# 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.

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.”

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.

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.

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.

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.

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.

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.

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. # Zoe Saldana Answers the Quantum Call Stephen Hawking & Zoe Saldana try to save Simon Pegg’s cat Watch Quantum Is Calling with Zoe Saldana, Stephen Hawking, Keanu Reeves, Paul Rudd, Simon Pegg, and John Cho. We are on the verge of a quantum revolution. Like in the days of the space race, technology has brought an impossibly distant frontier to our doorstep. Just over 17 years ago Michael Crichton wrote a parallel universe-hopping adventure, Timeline, whose fundamental transportation technology required the advent of quantum computing – a concept that was still only theoretical at the time. Today, IBM’s five-quantum bit (or qubit) array is at the fingertips of anyone within reach of the cloud. Google is building a fifty-qubit array. Microsoft is bankrolling a brain trust that will build a quantum computer based on topological qubits. Intel is investing$50 million on spin qubit technology. The UK has announced a £270 million program, and the EU a €1 billion program, to develop quantum technologies. And even more quantum circuits are on the way; the equivalent of competing classes of space shuttles. Only these crafts aren’t meant to travel through space, or even time. They travel through the complete unknown. Qubits fluctuate between the infinite universes of possibility, their quantum states based inherently on uncertainty. And the best way to harness that seemingly unlimited computing power, and take the first steps into the quantum frontier, is through the elusive concept of entanglement.

So then, the quantum crafts are ready; the standby lights on their consoles blinking in a steady yellow cadence. What we’re missing are the curiosity-driven pilots willing to grapple with the uncertain and unpredictable.

The quantum mechanics property of entanglement was discovered by Albert Einstein, Boris Podolsky, and Nathan Rosen and soon after described in a famous 1935 paper. Einstein called it “spooky action at a distance.” Virtually all of his contemporaries, including Edwin Schrödinger who coined the term “entanglement”, and the entire subsequent generation of physicists would struggle with this paradox. Although their struggles would be necessary to arrive at this particular moment in time, this precipice, their collective and prodigious minds were, and remain to be, handcuffed by training and experiences rooted in a classical understanding of the laws of nature – derived from phenomena that can be seen or felt, either directly or indirectly. Quantum entanglement, on the other hand, presents a puzzle of a fundamentally abstract nature.

Paul Rudd & Stephen Hawking chatting it up

When Paul Rudd defeated Stephen Hawking in a game of quantum chess – a game built from the ground up with a quantum mechanical set of moves leveraging superposition and entanglement – our intent was to suggest that an entirely new generation of physicists can emerge with an intuitive understanding of entanglement, even before having to dip their toes in mathematics.

Following up on Anyone Can Quantum, the challenges were to (1) further introduce and elaborate on quantum entanglement and (2) reach a wider audience, particularly women. Coming from a writer’s perspective, my primary concern was to make the abstract concept of entanglement somehow relatable. Popular stories, at their most basic, are told through interactions between people in relationships. Only through relational interactions can characters be challenged enough to affect a change in behavior, and as a result support a theme. Early story concepts evolved from the idea that any interaction with entanglement would result in a primary problem of miscommunication. Entanglement, in any form approaching personification, would be fully alien and incomprehensible. Language then, I decided, would become the fabric by which we could create a set of interactions between a human and entanglement.

Dr. Louise Banks (Amy Adams) & Ian Donnelly (Jeremy Renner) in Arrival

This particular dynamic was tackled in the recent movie Arrival. There, the fictional linguist Dr. Louise Banks is tasked with translating the coffee-ring-stain sign language of a visiting alien civilization before one of the world’s many nervous armies attacks them and causes an intergalactic incident. In the process of decoding the dense script, the controversial Sapir-Whorf theory is brought up introducing the idea that language shapes the way people think. While this theory may or may not hold snow, I am still impressed with the notion that a shared, specific, and descriptive language is necessary to collaborate and innovate. This impression is supported by my own experience in molecular and cell biology research in which communicating new findings always requires expending a tremendous amount of energy crafting a new and appropriate set of terms, or in other words, an expansion of the language.

