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 Information: Episode II: The Tools’ Applications

Monday dawns. Headlines report that “Star Wars: Episode VII” has earned more money, during its opening weekend, than I hope to earn in my lifetime. Trading the newspaper for my laptop, I learn that a friend has discovered ThinkGeek’s BB-8 plushie. “I want one!” she exclaims in a Facebook post. “Because BB-8 definitely needs to be hugged.”

BB-8 plays sidekick to Star Wars hero Poe Dameron. The droid has a spherical body covered with metallic panels and lights.Mr. Gadget and Frosty the Snowman could have spawned such offspring. BB-8 has captured viewers’ hearts, and its chirps have captured cell-phone ringtones.

BB-8

ThinkGeek’s BB-8 plushie

Still, I scratch my head over my friend’s Facebook post. Hugged? Why would she hug…

Oh. Oops.

I’ve mentally verbalized “BB-8” as “BB84.” BB84 denotes an application of quantum theory to cryptography. Cryptographers safeguard information from eavesdroppers and tampering. I’ve been thinking that my friend wants to hug a safety protocol.

Charles Bennett and Gilles Brassard invented BB84 in 1984. Imagine wanting to tell someone a secret. Suppose I wish to coordinate, with a classmate, the purchase of a BB-8 plushie for our friend the droid-hugger. Suppose that the classmate and I can communicate only via a public channel on which the droid-hugger eavesdrops.

Cryptographers advise me to send my classmate a key. A key is a random string of letters, such as CCCAAACCABACA. I’ll encode my message with the string, with which my classmate will decode the message.

Key 2

I have to transmit the key via the public channel. But the droid-hugger eavesdrops on the public channel. Haven’t we taken one step forward and one step back? Why would the key secure our information?

Because quantum-information science enables me to to transmit the key without the droid-hugger’s obtaining it. I won’t transmit random letters; I’ll transmit quantum states. That is, I’ll transmit physical systems, such as photons (particles of light), whose properties encode quantum information.

A nonquantum letter has a value, such as A or B or C.  Each letter has one and only one value, regardless of whether anyone knows what value the letter has. You can learn the value by measuring (looking at) the letter. We can’t necessarily associate such a value with a quantum state. Imagine my classmate measuring a state I send. Which value the measurement device outputs depends on chance and on how my classmate looks at the state.

If the droid-hugger intercepts and measures the state, she’ll change it. My classmate and I will notice such changes. We’ll scrap our key and repeat the BB84 protocol until the droid-hugger quits eavesdropping.

BB84 launched quantum cryptography, the safeguarding of information with quantum physics. Today’s quantum cryptographers rely on BB84 as you rely, when planning a holiday feast, on a carrot-cake recipe that passed your brother’s taste test on his birthday. Quantum cryptographers construct protocols dependent on lines like “The message sender and receiver are assumed to share a key distributed, e.g., via the BB84 protocol.”

BB84 has become a primitive task, a solved problem whose results we invoke in more-complicated problems. Other quantum-information primitives include (warning: jargon ahead) entanglement distillation, entanglement dilution, quantum data compression, and quantum-state merging. Quantum-information scientists solved many primitive problems during the 1990s and early 2000s. You can apply those primitives, even if you’ve forgotten how to prove them.

Caveman

A primitive task, like quantum-entanglement distillation

Those primitives appear to darken quantum information’s horizons. The spring before I started my PhD, an older physicist asked me why I was specializing in quantum information theory. Haven’t all the problems been solved? he asked. Isn’t quantum information theory “dead”?

Imagine discovering how to power plasma blades with kyber crystals. Would you declare, “Problem solved” and relegate your blades to the attic? Or would you apply your tool to defending freedom?

Saber + what to - small

Primitive quantum-information tools are unknotting problems throughout physics—in computer science; chemistry; optics (the study of light); thermodynamics (the study of work, heat, and efficiency); and string theory. My advisor has tracked how uses of “entanglement,” a quantum-information term, have swelled in high-energy-physics papers.

