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.

Local operations and Chinese communications

The workshop spotlighted entanglement. It began in Shanghai, paused as participants hopped the Taiwan Strait, and resumed in Taipei. We discussed quantum operations and chaos, thermodynamics and field theory.1 I planned to return from Taipei to Shanghai to Los Angeles.

Quantum thermodynamicist Nelly Ng and I drove to the Taipei airport early. News from Air China curtailed our self-congratulations: China’s military was running an operation near Shanghai. Commercial planes couldn’t land. I’d miss my flight to LA.

nelly-and-me

Two quantum thermodynamicists in Shanghai

An operation?

Quantum information theorists use a mindset called operationalism. We envision experimentalists in separate labs. Call the experimentalists Alice, Bob, and Eve (ABE). We tell stories about ABE to formulate and analyze problems. Which quantum states do ABE prepare? How do ABE evolve, or manipulate, the states? Which measurements do ABE perform? Do they communicate about the measurements’ outcomes?

Operationalism concretizes ideas. The outlook checks us from drifting into philosophy and into abstractions difficult to apply physics tools to.2 Operationalism infuses our language, our framing of problems, and our mathematical proofs.

Experimentalists can perform some operations more easily than others. Suppose that Alice controls the magnets, lasers, and photodetectors in her lab; Bob controls the equipment in his; and Eve controls the equipment in hers. Each experimentalist can perform local operations (LO). Suppose that Alice, Bob, and Eve can talk on the phone and send emails. They exchange classical communications (CC).

You can’t generate entanglement using LOCC. Entanglement consists of strong correlations that quantum systems can share and that classical systems can’t. A quantum system in Alice’s lab can hold more information about a quantum system of Bob’s than any classical system could. We must create and control entanglement to operate quantum computers. Creating and controlling entanglement poses challenges. Hence quantum information scientists often model easy-to-perform operations with LOCC.

Suppose that some experimentalist Charlie loans entangled quantum systems to Alice, Bob, and Eve. How efficiently can ABE compute some quantity, exchange quantum messages, or perform other information-processing tasks, using that entanglement? Such questions underlie quantum information theory.

ca

Taipei’s night market. Or Caltech’s neighborhood?

Local operations.

Nelly and I performed those, trying to finagle me to LA. I inquired at Air China’s check-in desk in English. Nelly inquired in Mandarin. An employee smiled sadly at each of us.

We branched out into classical communications. I called Expedia (“No, I do not want to fly to Manila”), United Airlines (“No flights for two days?”), my credit-card company, Air China’s American reservations office, Air China’s Chinese reservations office, and Air China’s Taipei reservations office. I called AT&T to ascertain why I couldn’t reach Air China (“Yes, please connect me to the airline. Could you tell me the number first? I’ll need to dial it after you connect me and the call is then dropped”).

As I called, Nelly emailed. She alerted Bob, aka Janet (Ling-Yan) Hung, who hosted half the workshop at Fudan University in Shanghai. Nelly emailed Eve, aka Feng-Li Lin, who hosted half the workshop at National Taiwan University in Taipei. Janet twiddled the magnets in her lab (investigated travel funding), and Feng-Li cooled a refrigerator in his.

ABE can process information only so efficiently, using LOCC. The time crept from 1:00 PM to 3:30.

nelly-2-001

Nelly Ng uses classical communications.

What could we have accomplished with quantum communication? Using LOCC, Alice can manipulate quantum states (like an electron’s orientation) in her lab. She can send nonquantum messages (like “My flight is delayed”) to Bob. She can’t send quantum information (like an electron’s orientation).

Alice and Bob can ape quantum communication, given entanglement. Suppose that Charlie strongly correlates two electrons. Suppose that Charlie gives Alice one electron and gives Bob the other. Alice can send one qubit–one unit of quantum information–to Bob. We call that sending quantum teleportation.

Suppose that air-traffic control had loaned entanglement to Janet, Feng-Li, and me. Could we have finagled me to LA quickly?

Quantum teleportation differs from human teleportation.

xkcd

xkcd.com/465

We didn’t need teleportation. Feng-Li arranged for me to visit Taiwan’s National Center for Theoretical Sciences (NCTS) for two days. Air China agreed to return me to Shanghai afterward. United would fly me to LA, thanks to help from Janet. Nelly rescued my luggage from leaving on the wrong flight.

