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.

Hamiltonian: An American Musical (without Americana or music)

Author’s note: I intended to post this article three months ago. Other developments delayed the release. Thanks in advance for pardoning the untimeliness.

Critics are raving about it. Barak Obama gave a speech about it. It’s propelled two books onto bestseller lists. Committees have showered more awards on it than clouds have showered rain on California this past decade.

What is it? The Hamiltonian, represented by \hat{H}. It’s an operator (a mathematical object) that basically represents a system’s energy. Hamiltonians characterize systems classical and quantum, from a brick in a Broadway theater to the photons that form a spotlight. \hat{H} determines how a system evolves, or changes in time.


I lied: Obama didn’t give a speech about the Hamiltonian. He gave a speech about Hamilton. Hamilton: An American Musical spotlights 18th-century revolutionary Alexander Hamilton. Hamilton conceived the United States’s national bank. He nurtured the economy as our first Secretary of the Treasury. The year after Alexander Hamilton died, William Rowan Hamilton was born. Rowan Hamilton conceived four-dimensional numbers called quaternions. He nurtured the style of physics, Hamiltonian mechanics, used to model quantum systems today.


Hamilton has enchanted audiences and critics. Ticket sell out despite costing over $1,000. Tonys, Grammys, and Pulitzers have piled up. Lawmakers, one newspaper reported, ridicule colleagues who haven’t seen the show. One political staff member confessed that “dodging ‘Hamilton’ barbs has affected her work—so much so that she hasn’t returned certain phone calls ‘because I couldn’t handle the anxiety’ of being harangued for her continued failure to see the show.”

Musical-theater fans across the country are applauding Alexander. Hamilton forbid that William Rowan should envy him. Let’s celebrate Hamiltonians.


I’ve been pondering the Hamiltonian


It describes a chain of L sites. L ranges from 10 to 30 in most computer simulations. The cast consists of quantum particles. Each site houses one particle or none. \hat{n}_j represents the number of particles at site j. c_j represents the removal of a particle from site j, and c_j^\dag represents the adding of a particle.

The last term in \hat{H} represents the repulsion between particles that border each other. The “nn” in “E_{\rm nn}” stands for “nearest-neighbor.” The J term encodes particles’ hopping between sites. \hat{c}_j^\dag \hat{c}_{j+1} means, “A particle jumps from site j+1 to site j.”

The first term in \hat{H}, we call disorder. Imagine a landscape of random dips and hills. Imagine, for instance, crouching on the dirt and snow in Valley Forge. Boots and hooves have scuffed the ground. Zoom in; crouch lower. Imagine transplanting the row of sites into this landscape. h_j denotes the height of site j.

Say that the dips sink low and the hills rise high. The disorder traps particles like soldiers behind enemy lines. Particles have trouble hopping. We call this system many-body localized.

Imagine flattening the landscape abruptly, as by stamping on the snow. This flattening triggers a phase transition.  Phase transitions are drastic changes, as from colony to country. The flattening frees particles to hop from site to site. The particles spread out, in accordance with the Hamiltonian’s J term. The particles come to obey thermodynamics, a branch of physics that I’ve effused about.

The Hamiltonian encodes repulsion, hopping, localization, thermalization, and more behaviors. A richer biography you’ll not find amongst the Founding Fathers.


As Hamiltonians constrain particles, politics constrain humans. A play has primed politicians to smile upon the name “Hamilton.” Physicists study Hamiltonians and petition politicians for funding. Would politicians fund us more if we emphasized the Hamiltonians in our science?

Gold star for whoever composes the most rousing lyrics about many-body localization. Or, rather, fifty white stars.

The weak shall inherit the quasiprobability.

Justin Dressel’s office could understudy for the archetype of a physicist’s office. A long, rectangular table resembles a lab bench. Atop the table perches a tesla coil. A larger tesla coil perches on Justin’s desk. Rubik’s cubes and other puzzles surround a computer and papers. In front of the desk hangs a whiteboard.