A colleague of that older physicist views quantum information theory as a toolkit, a perspective, a lens through which to view science. During the 1700s, the calculus invented by Isaac Newton and Gottfried Leibniz revolutionized physics. Emmy Noether (1882—1935) recast physics in terms of symmetries and conservation laws. (If the forces acting on a system don’t change in time, for example, the system doesn’t gain or lose energy. A constant force is invariant under, or symmetric with respect to, the progression of time. This symmetry implies that the system’s energy is conserved.) We can cast physics instead (jargon ahead) in terms of the minimization of a free energy or an action.

Quantum information theory, this physicist predicted, will revolutionize physics as calculus, symmetries, conservation, and free energy have. Quantum-information tools such as entropies, entanglement, and qubits will bleed into subfields of physics as Lucasfilm has bled into the fanfiction, LEGO, and Halloween-costume markets.

BB84, and the solution of other primitives, have not killed quantum information. They’ve empowered it to spread—thankfully, to this early-career quantum information scientist. Never mind BB-8; I’d rather hug BB84. Perhaps I shall. Engineers have realized technologies that debuted on Star Trek; quantum mechanics has secured key sharing; bakers have crafted cakes shaped like the Internet; and a droid’s popularity rivals R2D2’s. Maybe next Monday will bring a BB84 plushie.

Plushie

The author hugging the BB84 paper and a plushie. On my wish list: a combination of the two.

Discourse in Delft

A camel strolled past, yards from our window in the Applied-Sciences Building.

I hadn’t expected to see camels at TU Delft, aka the Delft University of Technology, in Holland. I breathed, “Oh!” and turned to watch until the camel followed its turbaned leader out of sight. Nelly Ng, the PhD student with whom I was talking, followed my gaze and laughed.

Nelly works in Stephanie Wehner’s research group. Stephanie—a quantum cryptographer, information theorist, thermodynamicist, and former Caltech postdoc—was kind enough to host me for half August. I arrived at the same time as TU Delft’s first-year undergrads. My visit coincided with their orientation. The orientation involved coffee hours, team-building exercises, and clogging the cafeteria whenever the Wehner group wanted lunch.

And, as far as I could tell, a camel.

Not even a camel could unseat Nelly’s and my conversation. Nelly, postdoc Mischa Woods, and Stephanie are the Wehner-group members who study quantum and small-scale thermodynamics. I study quantum and small-scale thermodynamics, as Quantum Frontiers stalwarts might have tired of hearing. The four of us exchanged perspectives on our field.

Mischa knew more than Nelly and I about clocks; Nelly knew more about catalysis; and I knew more about fluctuation relations. We’d read different papers. We’d proved different theorems. We explained the same phenomena differently. Nelly and I—with Mischa and Stephanie, when they could join us—questioned and answered each other almost perpetually, those two weeks.

We talked in our offices, over lunch, in the group discussion room, and over tea at TU Delft’s Quantum Café. We emailed. We talked while walking. We talked while waiting for Stephanie to arrive so that she could talk with us.

IMG_0125

The site of many a tête-à-tête.

The copiousness of the conversation drained me. I’m an introvert, formerly “the quiet kid” in elementary school. Early some mornings in Delft, I barricaded myself in the visitors’ office. Late some nights, I retreated to my hotel room or to a canal bank. I’d exhausted my supply of communication; I had no more words for anyone. Which troubled me, because I had to finish a paper. But I regret not one discussion, for three reasons.

First, we relished our chats. We laughed together, poked fun at ourselves, commiserated about calculations, and confided about what we didn’t understand.

We helped each other understand, second. As I listened to Mischa or as I revised notes about a meeting, a camel would stroll past a window in my understanding. I’d see what I hadn’t seen before. Mischa might be explaining which quantum states represent high-quality clocks. Nelly might be explaining how a quantum state ξ can enable a state ρ to transform into a state σ. I’d breathe, “Oh!” and watch the mental camel follow my interlocutor through my comprehension.