Marvel To The Rescue

The Tesseract & Groot in Guardians of the Galaxy

To drive their building, multi-threaded Infinity Stones storyline, the Marvel Cinematic Universe (MCU) has been fortuitously bold in broaching quantum physics concepts and attempting to ground them in real science, taking advantage of the contacts available through the Science & Entertainment Exchange. Through these consultations, movies like Thor and Ant-Man have already delivered to a wide and diverse audience complex concepts such as Einstein-Rosen bridges (wormholes) and the Quantum Realm.

The Ant-Man consultation, in particular, resulted in a relationship between IQIM’s own Spyridon Michalakis (aka Spiros) and Ant-Man himself, Paul Rudd. This relationship was not only responsible for Anyone Can Quantum, but it was also the reason why Spiros was invited to be a panelist at the Silicon Valley Comic Con earlier this year, where he was interviewed by science journalist Zuberoa “Zube” Marcos of the global press outfit, El Pais, a woman who would end up playing a central role in getting Quantum Is Calling off the ground.

So the language of quantum physics was being slowly introduced to a wider, global population thanks to the Marvel films. It occurred to us that we had the opportunity to explain some of the physics concepts brought up by the MCU through the lens of quantum physics, and entanglement in particular. The one element of the MCU storylines that was most attractive to us was the Tesseract and its encased Space Stone. It was the first of the Infinity Stones introduced (in Captain America: The First Avenger) and the one that drove the plot of The Avengers, culminating in the creation of a wormhole over Manhattan. For Spiros, the solution was simple: In order to create wormholes, the exotic matter comprising the Space Stone would likely have to exploit entanglement, as described in a conjecture, dubbed “ER=EPR”, published by Leonard Susskind and Juan Maldacena in 2013.

The USS Enterprise (NCC-1701) in the Star Trek TOS episode “The Immunity Syndrome”

Finding Our Star

The remaining challenge was to find the right actress to deliver the new story. The earliest version of our story (back in June, 2016) was based on the crew of the Starship Enterprise encountering an alien creature that was the embodiment of entanglement (a.k.a The Flying Spaghetti Monster), a creature that attempted communication with Earthlings by reciting sound bytes originating from past Earth radio transmissions. In this story iteration, Chief communications officer Uhura would have used her skills to translate the monster’s message amidst rising tension (just like in Arrival).

Zoe Saldana as Lt. Nyota Uhura

In the subsequent revisions to the story we had to simplify the script and winnow down the cast. We opted to lean on Zoe Saldana’s Uhura. Her character could take on the role of captain, communications officer, and engineer. Zoe was already widely known across multiple sci-fi franchises featuring aliens (namely Star Trek, Guardians of the Galaxy, and Avatar) and her characters have had to speak in or translate those languages.

Zoe = Script

But before approaching Zoe Saldana – and at that point in time, we had no idea how to go about that – we needed to complete a script. Two other incredible resources were available to us: the voices of Dr. Hawking and Keanu Reeves; and we had to make all three work together in a unique comedy – one that did not squander the involvement of either voice, but also served to elevate the role of Zoe.

Even in the first version of the story it was my intent to have Keanu Reeves provide the voice for entanglement, expressed through the most alien sounding languages I could imagine. To compress the story to fit our budget we were forced to narrow the list of languages to two, and I chose Dothraki and Navajo. The role of Keanu’s character was to test, recruit, and ultimately invite Zoe Saldana to enter and experience entanglement in the Quantum Realm. Dr. Stephen Hawking would be the reluctant guide that helps Zoe interpret the confusing clues embedded within the Dothraki and Navajo to arrive at the ER=EPR conjecture.

As for the riddle itself, I chose to use two poems from Through the Looking Glass (and What Alice Found There), The Walrus and The Carpenter as well as Haddock’s Eyes, as the reference material, so that those savvy enough to solve even half the riddle on their own would have a further clue pointing them to the final answer.

Simon Pegg’s cat, Schrodinger (not his actual cat)

The disappearance of Simon’s cat, Schrödinger, had a tripartite function of (a) presenting an inciting incident that urged Zoe to subject herself to the puzzle-solving trial, which we called the Riddle of the Tesseract, (b) to demonstrate the risk of touching the Tesseract and the gravity of her climactic choice, and (c) invoking Schrödinger’s famous thought experiment to present the idea that, in the Quantum Realm, the cat and Zoe are both dead and alive, an uncertainty.