A puzzle filled the whiteboard in August. Justin had written a model for a measurement of a quasiprobability. I introduced quasiprobabilities here last Halloween. Quasiprobabilities are to probabilities as ebooks are to books: Ebooks resemble books but can respond to touchscreen interactions through sounds and animation. Quasiprobabilities resemble probabilities but behave in ways that probabilities don’t.


A tesla coil of Justin Dressel’s


Let p denote the probability that any given physicist keeps a tesla coil in his or her office. p ranges between zero and one. Quasiprobabilities can dip below zero. They can assume nonreal values, dependent on the imaginary number i = \sqrt{-1}. Probabilities describe nonquantum phenomena, like tesla-coil collectors,1 and quantum phenomena, like photons. Quasiprobabilities appear nonclassical.2,3

We can infer the tesla-coil probability by observing many physicists’ offices:

\text{Prob(any given physicist keeps a tesla coil in his/her office)}  =  \frac{ \text{\# physicists who keep tesla coils in their offices} }{ \text{\# physicists} } \, . We can infer quasiprobabilities from weak measurements, Justin explained. You can measure the number of tesla coils in an office by shining light on the office, correlating the light’s state with the tesla-coil number, and capturing the light on photographic paper. The correlation needn’t affect the tesla coils. Observing a quantum state changes the state, by the Uncertainty Principle heralded by Heisenberg.

We could observe a quantum system weakly. We’d correlate our measurement device (the analogue of light) with the quantum state (the analogue of the tesla-coil number) unreliably. Imagining shining a dull light on an office for a brief duration. Shadows would obscure our photo. We’d have trouble inferring the number of tesla coils. But the dull, brief light burst would affect the office less than a strong, long burst would.

Justin explained how to infer a quasiprobability from weak measurements. He’d explained on account of an action that others might regard as weak: I’d asked for help.


Chaos had seized my attention a few weeks earlier. Chaos is a branch of math and physics that involves phenomena we can’t predict, like weather. I had forayed into quantum chaos for reasons I’ll explain in later posts. I was studying a function F(t) that can flag chaos in cold atoms, black holes, and superconductors.

I’d derived a theorem about F(t). The theorem involved a UFO of a mathematical object: a probability amplitude that resembled a probability but could assume nonreal values. I presented the theorem to my research group, which was kind enough to provide feedback.

“Is this amplitude physical?” John Preskill asked. “Can you measure it?”

“I don’t know,” I admitted. “I can tell a story about what it signifies.”

“If you could measure it,” he said, “I might be more excited.”

You needn’t study chaos to predict that private clouds drizzled on me that evening. I was grateful to receive feedback from thinkers I respected, to learn of a weakness in my argument. Still, scientific works are creative works. Creative works carry fragments of their creators. A weakness in my argument felt like a weakness in me. So I took the step that some might regard as weak—by seeking help.



Some problems, one should solve alone. If you wake me at 3 AM and demand that I solve the Schrödinger equation that governs a particle in a box, I should be able to comply (if you comply with my demand for justification for the need to solve the Schrödinger equation at 3 AM).One should struggle far into problems before seeking help.

Some scientists extend this principle into a ban on assistance. Some students avoid asking questions for fear of revealing that they don’t understand. Some boast about passing exams and finishing homework without the need to attend office hours. I call their attitude “scientific machismo.”

I’ve all but lived in office hours. I’ve interrupted lectures with questions every few minutes. I didn’t know if I could measure that probability amplitude. But I knew three people who might know. Twenty-five minutes after I emailed them, Justin replied: “The short answer is yes!”


I visited Justin the following week, at Chapman University’s Institute for Quantum Studies. I sat at his bench-like table, eyeing the nearest tesla coil, as he explained. Justin had recognized my probability amplitude from studies of the Kirkwood-Dirac quasiprobability. Experimentalists infer the Kirkwood-Dirac quasiprobability from weak measurements. We could borrow these experimentalists’ techniques, Justin showed, to measure my probability amplitude.

The borrowing grew into a measurement protocol. The theorem grew into a paper. I plunged into quasiprobabilities and weak measurements, following Justin’s advice. John grew more excited.