Nelly’s, Mischa’s, and Stephanie’s names appear in the acknowledgements of the paper I’d worried about finishing. The paper benefited from their explanations and feedback.

Third, I left Delft with more friends than I’d had upon arriving. Nelly, Mischa, and I grew to know each other, to trust each other, to enjoy each other’s company. At the end of my first week, Nelly invited Mischa and me to her apartment for dinner. She provided pasta; I brought apples; and Mischa brought a sweet granola-and-seed mixture. We tasted and enjoyed more than we would have separately.

IMG_0050

Dinner with Nelly and Mischa.

I’ve written about how Facebook has enhanced my understanding of, and participation in, science. Research involves communication. Communication can challenge us, especially many of us drawn to science. Let’s shoulder past the barrier. Interlocutors point out camels—and hot-air balloons, and lemmas and theorems, and other sources of information and delight—that I wouldn’t spot alone.

With gratitude to Stephanie, Nelly, Mischa, the rest of the Wehner group (with whom I enjoyed talking), QuTech and TU Delft.

During my visit, Stephanie and Delft colleagues unveiled the “first loophole-free Bell test.” Their paper sent shockwaves (AKA camels) throughout the quantum community. Scott Aaronson explains the experiment here.

“Experimenting” with women-in-STEM stereotypes

When signing up for physics grad school, I didn’t expect to be interviewed by a comedienne on a spoof science show about women in STEM.

Last May, I received an email entitled “Amy Poehler’s Smart Girls.” The actress, I read, had co-founded the Smart Girls organization to promote confidence and creativity in preteens and teens. Smart Girls was creating a webseries hosted by Megan Amram, author of Science…for Her! The book parodies women’s magazines and ridicules stereotypes of women as unsuited for science.

Megan would host the webseries, “Experimenting with Megan,” in character as an airhead. She planned to interview “kick-ass lady scientists/professors/doctors” in a parody of a talk show. Would I, the email asked, participate?

I’m such a straitlaced fogey, I never say “kick-ass.” I’m such a workaholic, I don’t watch webshows. I’ve not seen Parks and Recreation, the TV series that starred Amy Poehler and for which Megan wrote. The Hollywood bug hasn’t bitten me, though I live 30 minutes from Studio City.

But I found myself in a studio the next month. Young men and women typed on laptops and chattered in the airy, bright waiting lounge. Beyond a doorway lay the set, enclosed by fabric-covered walls that prevented sounds from echoing. Script-filled binders passed from hand to hand, while makeup artists, cameramen, and gophers scurried about.

APSG-Nicole-Yunger-Halpern1

Being interviewed on “Experimenting with Megan.”

Disney’s Mouseketeers couldn’t have exuded more enthusiasm or friendliness than the “Experimenting” team. “Can I bring you a bottle of water?” team members kept asking me and each other. “Would you like a chair?” The other women who interviewed that day—two biologist postdocs—welcomed me into their powwow. Each of us, we learned, is outnumbered by men at work. None of us wears a lab coat, despite stereotypes of scientists as white-coated. Each pours herself into her work: One postdoc was editing a grant proposal while off-set.

I watched one interview, in which Megan asked why biologists study fruit flies instead of “cuter” test subjects. Then I stepped on-set beside her. I perched on an armchair that threatened to swallow my 5’ 3.5” self.* Textbooks, chemistry flasks, and high-heeled pumps stood on the bookshelves behind Megan.

The room quieted. A clapperboard clapped: “Take one.” Megan thanked me for coming, then launched into questions.

Megan hadn’t warned me what she’d ask. We began with “Do you like me?” and “What is the ‘information’ [in ‘quantum information theory’], and do you ever say, ‘Too much information’?” Each question rode hot on the heels of the last. The barrage reminded me of interviews for not-necessarily-scientific scholarships. Advice offered by one scholarship-committee member, the year before I came to Caltech, came to mind: Let loose. Act like an athlete tearing down the field, the opposing team’s colors at the edges of your vision. Savor the challenge.

