My maternal grandfather gave me an antique key when I was in middle school. I loved the workmanship: The handle consisted of intertwined loops. I loved the key’s gold color and how the key weighed on my palm. Even more, I loved the thought that the key opened something. I accompanied my mother to antique shops, where I tried unlocking chests, boxes, and drawers.
My grandfather’s antique key
I found myself holding another such key, metaphorically, during the autumn of 2018. MIT’s string theorists had requested a seminar, so I presented about quasiprobabilities. Quasiprobabilities represent quantum states similarly to how probabilities represent a swarm of classical particles. Consider the steam rising from asphalt on a summer day. Calculating every steam particle’s position and momentum would require too much computation for you or me to perform. But we can predict the probability that, if we measure every particle’s position and momentum, we’ll obtain such-and-such outcomes. Probabilities are real numbers between zero and one. Quasiprobabilities can assume negative and nonreal values. We call these values “nonclassical,” because they’re verboten to the probabilities that describe classical systems, such as steam. I’d defined a quasiprobability, with collaborators, to describe quantum chaos.
David Arvidsson-Shukur was sitting in the audience. David is a postdoctoral fellow at the University of Cambridge and a visiting scholar in the other Cambridge (at MIT). He has a Swedish-and-southern-English accent that I’ve heard only once before and, I learned over the next two years, an academic intensity matched by his kindliness.1 Also, David has a name even longer than mine: David Roland Miran Arvidsson-Shukur. We didn’t know then, but we were destined to journey together, as postdoctoral knights-errant, on a quest for quantum truth.
David studies the foundations of quantum theory: What distinguishes quantum theory from classical? David suspected that a variation on my quasiprobability could unlock a problem in metrology, the study of measurements.
Suppose that you’ve built a quantum computer. It consists of gates—uses of, e.g., magnets or lasers to implement logical operations. A classical gate implements operations such as “add 11.” A quantum gate can implement an operation that involves some number more general than 11. You can try to build your gate correctly, but it might effect the wrong value. You need to measure .
How? You prepare some quantum state and operate on it with the gate. imprints itself on the state, which becomes . Measure some observable . You repeat this protocol in each of many trials. The measurement yields different outcomes in different trials, according to quantum theory. The average amount of information that you learn about per trial is called the Fisher information.
Let’s change this protocol. After operating with the gate, measure another observable, , and postselect: If the measurement yields a desirable outcome , measure . If the -measurement doesn’t yield the desirable outcome, abort the trial, and begin again. If you choose and adroitly, you’ll measure only when the trial will provide oodles of information about . You’ll save yourself many measurements that would have benefited you little.2
Why does postselection help us? We could understand easily if the system were classical: The postselection would effectively improve the input state. To illustrate, let’s suppose that (i) a magnetic field implemented the gate and (ii) the input were metal or rubber. The magnetic field wouldn’t affect the rubber; measuring in rubber trials would provide no information about the field. So you could spare yourself measurements by postselecting on the system’s consisting of metal.
Postselection on a quantum system can defy this explanation. Consider optimizing your input state , beginning each trial with the quantum equivalent of metal. Postselection could still increase the average amount of information information provided about per trial. Postselection can enhance quantum metrology even when postselection can’t enhance the classical analogue.
David suspected that he could prove this result, using, as a mathematical tool, the quasiprobability that collaborators and I had defined. We fulfilled his prediction, with Hugo Lepage, Aleks Lasek, Seth Lloyd, and Crispin Barnes. Nature Communications published our paper last month. The work bridges the foundations of quantum theory with quantum metrology and quantum information theory—and, through that quasiprobability, string theory. David’s and my quantum quest continues, so keep an eye out for more theory from us, as well as a photonic experiment based on our first paper.
I still have my grandfather’s antique key. I never found a drawer, chest, or box that it opened. But I don’t mind. I have other mysteries to help unlock.
1The morning after my wedding this June, my husband and I found a bouquet ordered by David on our doorstep.
2Postselection has a catch: The measurement has a tiny probability of yielding the desirable outcome. But, sometimes, measuring costs more than preparing , performing the gate, and postselecting. For example, suppose that the system is a photon. A photodetector will measure . Some photodetectors have a dead time: After firing, they take a while to reset, to be able to fire again. The dead time can outweigh the cost of the beginning of the experiment.