Would I rather have teleported? I would have avoided a bushel of stress. But I wouldn’t have learned from Janet about Chinese science funding, wouldn’t have heard Feng-Li’s views about gravitational waves, wouldn’t have glimpsed Taiwanese countryside flitting past the train we rode to the NCTS.

According to some metrics, classical resources outperform quantum.

einstein-2-001

At Taiwan’s National Center for Theoretical Sciences

The workshop organizers have generously released videos of the lectures. My lecture about quantum chaos and fluctuation relations appears here and here. More talks appear here.

With gratitude to Janet Hung, Feng-Li Lin, and Nelly Ng; to Fudan University, National Taiwan University, and Taiwan’s National Center for Theoretical Sciences for their hospitality; and to Xiao Yu for administrative support.

Glossary and other clarifications:

1Field theory describes subatomic particles and light.

2Physics and philosophy enrich each other. But I haven’t trained in philosophy. I benefit from differentiating physics problems that I’ve equipped to solve from philosophy problems that I haven’t.

It’s CHAOS!

My brother and I played the video game Sonic the Hedgehog on a Sega Dreamcast. The hero has spiky electric-blue fur and can run at the speed of sound.1 One of us, then the other, would battle monsters. Monster number one oozes onto a dark city street as an aquamarine puddle. The puddle spreads, then surges upward to form limbs and claws.2 The limbs splatter when Sonic attacks: Aqua globs rain onto the street.

chaos-vs-sonic

The monster’s master, Dr. Eggman, has ginger mustachios and a body redolent of his name. He scoffs as the heroes congratulate themselves.

“Fools!” he cries, the pauses in his speech heightening the drama. “[That monster is] CHAOS…the GOD…of DE-STRUC-TION!” His cackle could put a Disney villain to shame.

Dr. Eggman’s outburst comes to mind when anyone asks what topic I’m working on.

“Chaos! And the flow of time, quantum theory, and the loss of information.”

eggman

Alexei Kitaev, a Caltech physicist, hooked me on chaos. I TAed his spring-2016 course. The registrar calls the course Ph 219c: Quantum Computation. I call the course Topics that Interest Alexei Kitaev.

“What do you plan to cover?” I asked at the end of winter term.

Topological quantum computation, Alexei replied. How you simulate Hamiltonians with quantum circuits. Or maybe…well, he was thinking of discussing black holes, information, and chaos.

If I’d had a tail, it would have wagged.

“What would you say about black holes?” I asked.

untitled-2

Sonic’s best friend, Tails the fox.

I fwumped down on the couch in Alexei’s office, and Alexei walked to his whiteboard. Scientists first noticed chaos in classical systems. Consider a double pendulum—a pendulum that hangs from the bottom of a pendulum that hangs from, say, a clock face. Imagine pulling the bottom pendulum far to one side, then releasing. The double pendulum will swing, bend, and loop-the-loop like a trapeze artist. Imagine freezing the trapeze artist after an amount t of time.

What if you pulled another double pendulum a hair’s breadth less far? You could let the pendulum swing, wait for a time t, and freeze this pendulum. This pendulum would probably lie far from its brother. This pendulum would probably have been moving with a different speed than its brother, in a different direction, just before the freeze. The double pendulum’s motion changes loads if the initial conditions change slightly. This sensitivity to initial conditions characterizes classical chaos.

A mathematical object F(t) reflects quantum systems’ sensitivities to initial conditions. [Experts: F(t) can evolve as an exponential governed by a Lyapunov-type exponent: \sim 1 - ({\rm const.})e^{\lambda_{\rm L} t}.] F(t) encodes a hypothetical process that snakes back and forth through time. This snaking earned F(t) the name “the out-of-time-ordered correlator” (OTOC). The snaking prevents experimentalists from measuring quantum systems’ OTOCs easily. But experimentalists are trying, because F(t) reveals how quantum information spreads via entanglement. Such entanglement distinguishes black holes, cold atoms, and specially prepared light from everyday, classical systems.

Alexei illustrated, on his whiteboard, the sensitivity to initial conditions.

“In case you’re taking votes about what to cover this spring,” I said, “I vote for chaos.”