I savored it. I’d received instructions to play the straight man, answering Megan’s absurdity with science. To “Too much information?” I parried that we can never know enough. When I mentioned that quantum mechanics describes electrons, Megan asked about the electricity she feels upon seeing Chris Hemsworth. (I hadn’t heard of Chris Hemsworth. After watching the interview online, a friend reported that she’d enjoyed the reference to Thor. “What reference to Thor?” I asked. Hemsworth, she informed me, plays the title character.) I dodged Chris Hemsworth; caught “electricity”; and stretched to superconductors, quantum devices whose charges can flow forever.

Academic seminars conclude with question-and-answer sessions. If only those Q&As zinged with as much freshness and flexibility as Megan’s.

The “Experimenting” approach to stereotype-blasting diverges from mine. High-heeled pumps, I mentioned, decorated the set. The “Experimenting” team was parodying the stereotype of women as shoe-crazed. “Look at this stereotype!” the set shouts. “Isn’t it ridiculous?”

As a woman who detests high heels and shoe shopping, I prefer to starve the stereotype of justification. I’ve preferred reading to shopping since before middle school, when classmates began frequenting malls. I feel more comfortable demonstrating, through silence, how little shoes interest me. I’d rather offer no reason for anyone to associate me with shoes.**

I scarcely believe that I appear just after a “sexy science” tagline and a hot-or-not quiz. Before my interview on her quantum episode, Megan discussed the relationship between atoms and Adams. Three guests helped her, three Hollywood personalities named “Adam.”*** Megan held up cartoons of atoms, and photos of Adams, and asked her guests to rate their hotness. I couldn’t have played Megan’s role, couldn’t imagine myself in her (high-heeled) shoes.

But I respect the “Experimenting” style. Megan’s character serves as a foil for the interviewee I watched. Megan’s ridiculousness underscored the postdoc’s professionalism and expertise.

According to online enthusiasm, “Experimenting” humor resonates with many viewers. So diverse is the community that needs introducing to STEM, diverse senses of humor have roles to play. So deep run STEM’s social challenges, multiple angles need attacking.

Just as diverse perspectives can benefit women-in-STEM efforts, so can diverse perspectives benefit STEM. Which is why STEM needs women, Adams, shoe-lovers, shoe-haters…and experimentation.

With gratitude to the “Experimenting” team for the opportunity to contribute to its cause. The live-action interview appears here (beginning at 2:42), and a follow-up personality quiz appears here.

*If you’re 5′ 3.5″, every half-inch matters.

**Except when I blog about how little I wish to associate with shoes.

***Megan introduced her guests as “Adam Shankman, Adam Pally, and an intern that we made legally change his name to Adam to be on the show.” The “intern” is Adam Rymer, president of Legendary Digital Networks. Legendary owns Amy Poehler’s Smart Girls.

Toward physical realizations of thermodynamic resource theories

“This is your arch-nemesis.”

The thank-you slide of my presentation remained onscreen, and the question-and-answer session had begun. I was presenting a seminar about thermodynamic resource theories (TRTs), models developed by quantum-information theorists for small-scale exchanges of heat and work. The audience consisted of condensed-matter physicists who studied graphene and photonic crystals. I was beginning to regret my topic’s abstractness.

The question-asker pointed at a listener.

“This is an experimentalist,” he continued, “your arch-nemesis. What implications does your theory have for his lab? Does it have any? Why should he care?”

I could have answered better. I apologized that quantum-information theorists, reared on the rarefied air of Dirac bras and kets, had developed TRTs. I recalled the baby steps with which science sometimes migrates from theory to experiment. I could have advocated for bounding, with idealizations, efficiencies achievable in labs. I should have invoked the connections being developed with fluctuation results, statistical mechanical theorems that have withstood experimental tests.

The crowd looked unconvinced, but I scored one point: The experimentalist was not my arch-nemesis.