The meek might inherit the Earth. But the weak shall measure the quasiprobability.

With gratitude to Justin for sharing his expertise and time; and to Justin, Matt Leifer, and Chapman University’s Institute for Quantum Studies for their hospitality.

Chapman’s community was gracious enough to tolerate a seminar from me about thermal states of quantum systems. You can watch the seminar here.

1Tesla-coil collectors consists of atoms described by quantum theory. But we can describe tesla-coil collectors without quantum theory.

2Readers foreign to quantum theory can interpret “nonclassical” roughly as “quantum.”

3Debate has raged about whether quasiprobabilities govern classical phenomena.

4I should be able also to recite the solutions from memory.

The mechanics of thanksgiving

You and a friend are driving in a car. You’ve almost reached an intersection. The stoplight turns red. 

My teacher had handwritten the narrative on my twelfth-grade physics midterm. Many mechanics problems involve cars: Drivers smash into each other at this angle or that angle, interlocking their vehicles. The Principle of Conservation of Linear Momentum governs how the wreck moves. Students deduce how quickly the wreck skids, and in which direction.

Few mechanics problems involve the second person. I have almost reached an intersection?

You’re late for an event, and stopping would cost you several minutes. What do you do?

We’re probably a few meters from the light, I thought. How quickly are we driving? I could calculate the acceleration needed to—

(a) Speed through the red light.

(b) Hit the brakes. Fume about missing the light while you wait.

(c) Stop in front of the intersection. Chat with your friend, making the most of the situation. Resolve to leave your house earlier next time.

Pencils scritched, and students shifted in their chairs. I looked up from the choices.


Our classroom differed from most high-school physics classrooms. Sure, posters about Einstein and Nobel prizes decorated the walls. Circuit elements congregated in a corner. But they didn’t draw the eye the moment one stepped through the doorway.

A giant yellow smiley face did.

It sat atop the cupboards that faced the door. Next to the smiley stood a placard that read, “Say please and thank you.” Another placard hung above the chalkboard: “Are you showing your good grace and character?”

Our instructor taught mechanics and electromagnetism. He wanted to teach us more. He pronounced the topic in a southern sing-song: “an attitude of gratitude.”

Teenagers populate high-school classrooms. The cynicism in a roomful of teenagers could have rivaled the cynicism in Hemingway’s Paris. Students regarded his digressions as oddities. My high school fostered more manners than most. But a “Can you believe…?” tone accompanied recountings of the detours.

Yet our teacher’s drawl held steady as he read students’ answers to a bonus question on a test (“What are you grateful for?”). He bade us gaze at a box of Wheaties—the breakfast of champions—on whose front hung a mirror. He awarded Symbolic Lollipops for the top grades on tests and for acts of kindness. All with a straight face.

Except, once or twice over the years, I thought I saw his mouth tweak into a smile.

I’ve puzzled out momentum problems since graduating from that physics class. I haven’t puzzled out how to regard the class. As mawkish or moral? Heroic or humorous? I might never answer those questions. But the class led me toward a career in physics, and physicists value data. One datum stands out: I didn’t pack my senior-year high-school physics midterm when moving to Pasadena. But the midterm remains with me.


Happy Halloween from…the discrete Wigner function?

Do you hope to feel a breath of cold air on the back of your neck this Halloween? I’ve felt one literally: I earned my Masters in the icebox called “Ontario,” at the Perimeter Institute for Theoretical Physics. Perimeter’s colloquia1 take place in an auditorium blacker than a Quentin Tarantino film. Aephraim Steinberg presented a colloquium one air-conditioned May.

Steinberg experiments on ultracold atoms and quantum optics2 at the University of Toronto. He introduced an idea that reminds me of biting into an apple whose coating you’d thought consisted of caramel, then tasting blood: a negative (quasi)probability.

Probabilities usually range from zero upward. Consider Shirley Jackson’s short story The Lottery. Villagers in a 20th-century American village prepare slips of paper. The number of slips equals the number of families in the village. One slip bears a black spot. Each family receives a slip. Each family has a probability p > 0  of receiving the marked slip. What happens to the family that receives the black spot? Read Jackson’s story—if you can stomach more than a Tarantino film.