We covered chaos. A guest attended one lecture: Beni Yoshida, a former IQIM postdoc. Beni and colleagues had devised quantum error-correcting codes for black holes.3 Beni’s foray into black-hole physics had led him to F(t). He’d written an OTOC paper that Alexei presented about. Beni presented about a follow-up paper. If I’d had another tail, it would have wagged.

tails-2

Sonic’s friend has two tails.

Alexei’s course ended. My research shifted to many-body localization (MBL), a quantum phenomenon that stymies the spread of information. OTOC talk burbled beyond my office door.

At the end of the summer, IQIM postdoc Yichen Huang posted on Facebook, “In the past week, five papers (one of which is ours) appeared . . . studying out-of-time-ordered correlators in many-body localized systems.”

I looked down at the MBL calculation I was performing. I looked at my computer screen. I set down my pencil.

“Fine.”

I marched to John Preskill’s office.

boss

The bosses. Of different sorts, of course.

The OTOC kept flaring on my radar, I reported. Maybe the time had come for me to try contributing to the discussion. What might I contribute? What would be interesting?

We kicked around ideas.

“Well,” John ventured, “you’re interested in fluctuation relations, right?”

Something clicked like the “power” button on a video-game console.

Fluctuation relations are equations derived in nonequilibrium statistical mechanics. They describe systems driven far from equilibrium, like a DNA strand whose ends you’ve yanked apart. Experimentalists use fluctuation theorems to infer a difficult-to-measure quantity, a difference \Delta F between free energies. Fluctuation relations imply the Second Law of Thermodynamics. The Second Law relates to the flow of time and the loss of information.

Time…loss of information…Fluctuation relations smelled like the OTOC. The two had to join together.

on-button

I spent the next four days sitting, writing, obsessed. I’d read a paper, three years earlier, that casts a fluctuation relation in terms of a correlator. I unearthed the paper and redid the proof. Could I deform the proof until the paper’s correlator became the out-of-time-ordered correlator?

Apparently. I presented my argument to my research group. John encouraged me to clarify a point: I’d defined a mathematical object A, a probability amplitude. Did A have physical significance? Could anyone measure it? I consulted measurement experts. One identified A as a quasiprobability, a quantum generalization of a probability, used to model light in quantum optics. With the experts’ assistance, I devised two schemes for measuring the quasiprobability.

The result is a fluctuation-like relation that contains the OTOC. The OTOC, the theorem reveals, is a combination of quasiprobabilities. Experimentalists can measure quasiprobabilities with weak measurements, gentle probings that barely disturb the probed system. The theorem suggests two experimental protocols for inferring the difficult-to-measure OTOC, just as fluctuation relations suggest protocols for inferring the difficult-to-measure \Delta F. Just as fluctuation relations cast \Delta F in terms of a characteristic function of a probability distribution, this relation casts F(t) in terms of a characteristic function of a (summed) quasiprobability distribution. Quasiprobabilities reflect entanglement, as the OTOC does.

pra-image

Collaborators and I are extending this work theoretically and experimentally. How does the quasiprobability look? How does it behave? What mathematical properties does it have? The OTOC is motivating questions not only about our quasiprobability, but also about quasiprobability and weak measurements. We’re pushing toward measuring the OTOC quasiprobability with superconducting qubits or cold atoms.

Chaos has evolved from an enemy to a curiosity, from a god of destruction to an inspiration. I no longer play the electric-blue hedgehog. But I remain electrified.

 

1I hadn’t started studying physics, ok?

2Don’t ask me how the liquid’s surface tension rises enough to maintain the limbs’ shapes.

3Black holes obey quantum mechanics. Quantum systems can solve certain problems more quickly than ordinary (classical) computers. Computers make mistakes. We fix mistakes using error-correcting codes. The codes required by quantum computers differ from the codes required by ordinary computers. Systems that contain black holes, we can regard as performing quantum computations. Black-hole systems’ mistakes admit of correction via the code constructed by Beni & co. 

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.

obama-h

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.

2-hamiltons

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.

playbill-h

I’ve been pondering the Hamiltonian

mbl-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.

cast-h

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.

tesla-coil-2

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.

whiteboard

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.

 

drizzle

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!”

sun

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.

stoplight

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.

smiley

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.

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.”

too-cute

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.

discrete

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.

woo

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.

2Light.

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.

cactus-garden

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.

instructions

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} ).

google-maps

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

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.

cars

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.

perp-to-self

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.

table-corner

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.