which there were ordinarily four judges who applied the law set down by the owner of the river, the bridge, and the lands, which was as follows: ‘If anyone crosses this bridge from one side to the other, he must first take an oath as to where he is going and why; and if he swears the truth, let him pass; and if he tells a lie, let him die by hanging on the gallows displayed there, with no chance of pardon.’ Knowing this law and its rigorous conditions, many people crossed the bridge, and then, when it was clear that what they swore was true, the judges let them pass freely. It so happened, then, that a man once took the oath, and he swore and said that because of the oath he was going to die on the gallows, and he swore to nothing else. The judges studied the oath and said: ‘If we allow this man to pass freely, he lied in his oath, and according to the law he must die; and if we hang him, he swore that he was going to die on this gallows, and having sworn the truth, according to the same law he must go free.’ Señor Governor, the question for your grace is what should the judges do with the man.”

Cervantes formulated a paradox that looks, to me, equivalent to Russell’s barber paradox. Bertrand Russell contributed to philosophy during the early 1900s. He concocted an argument called the Russell-Zermelo paradox, which I’ll describe later. An acquaintance tried to encapsulate the paradox as follows: Consider an adult male barber who shaves all men who do not shave themselves. Does the barber shave himself?

Suppose that the barber doesn’t. (Analogously, suppose that the smart aleck in Panza’s paradox doesn’t lie.) The barber is a man whom the barber shaves. (The smart aleck must survive.) Hence the barber must shave himself. (Hence the traveler lies.) But we assumed that the barber doesn’t shave himself. (But we assumed that the traveler doesn’t lie.) Stalemate.

Barber

A barber plays a role in Don Quixote as in the Russell-Zermelo-like paradox. But the former barber has a wash basin that Don Quixote mistakes for a helmet.

Philosophers and mathematicians have debated to what extent the barber paradox illustrates the Russell-Zermelo paradox. Russell formulated the paradox in 1902. The mathematician Ernst Zermelo formulated the paradox around the same time. Mathematicians had just developed the field of set theory. A set is a collection of objects. Examples include the set of positive numbers, the set of polygons, and the set of readers who’ve looked at a Quantum Frontiers post.

Russell and Zermelo described a certain set \mathcal{S} of sets, a certain collection of sets. Let’s label the second-tier sets S_j  =  S_1, S_2, S_3, etc.

Russell 1

Each second-tier set S_j can contain elements. The elements can include third-tier sets s^{(j)}_k = s^{(j)}_1 ,  s^{(j)}_2, s^{(j)}_3, etc.

Russell 2

But no third-tier set s^{(j)}_k equals the second-tier set S_j. That is, no second-tier set S_j is element of itself.

Russell 3

Let \mathcal{S} contain every set that does not contain itself. Does the first-tier set \mathcal{S} contain itself?

Russell 4

Suppose that it does: \mathcal{S} = S_j for some j. \mathcal{S} is an element of itself. But, we said, “no second-tier set S_j is an element of itself.” So \mathcal{S} must not be an element of itself. But \mathcal{S} “contain[s] every set that does not contain itself.” So \mathcal{S} must contain itself. But we just concluded that \mathcal{S} doesn’t. Stalemate.

The Stanford Encyclopedia of Philosophy synopsizes the Russell-Zermelo paradox: “the paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself.”

One might resolve the Russell-Zermelo paradox by concluding that no set \mathcal{S} exists. One might resolve the barber paradox by concluding that no such barber exists. How does Sancho resolve what I’ll dub Panza’s paradox?

He initially decrees, “‘let the part of the man that swore the truth pass freely, and hang the part that told a lie.’”1 The petitioner protests: Dividing the smart aleck will kill him. But the law suggests that the smart aleck should live.

Sancho revises his decree:

“since the reasons for condemning him or sparing him are balanced perfectly, they should let him pass freely, for doing good is always more praiseworthy than doing evil, and I’d sign this with my own name if I knew how to write, and in this case I haven’t said my own idea but a precept that came to mind, one of many that was given to me by my master, Don Quixote [ . . . ] when the law is in doubt, I should favor and embrace mercy.”

One can resolve the barber’s paradox by concluding that such a barber cannot exist. Sancho resolves Panza’s paradox by concluding that the landowner’s law cannot govern all bridge-crossings. The law lacks what computer scientists would call an “edge case.” An edge case falls outside the jurisdiction of the most-often-used part of a rule. One must specify explicitly how to process edge cases, when writing computer programs. Sancho codes the edge case, supplementing the law.

Bridge

Upon starting to read about Sancho’s trial, I sat bolt upright. I ran to my laptop upon finishing. Miguel de Cervantes had intuited, during the 1600s, a paradox not articulated by mathematicians until the 1900s. Surely, the literati had pounced on Cervantes’s foresight?

Mathematics writer Martin Gardner had. I found also two relevant slides in a Powerpoint and three relevant pages in an article. But more critics classified Panza’s paradox as an incarnation of the liar’s paradox than invoked Russell.

Scholars have credited Cervantes with anticipating, and initiating, literary waves that have propagated for four centuries. Perhaps we should credit him with anticipating mathematics not formalized for three.

 

1This decree evokes the story of King Solomon and the baby. Two women bring a baby to King Solomon. Each woman claims the baby as hers. “Cut the baby in two,” Solomon rules, “and give half to each woman.” One woman assents. The other cries, “No! Let her have the child; just don’t kill the baby.” The baby, Solomon discerns, belongs to the second woman. A mother would rather see her child in someone else’s home than see her child killed. Sancho, like Solomon, rules that someone be divided in two. But Sancho, lacking Solomon’s wisdom, misapplies Solomon’s tactic.