Jackson peeled off skin to reveal the offal of human nature. Steinberg’s experiments reveal the offal of Nature. I’d expect humaneness of Jackson’s villagers and nonnegativity of probabilities. But what looks like a probability and smells like a probability might be hiding its odor with Special-Edition Autumn-Harvest Febreeze.


A quantum state resembles a set of classical3 probabilities. Consider a classical system that has too many components for us to track them all. Consider, for example, the cold breath on the back of your neck. The breath consists of air molecules at some temperature T. Suppose we measured the molecules’ positions and momenta. We’d have some probability p_1 of finding this particle here with this momentum, that particle there with that momentum, and so on. We’d have a probability p_2 of finding this particle there with that momentum, that particle here with this momentum, and so on. These probabilities form the air’s state.

We can tell a similar story about a quantum system. Consider the quantum light prepared in a Toronto lab. The light has properties analogous to position and momentum. We can represent the light’s state with a mathematical object similar to the air’s probability density.4 But this probability-like object can sink below zero. We call the object a quasiprobability, denoted by \mu.

If a \mu sinks below zero, the quantum state it represents encodes entanglement. Entanglement is a correlation stronger than any achievable with nonquantum systems. Quantum information scientists use entanglement to teleport information, encrypt messages, and probe the nature of space-time. I usually avoid this cliché, but since Halloween is approaching: Einstein called entanglement “spooky action at a distance.”


Eugene Wigner and others defined quasiprobabilities shortly before Shirley Jackson wrote The Lottery. Quantum opticians use these \mu’s, because quantum optics and quasiprobabilities involve continuous variables. Examples of continuous variables include position: An air molecule can sit at this point (e.g., x = 0) or at that point (e.g., x = 1) or anywhere between the two (e.g., x = 0.001). The possible positions form a continuous set. Continuous variables model quantum optics as they model air molecules’ positions.

Information scientists use continuous variables less than we use discrete variables. A discrete variable assumes one of just a few possible values, such as 0 or 1, or trick or treat.


How a quantum-information theorist views Halloween.

Quantum-information scientists study discrete systems, such as electron spins. Can we represent discrete quantum systems with quasiprobabilities \mu as we represent continuous quantum systems? You bet your barmbrack.

Bill Wootters and others have designed quasiprobabilities for discrete systems. Wootters stipulated that his \mu have certain properties. The properties appear in this review.  Most physicists label properties “1,” “2,” etc. or “Prop. 1,” “Prop. 2,” etc. The Wootters properties in this review have labels suited to Halloween.


Seeing (quasi)probabilities sink below zero feels like biting into an apple that you think has a caramel coating, then tasting blood. Did you eat caramel apples around age six? Caramel apples dislodge baby teeth. When baby teeth fall out, so does blood. Tasting blood can mark growth—as does the squeamishness induced by a colloquium that spooks a student. Who needs haunted mansions when you have negative quasiprobabilities?


For nonexperts:

1Weekly research presentations attended by a department.


3Nonquantum (basically).

4Think “set of probabilities.”

Tripping over my own inner product

A scrape stood out on the back of my left hand. The scrape had turned greenish-purple, I noticed while opening the lecture-hall door. I’d jounced the hand against my dining-room table while standing up after breakfast. The table’s corners form ninety-degree angles. The backs of hands do not.

Earlier, when presenting a seminar, I’d forgotten to reference papers by colleagues. Earlier, I’d offended an old friend without knowing how. Some people put their feet in their mouths. I felt liable to swallow a clog.

The lecture was for Ph 219: Quantum ComputationI was TAing (working as a teaching assistant for) the course. John Preskill was discussing quantum error correction.

Computers suffer from errors as humans do: Imagine setting a hard drive on a table. Coffee might spill on the table (as it probably would have if I’d been holding a mug near the table that week). If the table is in my California dining room, an earthquake might judder the table. Juddering bangs the hard drive against the wood, breaking molecular bonds and deforming the hardware. The information stored in computers degrades.