In elementary and middle school, I felt an affinity for the class three years above mine. Five of my peers had siblings in that year. I carpooled with a student in that class, which partnered with mine in holiday activities and Grandparents’ Day revues. Two students in that class stood out. They won academic-achievement awards, represented our school in science fairs and speech competitions, and enrolled in rigorous high-school programs.
Those students came to mind as I grew to know David Limmer. David is an assistant professor of chemistry at the University of California, Berkeley. He studies statistical mechanics far from equilibrium, using information theory. Though a theorist ardent about mathematics, he partners with experimentalists. He can pass as a physicist and keeps an eye on topics as far afield as black holes. According to his faculty page, I discovered while writing this article, he’s even three years older than I.
I met David in the final year of my PhD. I was looking ahead to postdocking, as his postdoc fellowship was fading into memory. The more we talked, the more I thought, I’d like to be like him.
We studied a molecule prevalent across nature and technologies. Such molecules feature in your eyes, solar-fuel-storage devices, and more. The molecule has two clumps of atoms. One clump may rotate relative to the other if the molecule absorbs light. The rotation switches the molecule from a “closed” configuration to an “open” configuration.
These molecular switches are small, quantum, and far from equilibrium; so modeling them is difficult. Making assumptions offers traction, but many of the assumptions disagreed with David. He wanted general, thermodynamic-style bounds on the probability that one of these molecular switches would switch. Then, he ran into me.
I traffic in mathematical models, developed in quantum information theory, called resource theories. We use resource theories to calculate which states can transform into which in thermodynamics, as a dime can transform into ten pennies at a bank. David and I modeled his molecule in a resource theory, then bounded the molecule’s probability of switching from “closed” to “open.” I accidentally composed a theme song for the molecule; you can sing along with this post.
That post didn’t mention what David and I discovered about quantum clocks. But what better backdrop for a mental trip to elementary school or to three years into the future?
I’ve blogged about autonomous quantum clocks (and ancient Assyria) before. Autonomous quantum clocks differ from quantum clocks of another type—the most precise clocks in the world. Scientists operate the latter clocks with lasers; autonomous quantum clocks need no operators. Autonomy benefits you if you want for a machine, such as a computer or a drone, to operate independently. An autonomous clock in the machine ensures that, say, the computer applies the right logical gate at the right time.
What’s an autonomous quantum clock? First, what’s a clock? A clock has a degree of freedom (e.g., a pair of hands) that represents the time and that moves steadily. When the clock’s hands point to 12 PM, you’re preparing lunch; when the clock’s hands point to 6 PM, you’re reading Quantum Frontiers. An autonomous quantum clock has a degree of freedom that represents the time fairly accurately and moves fairly steadily. (The quantum uncertainty principle prevents a perfect quantum clock from existing.)
Suppose that the autonomous quantum clock constitutes one part of a machine, such as a quantum computer, that the clock guides. When the clock is in one quantum state, the rest of the machine undergoes one operation, such as one quantum logical gate. (Experts: The rest of the machine evolves under one Hamiltonian.) When the clock is in another state, the rest of the machine undergoes another operation (evolves under another Hamiltonian).
Physicists have been modeling quantum clocks using the resource theory with which David and I modeled our molecule. The math with which we represented our molecule, I realized, coincided with the math that represents an autonomous quantum clock.
Think of the molecular switch as a machine that operates (mostly) independently and that contains an autonomous quantum clock. The rotating clump of atoms constitutes the clock hand. As a hand rotates down a clock face, so do the nuclei rotate downward. The hand effectively points to 12 PM when the switch occupies its “closed” position. The hand effectively points to 6 PM when the switch occupies its “open” position.
The nuclei account for most of the molecule’s weight; electrons account for little. They flit about the landscape shaped by the atomic clumps’ positions. The landscape governs the electrons’ behavior. So the electrons form the rest of the quantum machine controlled by the nuclear clock.