The story was done. And it looked good on paper. But the script was just a piece of paper unless we got Zoe Saldana to sign on.

Zuberoa Marcos

Zoe = Zube

For weeks, Spiros worked all of his connections only to come up empty. It wasn’t until he mentioned our holy quest to Zube (from El Pais and Silicon Valley Comic Con) during an unrelated Skype session that he had the first glimmer of hope, even kismet. Zube had been working on arranging an interview with Zoe for months, an interview that would be taking place three days later in Atlanta. Without even a second thought, Spiros purchased a plane ticket and was on his way to Atlanta two days later. Watching the interview take place, he heard Zoe answer one of Zube’s question about what kind of technology interested her the most. It was the transporter, the teleportation machine used by the crew of the Enterprise to shift matter to and from surfaces of alien planets. This was precisely the kind of technology we were interested in describing at a quantum level! Realizing this was the opening we needed, Zube nodded over to Spiros and made the introductions.

It turns out Zoe had been fascinated by science fiction since her early childhood, being particularly obsessed with Frank Herbert’s Dune. Moreover, she was interested in playing the role of our lead character. In the weeks that followed, communication proceeded through managers in an attempt to nail down a filming date.

Mariel, Zoe, and Cicely Saldana

The Dangers of Miscommunication

I probably don’t need to remind you that Zoe Saldana is a core component of three gigantic franchises. That means tight schedules, press conferences, and international travel. Ultimately Zoe said that her travel commitments wouldn’t allow her to film our short. It was back to square one. We were dead in the water. The script was just a piece of paper.

However, for some reason, Spiros and Zube were not willing to concede. Zube found out about Zoe Saldana’s production company Cinestar and got in contact with coordinator Diego Gonzalez, to set up a lunch meeting. At lunch, Diego informed Zube and Spiros that Zoe really wanted to do this, but her team was under the impression that filming for our short video had to take place the week Star Trek: Beyond was to be released (Zoe was arguably busier than the POTUS during that week). Spiros informed Cinestar that we would accommodate whatever date Zoe could be available. Having that hurdle removed paved the way for a concrete film date to be set, October 25th. And now the real work began.

Simon Pegg in Shaun of the Dead

Finding Common Language

We had set the story inside Simon Pegg’s house and the script included voice-over dialogue for the superstar, but we had yet to even contact Simon. We had written in a part with Paul Rudd on a voicemail message. And we had also included a sixth character that would knock on the door and force Zoe to make her big decision. On top of that I had incorporated Dothraki and Navajo versions of century-old poems that had yet to be translated into those two languages. While Spiros worked on chasing down the talent, I nervously attempted to make contact with experts in the two languages.

David J. Peterson

I remember watching a video of Prof. David J. Peterson, creator of the Dothraki language for HBO’s Game of Thrones, speaking at Google about the process of crafting the language. Some unknown courage surfaced and I hunted down contact information for the famous linguist. I found an old website of his, an email address, and sent and inquiry at about midnight pacific standard time on October 14th, the day before my birthday. Within 45 minutes David had responded with interest in helping out. I was floored. And I couldn’t help geeking out. But more importantly this meant we would have the most accurate translation humanly possible. And when one is working on behalf of Caltech you definitely feel the pressure to be above reproach, or unsullied ;).

Keanu Reeves, Jennifer Wheeler, a pumpkin, a highlighter & my left arm

Finding a Navajo translator was comparatively difficult. A couple days after receiving Dr. Peterson’s email, I was in Scottsdale, AZ with my brother. I had previously scheduled the trip so that I could be in attendance at a book-signing featuring two of my favorite authors, as a birthday gift to myself. The event was held at the Poisoned Pen bookstore where many other local authors would regularly hold book-signings. While I was geeking out over meeting my favorite writing duo, as well as over my recent interaction with David Peterson, I was also stressed by the pressure to come through on an authentic Navajo translation. My brother urged me to ask the proprietors of the Poisoned Pen for any leads. And wouldn’t you know it, they had recently hosted a book-signing for the author of a Code Talkers book, and she was local. A morning of emails led to Jennifer Wheeler. We had struck gold. Jennifer had recently overseen Navajo translations of Star Wars: A New Hope and Finding Nemo, complete with voice-overs. There was probably nobody more qualified in the world.