How can we protect information? By encoding it—by translating the message into a longer, encrypted message. An earthquake might judder the encoded message. We can reverse some of the damage by error-correcting.

Different types of math describe different codes. John introduced a type of math called symplectic vector spaces. “Symplectic vector space” sounds to me like a garden of spiny cacti (on which I’d probably have pricked fingers that week). Symplectic vector spaces help us translate between the original and encoded messages.


Symplectic vector space?

Say that an earthquake has juddered our hard drive. We want to assess how the earthquake corrupted the encoded message and to error-correct. Our encryption scheme dictates which operations we should perform. Each possible operation, we represent with a mathematical object called a vector. A vector can take the form of a list of numbers.

We construct the code’s vectors like so. Say that our quantum hard drive consists of seven phosphorus nuclei atop a strip of silicon. Each nucleus has two observables, or measurable properties. Let’s call the observables Z and X.

Suppose that we should measure the first nucleus’s Z. The first number in our symplectic vector is 1. If we shouldn’t measure the first nucleus’s Z, the first number is 0. If we should measure the second nucleus’s Z, the second number is 1; if not, 0; and so on for the other nuclei. We’ve assembled the first seven numbers in our vector. The final seven numbers dictate which nuclei’s Xs we measure. An example vector looks like this: ( 1, \, 0, \, 1, \, 0, \, 1, \, 0, \, 1 \; | \; 0, \, 0, \, 0, \, 0, \, 0, \, 0, \, 0 ).

The vector dictates that we measure four Zs and no Xs.


Symplectic vectors represent the operations we should perform to correct errors.

A vector space is a collection of vectors. Many problems—not only codes—involve vector spaces. Have you used Google Maps? Google illustrates the step that you should take next with an arrow. We can represent that arrow with a vector. A vector, recall, can take the form of a list of numbers. The step’s list of twonumbers indicates whether you should walk ( \text{Northward or not} \; | \; \text{Westward or not} ).


I’d forgotten about my scrape by this point in the lecture. John’s next point wiped even cacti from my mind.

Say you want to know how similar two vectors are. You usually calculate an inner product. A vector v tends to have a large inner product with any vector w that points parallel to v.


Parallel vectors tend to have a large inner product.

The vector v tends to have an inner product of zero with any vector w that points perpendicularly. Such v and w are said to annihilate each other. By the end of a three-hour marathon of a research conversation, we might say that v and w “destroy” each other. v is orthogonal to w.


Two orthogonal vectors, having an inner product of zero, annihilate each other.

You might expect a vector v to have a huge inner product with itself, since v points parallel to v. Quantum-code vectors defy expectations. In a symplectic vector space, John said, “you can be orthogonal to yourself.”

A symplectic vector2 can annihilate itself, destroy itself, stand in its own way. A vector can oppose itself, contradict itself, trip over its own feet. I felt like I was tripping over my feet that week. But I’m human. A vector is a mathematical ideal. If a mathematical ideal could be orthogonal to itself, I could allow myself space to err.


Tripping over my own inner product.

Lloyd Alexander wrote one of my favorite books, the children’s novel The Book of Three. The novel features a stout old farmer called Coll. Coll admonishes an apprentice who’s burned his fingers: “See much, study much, suffer much.” We smart while growing smarter.

An ant-sized scar remains on the back of my left hand. The scar has been fading, or so I like to believe. I embed references to colleagues’ work in seminar Powerpoints, so that I don’t forget to cite anyone. I apologized to the friend, and I know about symplectic vector spaces. We all deserve space to err, provided that we correct ourselves. Here’s to standing up more carefully after breakfast.


1Not that I advocate for limiting each coordinate to one bit in a Google Maps vector. The two-bit assumption simplifies the example.

2Not only symplectic vectors are orthogonal to themselves, John pointed out. Consider a string of bits that contains an even number of ones. Examples include (0, 0, 0, 0, 1, 1). Each such string has a bit-wise inner product, over the field {\mathbb Z}_2, of zero with itself.