“My new friend,” I corrected the questioner.

His question has burned in my mind for two years. Experiments have inspired, but not guided, TRTs. TRTs have yet to drive experiments. Can we strengthen the connection between TRTs and the natural world? If so, what tools must resource theorists develop to predict outcomes of experiments? If not, are resource theorists doing physics?

http://everystevejobsvideo.com/steve-jobs-qa-session-excerpt-following-antennagate-2010/

A Q&A more successful than mine.

I explore answers to these questions in a paper released today. Ian Durham and Dean Rickles were kind enough to request a contribution for a book of conference proceedings. The conference, “Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge” took place at the University of Cambridge (including a graveyard thereof), thanks to FQXi (the Foundational Questions Institute).

What, I asked my advisor, does one write for conference proceedings?

“Proceedings are a great opportunity to get something off your chest,” John said.

That seminar Q&A had sat on my chest, like a pet cat who half-smothers you while you’re sleeping, for two years. Theorists often justify TRTs with experiments.* Experimentalists, an argument goes, are probing limits of physics. Conventional statistical mechanics describe these regimes poorly. To understand these experiments, and to apply them to technologies, we must explore TRTs.

Does that argument not merit testing? If experimentalists observe the extremes predicted with TRTs, then the justifications for, and the timeliness of, TRT research will grow.

http://maryqin.com/wp-content/uploads/2014/05/

Something to get off your chest. Like the contents of a conference-proceedings paper, according to my advisor.

You’ve read the paper’s introduction, the first eight paragraphs of this blog post. (Who wouldn’t want to begin a paper with a mortifying anecdote?) Later in the paper, I introduce TRTs and their role in one-shot statistical mechanics, the analysis of work, heat, and entropies on small scales. I discuss whether TRTs can be realized and whether physicists should care. I identify eleven opportunities for shifting TRTs toward experiments. Three opportunities concern what merits realizing and how, in principle, we can realize it. Six adjustments to TRTs could improve TRTs’ realism. Two more-out-there opportunities, though less critical to realizations, could diversify the platforms with which we might realize TRTs.

One opportunity is the physical realization of thermal embezzlement. TRTs, like thermodynamic laws, dictate how systems can and cannot evolve. Suppose that a state R cannot transform into a state S: R \not\mapsto S. An ancilla C, called a catalyst, might facilitate the transformation: R + C \mapsto S + C. Catalysts act like engines used to extract work from a pair of heat baths.

Engines degrade, so a realistic transformation might yield S + \tilde{C}, wherein \tilde{C} resembles C. For certain definitions of “resembles,”** TRTs imply, one can extract arbitrary amounts of work by negligibly degrading C. Detecting the degradation—the work extraction’s cost—is difficult. Extracting arbitrary amounts of work at a difficult-to-detect cost contradicts the spirit of thermodynamic law.

The spirit, not the letter. Embezzlement seems physically realizable, in principle. Detecting embezzlement could push experimentalists’ abilities to distinguish between close-together states C and \tilde{C}. I hope that that challenge, and the chance to violate the spirit of thermodynamic law, attracts researchers. Alternatively, theorists could redefine “resembles” so that C doesn’t rub the law the wrong way.

http://www.eoht.info/page/Laws+of+thermodynamics+(game+version)

The paper’s broadness evokes a caveat of Arthur Eddington’s. In 1927, Eddington presented Gifford Lectures entitled The Nature of the Physical World. Being a physicist, he admitted, “I have much to fear from the expert philosophical critic.” Specializing in TRTs, I have much to fear from the expert experimental critic. The paper is intended to point out, and to initiate responses to, the lack of physical realizations of TRTs. Some concerns are practical; some, philosophical. I expect and hope that the discussion will continue…preferably with more cooperation and charity than during that Q&A.

If you want to continue the discussion, drop me a line.

*So do theorists-in-training. I have.