Keanu Reeves as Ted “Theodore” Logan in Bill & Ted’s Excellent Adventure

So it turns out that Navajo is a much more difficult language to translate and speak than I had anticipated. For instance, there are over a hundred vowel sounds. So even though the translation was in good hands, I would be imposing on Keanu Reeves one of the greatest vocal challenges he would ever undertake. Eventually I arranged to have Jennifer on hand during Keanu’s voice recording. Here’s what he had to record (phonetically):

Tsee /da / a / ko / ho / di / say / tsaa, / a / nee / di

aɫ / tso / n’ / shay / ch’aa / go

Echo Papa Romeo / do / do / chxih / da

Bi / nee / yay / bi / zhay / ho / lo / nee / bay / do / bish / go.

Alex Winter & Zoe Saldana hard at work

Filming Day

After months of planning and weeks of script revisions, filming finally happened at an opulent, palatial residence in the Hollywood Hills (big props to Shaun Maguire and Liana Kadisha for securing the location). Six cats. Three trainers. Lights. Cameras. Zube. Zoe Saldana actually showed up! Along with her sisters, Cinestar, and even John Cho! Spiros had gotten assurances from Simon Pegg that he would lend his name and golden voice so we were able to use the ridiculous “Simon’s Peggs” wood sign that we had crafted just for the shoot. Within a few busy hours we were wrapped. All the cats and props were packed and back in LA traffic, where we all seem to exist more often than not. Now the story was left to the fate of editing and post-production.

In Post

Unlike the circumstances involved with Anyone Can Quantum, for which there was a fast approaching debut date, Spiros and myself actually had time to be an active part of the post-production process. Alex Winter, Trouper Productions, and STITCH graciously involved us through virtually every step.

One thing that became quite apparent through the edits was the lack of a strong conclusion. Zoe’s story was designed to be somewhat open-ended. Although her character arc was meant to reach a conclusion with the decision to enter the Quantum Realm, it was clear that the short still needed a clear resolution.

What Seraph looks like as code in the Matrix Reloaded

Through much debate and workshopping, Spiros and I finally arrived at bookend scenes that took advantage of Keanu Reeve’s emblematic representation of, and inescapable entanglement with, The Matrix. Our ultimate goal is to create stories that reflect the quantum nature of the universe, the underlying quantum code that is the fabric from which all things emerge, exist, and interact. So, in a way, The Matrix wasn’t that far off.

Language Is Fluid

LIQUi|> (“liquid”), or Language-Integrated Quantum Operations, is an architecture, programming language, and tools suite designed for quantum computing that is being developed by the Microsoft team at Quantum Architectures and Computation Group (or QuArC). Admittedly taking a few liberties, on Spiros’s advice I used actual LIQUi|> commands to create a short script that established a gate (or data structure) that I called Alice (which is meant to represent Zoe and her location), created an entanglement between Alice and the Tesseract, then teleported the Tesseract to Alice. You’ll notice that the visual and sound effects are ripped right from The Matrix.

This set up the possibility of adapting Neo’s famous monologue (from the end of the original Matrix) so we could hint that Zoe was somewhere adrift within the quantum code that defines the Quantum Realm. Yes, both Spiros and I were in the studio when Keanu recorded those lines (along with his lines in Dothraki and Navajo). Have I mentioned geeking out yet? An accompanying sequence of matrix code, or digital rain, had to be constructed that could accommodate examples of entanglement-related formulas. As you might have guessed, the equations highlighted in the digital rain at the end of the short are real, most of which came from this paper on emergent space (of which Spiros is a co-author).

Keanu Reeves & Keanu

Listen To Your Friend Keanu Reeves. He’s A Cool Dude.

With only a few days left before our debut date, Simon Pegg, Stephen Hawking and Paul Rudd all came through with their voice-over samples. Everything was then stitched together and the color correction, sound balancing, and visual effects were baked into the final video and phew. Finally, and impossibly, through the collaboration of a small army of unique individuals, the script had become a short movie. And hopefully it has become something unique, funny, and inspiring, especially to any young women (and men) who may be harboring an interest in, or a doubt preventing them from, delving into the quantum realm.

# 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.

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.

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.