Upending my equilibrium

Few settings foster equanimity like Canada’s Banff International Research Station (BIRS). Mountains tower above the center, softened by pines. Mornings have a crispness that would turn air fresheners evergreen with envy. The sky looks designed for a laundry-detergent label.


Doesn’t it?

One day into my visit, equilibrium shattered my equanimity.

I was participating in the conference “Beyond i.i.d. in information theory.” What “beyond i.i.d.” means is explained in these articles.  I was to present about resource theories for thermodynamics. Resource theories are simple models developed in quantum information theory. The original thermodynamic resource theory modeled systems that exchange energy and information.

Imagine a quantum computer built from tiny, cold, quantum circuits. An air particle might bounce off the circuit. The bounce can slow the particle down, transferring energy from particle to circuit. The bounce can entangle the particle with the circuit, transferring quantum information from computer to particle.

Suppose that particles bounced off the computer for ages. The computer would thermalize, or reach equilibrium: The computer’s energy would flatline. The computer would reach a state called the canonical ensemble. The canonical ensemble looks like this:  e^{ - \beta H }  / { Z }.

Joe Renes and I had extended these resource theories. Thermodynamic systems can exchange quantities other than energy and information. Imagine white-bean soup cooling on a stovetop. Gas condenses on the pot’s walls, and liquid evaporates. The soup exchanges not only heat, but also particles, with its environment. Imagine letting the soup cool for ages. It would thermalize to the grand canonical ensemble, e^{ - \beta (H - \mu N) } / { Z }. Joe and I had modeled systems that exchange diverse thermodynamic observables.*

What if, fellow beyond-i.i.d.-er Jonathan Oppenheim asked, those observables didn’t commute with each other?

Mathematical objects called operators represent observables. Let \hat{H} represent a system’s energy, and let \hat{N} represent the number of particles in the system. The operators fail to commute if multiplying them in one order differs from multiplying them in the opposite order: \hat{H}  \hat{N}  \neq  \hat{N}  \hat{H}.

Suppose that our quantum circuit has observables represented by noncommuting operators \hat{H} and \hat{N}. The circuit cannot have a well-defined energy and a well-defined particle number simultaneously. Physicists call this inability the Uncertainty Principle. Uncertainty and noncommutation infuse quantum mechanics as a Cashmere GlowTM infuses a Downy fabric softener.

Downy Cashmere Glow 2

Quantum uncertainty and noncommutation.

I glowed at Jonathan: All the coolness in Canada couldn’t have pleased me more than finding someone interested in that question.** Suppose that a quantum system exchanges observables \hat{Q}_1 and \hat{Q}_2 with the environment. Suppose that \hat{Q}_1 and \hat{Q}_2 don’t commute, like components \hat{S}_x and \hat{S}_y of quantum angular momentum. Would the system thermalize? Would the thermal state have the form e^{ \mu_1 \hat{Q}_1 + \mu_2 \hat{Q}_2 } / { Z }? Could we model the system with a resource theory?

Jonathan proposed that we chat.

The chat sucked in beyond-i.i.d.-ers Philippe Faist and Andreas Winter. We debated strategies while walking to dinner. We exchanged results on the conference building’s veranda. We huddled over a breakfast table after colleagues had pushed their chairs back. Information flowed from chalkboard to notebook; energy flowed in the form of coffee; food particles flowed onto the floor as we brushed crumbs from our notebooks.

Coffee, crumbs

Exchanges of energy and particles.

The idea morphed and split. It crystallized months later. We characterized, in three ways, the thermal state of a quantum system that exchanges noncommuting observables with its environment.

First, we generalized the microcanonical ensemble. The microcanonical ensemble is the thermal state of an isolated system. An isolated system exchanges no observables with any other system. The quantum computer and the air molecules can form an isolated system. So can the white-bean soup and its kitchen. Our quantum system and its environment form an isolated system. But they cannot necessarily occupy a microcanonical ensemble, thanks to noncommutation.