**A definition that involves the trace distance.

Bits, bears, and beyond in Banff

Another conference about entropy. Another graveyard.

Last year, I blogged about the University of Cambridge cemetery visited by participants in the conference “Eddington and Wheeler: Information and Interaction.” We’d lectured each other about entropy–a quantification of decay, of the march of time. Then we marched to an overgrown graveyard, where scientists who’d lectured about entropy decades earlier were decaying.

This July, I attended the conference “Beyond i.i.d. in information theory.” The acronym “i.i.d.” stands for “independent and identically distributed,” which requires its own explanation. The conference took place at BIRS, the Banff International Research Station, in Canada. Locals pronounce “BIRS” as “burrs,” the spiky plant bits that stick to your socks when you hike. (I had thought that one pronounces “BIRS” as “beers,” over which participants in quantum conferences debate about the Measurement Problem.) Conversations at “Beyond i.i.d.” dinner tables ranged from mathematical identities to the hiking for which most tourists visit Banff to the bears we’d been advised to avoid while hiking. So let me explain the meaning of “i.i.d.” in terms of bear attacks.

BIRS

The BIRS conference center. Beyond here, there be bears.

Suppose that, every day, exactly one bear attacks you as you hike in Banff. Every day, you have a probability p1 of facing down a black bear, a probability p2 of facing down a grizzly, and so on. These probabilities form a distribution {pi} over the set of possible events (of possible attacks). We call the type of attack that occurs on a given day a random variable. The distribution associated with each day equals the distribution associated with each other day. Hence the variables are identically distributed. The Monday distribution doesn’t affect the Tuesday distribution and so on, so the distributions are independent.

Information theorists quantify efficiencies with which i.i.d. tasks can be performed. Suppose that your mother expresses concern about your hiking. She asks you to report which bear harassed you on which day. You compress your report into the fewest possible bits, or units of information. Consider the limit as the number of days approaches infinity, called the asymptotic limit. The number of bits required per day approaches a function, called the Shannon entropy HS, of the distribution:

Number of bits required per day → HS({pi}).

The Shannon entropy describes many asymptotic properties of i.i.d. variables. Similarly, the von Neumann entropy HvN describes many asymptotic properties of i.i.d. quantum states.

But you don’t hike for infinitely many days. The rate of black-bear attacks ebbs and flows. If you stumbled into grizzly land on Friday, you’ll probably avoid it, and have a lower grizzly-attack probability, on Saturday. Into how few bits can you compress a set of nonasymptotic, non-i.i.d. variables?

We answer such questions in terms of ɛ-smooth α-Rényi entropies, the sandwiched Rényi relative entropy, the hypothesis-testing entropy, and related beasts. These beasts form a zoo diagrammed by conference participant Philippe Faist. I wish I had his diagram on a placemat.

Entropy zoo

“Beyond i.i.d.” participants define these entropies, generalize the entropies, probe the entropies’ properties, and apply the entropies to physics. Want to quantify the efficiency with which you can perform an information-processing task or a thermodynamic task? An entropy might hold the key.

Many highlights distinguished the conference; I’ll mention a handful.  If the jargon upsets your stomach, skip three paragraphs to Thermodynamic Thursday.

Aram Harrow introduced a resource theory that resembles entanglement theory but whose agents pay to communicate classically. Why, I interrupted him, define such a theory? The backstory involves a wager against quantum-information pioneer Charlie Bennett (more precisely, against an opinion of Bennett’s). For details, and for a quantum version of The Princess and the Pea, watch Aram’s talk.

Graeme Smith and colleagues “remove[d] the . . . creativity” from proofs that certain entropic quantities satisfy subadditivity. Subadditivity is a property that facilitates proofs and that offers physical insights into applications. Graeme & co. designed an algorithm for checking whether entropic quantity Q satisfies subadditivity. Just add water; no innovation required. How appropriate, conference co-organizer Mark Wilde observed. BIRS has the slogan “Inspiring creativity.”