Experimentalists can create and manipulate these molecular switches easily. For instance, experimentalists can set the atomic clump moving—can “wind up” the clock—with ultrafast lasers. In contrast, the only other autonomous quantum clocks that I’d read about live in theory land. Can these molecules bridge theory to experiment? Reach out if you have ideas!
Have you ever wondered what can be done in 48 hours? For instance, our heart beats around 200 000 times. One of the biggest supercomputers crunches petabytes (peta = 1015) of numbers to simulate an experiment that took Google’s quantum processor only 300 seconds to run. In 48 hours, one can also participate in the Sciathon with almost 500 young researchers from more than 80 countries!
Two weeks ago I participated in a scientific marathon, the Sciathon. The structure of this event roughly resembled a hackathon. I am sure many readers are familiar with the idea of a hackathon from personal experience. For those unfamiliar — a hackathon is an intense collaborative event, usually organized over the weekend, during which people with different backgrounds work in groups to create prototypes of functioning software or hardware. For me, it was the very first time to have firsthand experience with a hackathon-like event!
The Sciathon was organized by the Lindau Nobel Laureate Meetings (more about the meetings with Nobel laureates, which happen annually in the lovely German town of Lindau, in another blogpost, I promise!) This year, unfortunately, the face-to-face meeting in Lindau was postponed until the summer of 2021. Instead, the Lindau Nobel Laureate Meetings alumni and this year’s would-be attendees had an opportunity to gather for the Sciathon, as well as the Online Science Days earlier this week, during which the best Sciathon projects were presented.
The participants of the Sciathon could choose to contribute new views, perspectives and solutions to three main topics: Lindau Guidelines, Communicating Climate Change and Capitalism After Corona. The first topic concerned an open, cooperative science community where data and knowledge are freely shared, the second — how scientists could show that the climate crisis is just as big a threat as the SARS-CoV-19 virus, and the last — how to remodel our current economic systems so that they are more robust to unexpected sudden crises. More detailed descriptions of each topic can be found on the official Sciathon webpage.
My group of ten eager scientists, mostly physicists, from master students to postdoctoral researchers, focused on the first topic. In particular, our goal was to develop a method of familiarizing high school students with the basics of quantum information and computation. We envisioned creating an online notebook, where an engaging story would be intertwined with interactive blocks of Python code utilizing the open-source quantum computing toolkit Qiskit. This hands-on approach would enable students to play with quantum systems described in the story-line by simply running the pre-programmed commands with a click of the mouse and then observe how “experiment” matches “the theory”. We decided to work with a system comprising one or two qubits and explain such fundamental concepts in quantum physics as superposition, entanglement and measurement. The last missing part was a captivating story.
The story we came up with involved two good friends from the lab, Miss Schrödinger and Miss Pauli, as well as their kittens, Alice and Bob. At first, Alice and Bob seemed to be ordinary cats, however whenever they sipped quantum milk, they would turn into quantum cats, or as quantum physicists would say — kets. Do I have to remind the reader that a quantum cat, unlike an ordinary one, could be both awake and asleep at the same time?
Miss Schrödinger was a proud cat owner who not only loved her cat, but also would take hundreds of pictures of Alice and eagerly upload them on social media. Much to Miss Schrödinger’s surprise, none of the pictures showed Alice partly awake and partly asleep — the ket would always collapse to the cat awake or the cat asleep! Every now and then, Miss Pauli would come to visit Miss Schrödinger and bring her own cat Bob. While the good friends were chit-chatting over a cup of afternoon tea, the cats sipped a bit of quantum milk and started to play with a ball of wool, resulting in a cute mess of two kittens tangled up in wool. Every time after coming back home, Miss Pauli would take a picture of Bob and share it with Miss Schrödinger, who would obviously also take a picture of Alice. After a while, the young scientists started to notice some strange correlations between the states of their cats…
The adventures of Miss Schrödinger and her cat continue! For those interested, you can watch a short video about our project!