We generalized the microcanonical ensemble. The generalization involves approximation, unlikely measurement outcomes, and error tolerances. The microcanonical ensemble has a simple definition—sharp and clean as Banff air. We relaxed the definition to accommodate noncommutation. If the microcanonical ensemble resembles laundry detergent, our generalization resembles fabric softener.

Detergent vs. softener

Suppose that our system and its environment occupy this approximate microcanonical ensemble. Tracing out (mathematically ignoring) the environment yields the system’s thermal state. The thermal state basically has the form we expected, \gamma = e^{ \sum_j  \mu_j \hat{Q}_j } / { Z }.

This exponential state, we argued, follows also from time evolution. The white-bean soup equilibrates upon exchanging heat and particles with the kitchen air for ages. Our quantum system can exchange observables \hat{Q}_j with its environment for ages. The system equilibrates, we argued, to the state \gamma. The argument relies on a quantum-information tool called canonical typicality.

Third, we defined a resource theory for thermodynamic exchanges of noncommuting observables. In a thermodynamic resource theory, the thermal states are the worthless states: From a thermal state, one can’t extract energy usable to lift a weight or to power a laptop. The worthless states, we showed, have the form of \gamma.

Three path lead to the form \gamma of the thermal state of a quantum system that exchanges noncommuting observables with its environment. We published the results this summer.

Not only was Team Banff spilling coffee over \gamma. So were teams at Imperial College London and the University of Bristol. Our conclusions overlap, suggesting that everyone calculated correctly. Our methodologies differ, generating openings for exploration. The mountain passes between our peaks call out for mapping.

So does the path to physical reality. Do these thermal states form in labs? Could they? Cold atoms offer promise for realizations. In addition to experiments and simulations, master equations merit study. Dynamical typicality, Team Banff argued, suggests that \gamma results from equilibration. Master equations model equilibration. Does some Davies-type master equation have \gamma as its fixed point? Email me if you have leads!


Experimentalists, can you realize the thermal state e^{ \sum_j \mu_j \hat{Q}_j } / Z whose charges \hat{Q}_j don’t commute?


A photo of Banff could illustrate Merriam-Webster’s entry for “equanimity.” Banff equanimity deepened our understanding of quantum equilibrium. But we wouldn’t have understood quantum equilibrium if questions hadn’t shattered our tranquility. Give me the disequilibrium of recognizing problems, I pray, and the equilibrium to solve them.


*By “observable,” I mean “property that you can measure.”

**Teams at Imperial College London and Bristol asked that question, too. More pleasing than three times the coolness in Canada!

Bringing the heat to Cal State LA

John Baez is a tough act to follow.

The mathematical physicist presented a colloquium at Cal State LA this May.1 The talk’s title: “My Favorite Number.” The advertisement image: A purple “24” superimposed atop two egg cartons.


The colloquium concerned string theory. String theorists attempt to reconcile Einstein’s general relativity with quantum mechanics. Relativity concerns the large and the fast, like the sun and light. Quantum mechanics concerns the small, like atoms. Relativity and with quantum mechanics individually suggest that space-time consists of four dimensions: up-down, left-right, forward-backward, and time. String theory suggests that space-time has more than four dimensions. Counting dimensions leads theorists to John Baez’s favorite number.

His topic struck me as bold, simple, and deep. As an otherworldly window onto the pedestrian. John Baez became, when I saw the colloquium ad, a hero of mine.

And a tough act to follow.

I presented Cal State LA’s physics colloquium the week after John Baez. My title: “Quantum steampunk: Quantum information applied to thermodynamics.” Steampunk is a literary, artistic, and film genre. Stories take place during the 1800s—the Victorian era; the Industrial era; an age of soot, grime, innovation, and adventure. Into the 1800s, steampunkers transplant modern and beyond-modern technologies: automata, airships, time machines, etc. Example steampunk works include Will Smith’s 1999 film Wild Wild West. Steampunk weds the new with the old.