Patrick Hayden applied one-shot entropies to AdS/CFT and emergent spacetime, enthused about elsewhere on this blog. Debbie Leung discussed approximations to Haar-random unitaries. Gilad Gour compared resource theories.

Presentation

Conference participants graciously tolerated my talk about thermodynamic resource theories. I closed my eyes to symbolize the ignorance quantified by entropy. Not really; the photo didn’t turn out as well as hoped, despite the photographer’s goodwill. But I could have closed my eyes to symbolize entropic ignorance.

Thermodynamics and resource theories dominated Thursday. Thermodynamics is the physics of heat, work, entropy, and stasis. Resource theories are simple models for transformations, like from a charged battery and a Tesla car at the bottom of a hill to an empty battery and a Tesla atop a hill.

John

My advisor’s Tesla. No wonder I study thermodynamic resource theories.

Philippe Faist, diagrammer of the Entropy Zoo, compared two models for thermodynamic operations. I introduced a generalization of resource theories for thermodynamics. Last year, Joe Renes of ETH and I broadened thermo resource theories to model exchanges of not only heat, but also particles, angular momentum, and other quantities. We calculated work in terms of the hypothesis-testing entropy. Though our generalization won’t surprise Quantum Frontiers diehards, the magic tricks in my presentation might.

At twilight on Thermodynamic Thursday, I meandered down the mountain from the conference center. Entropies hummed in my mind like the mosquitoes I slapped from my calves. Rising from scratching a bite, I confronted the Banff Cemetery. Half-wild greenery framed the headstones that bordered the gravel path I was following. Thermodynamicists have associated entropy with the passage of time, with deterioration, with a fate we can’t escape. I seem unable to escape from brushing past cemeteries at entropy conferences.

Not that I mind, I thought while scratching the bite in Pasadena. At least I escaped attacks by Banff’s bears.

Cemetery

With thanks to the conference organizers and to BIRS for the opportunity to participate in “Beyond i.i.d. 2015.”

Of Supersoakers and squeezed states

“BBs,” the lecturer said. I was sitting in the center of my row of seats, the two yards between me and the whiteboard empty. But I fancied I hadn’t heard correctly. “You know, like in BB guns?”

I had heard correctly. I nodded.

“Did you play with BB guns when you were a kid?”

I nodded again.

“I had BB guns,” the lecturer ruminated. “I had to defend myself from my brothers.”

I nodded more vigorously. My brother and I love each other, but we’ve crossed toy pistols.

Photons are like BBs, like bullets.”

Light, the lecturer continued, behaves like BBs under certain conditions. Under other conditions, light behaves differently. Different behaviors correspond to different species of light. Some species, we can approximate with classical (nonquantum*) physics. Some species, we can’t.

Kids begged less for BB guns, in my experience, than for water guns. I grew up in Florida, where swimming season stretches from April till September. To reload a BB gun, you have to fetch spent BBs. But, toting a Supersoaker, you swim in ammunition.

Water guns brought to mind water waves, which resemble a species of classical light. If BBs resemble photons, I mused, what about Supersoaker sprays? Water balloons?

I resolved to draw as many parallels as I could between species of light and childhood weapons.

Under scrutiny, the Supersoaker analogy held little water (sorry). A Supersoaker releases water in a stream, rather than in a coherent wave. By coherent, I mean that the wave has a well-defined wavelength: The distance from the first crest to the second equals the distance from the second to the third, and so on. I can’t even identify crests in the Supersoaker photo below.

http://facstaff.gpc.edu/~pgore/PhysicalScience/Waves.html, http://www.mlive.com/living/grand-rapids/index.ssf/2009/07/happy_birthday_super_soaker_mo.html

Coherent waves vs. Supersoaker not-really-waves

Maybe Supersoaker sprays resemble incoherent light? Incoherent light is a mixture of waves of all different wavelengths. Classical physics approximates incoherent light, examples of which include sunlight. If you tease apart sunlight into coherent components, you’ll find waves with short wavelengths (such as ultraviolet rays), waves with medium (such as light we can see), and waves with long (such as microwaves). You can’t ascribe just one wavelength to incoherent light, just as I seemed unable to ascribe a wavelength to Supersoaker sprays.