So does quantum information applied to thermodynamics. Thermodynamics budded off from the Industrial Revolution: The steam engine crowned industrial technology. Thinkers wondered how efficiently engines could run. Thinkers continue to wonder. But the steam engine no longer crowns technology; quantum physics (with other discoveries) does. Quantum information scientists study the roles of information, measurement, and correlations in heat, energy, entropy, and time. We wed the new with the old.


What image could encapsulate my talk? I couldn’t lean on egg cartons. I proposed a steampunk warrior—cravatted, begoggled, and spouting electricity. The proposal met with a polite cough of an email. Not all department members, Milan Mijic pointed out, had heard of steampunk.

Steampunk warrior

Milan is a Cal State LA professor and my erstwhile host. We toured the palm-speckled campus around colloquium time. What, he asked, can quantum information contribute to thermodynamics?

Heat offers an example. Imagine a classical (nonquantum) system of particles. The particles carry kinetic energy, or energy of motion: They jiggle. Particles that bump into each other can exchange energy. We call that energy heat. Heat vexes engineers, breaking transistors and lowering engines’ efficiencies.

Like heat, work consists of energy. Work has more “orderliness” than the heat transferred by random jiggles. Examples of work exertion include the compression of a gas: A piston forces the particles to move in one direction, in concert. Consider, as another example, driving electrons around a circuit with an electric field. The field forces the electrons to move in the same direction. Work and heat account for all the changes in a system’s energy. So states the First Law of Thermodynamics.

Suppose that the system is quantum. It doesn’t necessarily have a well-defined energy. But we can stick the system in an electric field, and the system can exchange motional-type energy with other systems. How should we define “work” and “heat”?

Quantum information offers insights, such as via entropies. Entropies quantify how “mixed” or “disordered” states are. Disorder grows as heat suffuses a system. Entropies help us extend the First Law to quantum theory.

First slide

So I explained during the colloquium. Rarely have I relished engaging with an audience as much as I relished engaging with Cal State LA’s. Attendees made eye contact, posed questions, commented after the talk, and wrote notes. A student in a corner appeared to be writing homework solutions. But a presenter couldn’t have asked for more from the rest. One exclamation arrested me like a coin in the cogs of a grandfather clock.

I’d peppered my slides with steampunk art: paintings, drawings, stills from movies. The peppering had staved off boredom as I’d created the talk. I hoped that the peppering would stave off my audience’s boredom. I apologized about the trimmings.

“No!” cried a woman near the front. “It’s lovely!”

I was about to discuss experiments by Jukka Pekola’s group. Pekola’s group probes quantum thermodynamics using electronic circuits. The group measures heat by counting the electrons that hop from one part of the circuit to another. Single-electron transistors track tunneling (quantum movements) of single particles.

Heat complicates engineering, calculations, and California living. Heat scrambles signals, breaks devices, and lowers efficiencies. Quantum heat can evade definition. Thermodynamicists grind their teeth over heat.

“No!” the woman near the front had cried. “It’s lovely!”

She was referring to steampunk art. But her exclamation applied to my subject. Heat has not only practical importance, but also fundamental: Heat influences every law of thermodynamics. Thermodynamic law underpins much of physics as 24 underpins much of string theory. Lovely, I thought, indeed.

Cal State LA offered a new view of my subfield, an otherworldly window onto the pedestrian. The more pedestrian an idea—the more often the idea surfaces, the more of our world the idea accounts for—the deeper the physics. Heat seems as pedestrian as a Pokémon Go player. But maybe, someday, I’ll present an idea as simple, bold, and deep as the number 24.


A window onto Cal State LA.

With gratitude to Milan Mijic, and to Cal State LA’s Department of Physics and Astronomy, for their hospitality.

1For nonacademics: A typical physics department hosts several presentations per week. A seminar relates research that the speaker has undertaken. The audience consists of department members who specialize in the speaker’s subfield. A department’s astrophysicists might host a Monday seminar; its quantum theorists, a Wednesday seminar; etc. One colloquium happens per week. Listeners gather from across the department. The speaker introduces a subfield, like the correction of errors made by quantum computers. Course lectures target students. Endowed lectures, often named after donors, target researchers.