But Supersoaker sprays differ from incoherent light in other respects. I’d expect triggers, for instance, to introduce nonlinearity into the spray’s dynamics. Readers who know more than I about fluid mechanics can correct me.

http://www.parentdish.com/2010/07/14/water-balloon-volley-game/

Though far-reaching and forceful, Supersoakers weigh down combatants and are difficult to hide. If you need ammunition small enough for a sneak attack, I recommend water balloons. Water balloons resemble squeezed states, which form a quantum class of light related to the Uncertainty Principle.

Werner Heisenberg proposed that, the more you know about a quantum particle’s position, the less you can know about its momentum, and vice versa. Let’s represent your uncertainty about the position by Δx and your uncertainty about the momentum by Δp. The product of these uncertainties can’t dip below some number, represented by ћ/2:

\Delta_x \Delta_p \geq \frac{\hbar}{2}.

Neither uncertainty, for example, can equal zero. Heisenberg’s proposal has evolved into more rigorous, more general forms. But the story remains familiar: The lesser the “spread in the possible values” of some property (like position), the greater the “spread in the possible values” of another property (like momentum).

Imagine plotting the possible positions along a graph’s horizontal axis and the possible momenta along the vertical. The points that could characterize our quantum system form a blob of area ћ/2. Doesn’t the blob resemble a water balloon?

Imagine squeezing a water balloon along one direction. The balloon bulges out along another. Now, imagine squeezing most of the quantum uncertainty along one direction in the diagram. You’ve depicted a squeezed state.

http://www.rp-photonics.com/squeezed_states_of_light.html

Depiction of a squeezed state

Not all childhood weapons contain water or BBs, and not all states of light contain photons.** A vacuum is a state that consists of zero photons. Classical physics suggests that the vacuum is empty and lacks energy. A sliver of energy, called zero-point energy, pervades each quantum vacuum. The Uncertainty Principle offers one reason why.

The vacuum reminds me of the silent treatment. Silence sounds empty, but it can harbor malevolence as quantum vacua harbor energy. Middle-school outcasts beware zero-point malice.

Retreating up Memory Lane, I ran out of analogies between classes of light and childhood weapons. Children play with lasers (with laser pointers and laser-tag guns), and lasers emit (approximately) coherent light. But laser light’s resemblance to laser light doesn’t count as an analogy. The class of incoherent light includes thermal states. (Non-experts, I’m about to spew jargon. If you have the energy, I recommend Googling the italicized terms. If you haven’t, feel free to skip to the next paragraph.) Physicists model much of the natural world with thermal states. To whichever readers identify childhood weapons that resemble them, I offer ten points. I offer 20 for mimicry of solitons or solitary waves, and 25 for that of parametric down-conversion or photon antibunching.

But if sunshine and Supersoakers lure you away from your laptop, I can’t object. Happy summer.

With thanks to Bassam Helou for corrections and discussions.

*Pardon my simplifying inaccuracy. Some nonquantum physics is nonclassical.

**More precisely, not all Fock states correspond to particle numbers n > 0. Alternatively: Not all states of light correspond to positive expectation values \langle \hat{n} \rangle > 0 of the particle-number operator \hat{n}.

Hello, my name is QUANTUM MASTER EQUATION

“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.”

https://en.wikipedia.org/wiki/Name_tag

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.

http://thewatchers.adorraeli.com/2012/04/18/the-santorini-volcano-caldera-is-awake-again-and-rapidly-deforming/

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.

http://www.savingyoudinero.com/2012/06/05/chicken-fajita-casserole/

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?

http://www.wallcoo.net/movie/master_and_commander/html/image4.html

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.

http://fashions-cloud.com/pages/y/young-frankenstein-igor-brain/

*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.