Overall, I can say that I had a lot of fun participating in the Sciathon. It was an intense yet extremely gratifying event. In addition to the obvious difficulty of racing against the clock, our group also had to struggle with coordinating video calls between group members scattered across three almost equidistant time zones — Eastern Australian, Central European and Central US! During the Sciathon I had a chance to interact with other science enthusiasts from different backgrounds and work on something from outside my area of expertise. I would strongly encourage anyone to participate in hackathon-like events to break the daily routine, particularly monotonous during the lockdown, and unleash one’s creative spirit. Such events can also be viewed as an opportunity to communicate science and scientific progress to the public. Lastly, I would like to thank other members of my team — collaborating with you during the Sciathon was a blast!
Some of you may have wondered whether I have a life. I do. He’s a computer scientist, and we got married earlier this month.
Marrying a quantum information scientist comes with dangers not advertised in any Brides magazine (I assume; I’ve never opened a copy of Brides magazine). Never mind the perils of gathering together Auntie So-and-so and Cousin Such-and-such, who’ve quarreled since you were six; or spending tens of thousands of dollars on one day; or assembling two handfuls of humans during a pandemic. Beware the risks of marrying someone who unconsciously types “entropy” when trying to type “entry,” twice in a row.
1) She’ll introduce you to friends as “a classical computer scientist.” They’d assume, otherwise, that he does quantum computer science. Of course. Wouldn’t you?
2) The quantum punning will commence months before the wedding. One colleague wrote, “Many congratulations! Now you know the true meaning of entanglement.” Quantum particles can share entanglement. If you measure entangled particles, your outcomes can exhibit correlations stronger than any produceable by classical particles. As a card from another colleague read, “May you stay forever entangled, with no decoherence.”
I’d rather not dedicate much of a wedding article to decoherence, but suppose that two particles are maximally entangled (can generate the strongest correlations possible). Suppose that particle 2 heats up or suffers bombardment by other particles. The state of particle 2 decoheres as the entanglement between 1 and 2 frays. Equivalently, particle 2 entangles with its environment, and particle 2 can entangle only so much: The more entanglement 2 shares with the environment, the less entanglement 2 can share with 1. Physicists call entanglement—ba-duh-bum—monogamous.
The matron-of-honor toast featured another entanglement joke, as well as five more physics puns.1 (She isn’t a scientist, but she did her research.) She’ll be on Zoom till Thursday; try the virtual veal.
3) When you ask what sort of engagement ring she’d like, she’ll mention black diamonds. Experimentalists and engineers are building quantum computers from systems of many types, including diamond. Diamond consists of carbon atoms arranged in a lattice. Imagine expelling two neighboring carbon atoms and replacing one with a nitrogen atom. You’ll create a nitrogen-vacancy center whose electrons you can control with light. Such centers color the diamond black but let you process quantum information.
If I’d asked my fiancé for a quantum computer, we’d have had to wait 20 years to marry. He gave me an heirloom stone instead.
4) When a wedding-gown shopkeeper asks which sort of train she’d prefer, she’ll inquire about Maglevs. I dislike shopping, as the best man knows better than most people. In middle school, while our classmates spent their weekends at the mall, we stayed home and read books. But I filled out gown shops’ questionnaires.
“They want to know what kinds of material I like,” I told the best man over the phone, “and what styles, and what type of train. I had to pick from four types of train. I didn’t even know there were four types of train!”
“Steam?” guessed the best man. “Diesel?”
His suggestions appealed to me as a quantum thermodynamicist. Thermodynamics is the physics of energy, which engines process. Quantum thermodynamicists study how quantum phenomena, such as entanglement, can improve engines.
“Get the Maglev train,” the best man added. “Low emissions.”
“Ooh,” I said, “that’s superconducting.” Superconductors are quantum systems in which charge can flow forever, without dissipating. Labs at Yale, at IBM, and elsewhere are building quantum computers from superconductors. A superconductor consists of electrons that pair up with help from their positively charged surroundings—Cooper pairs. Separating Cooper-paired electrons requires an enormous amount of energy. What other type of train would better suit a wedding?
I set down my phone more at ease. Later, pandemic-era business closures constrained me to wearing a knee-length dress that I’d worn at graduations. I didn’t mind dodging the train.
5) When you ask what style of wedding dress she’ll wear, she’ll say that she likes her clothing as she likes her equations. Elegant in their simplicity.
6) You’ll plan your wedding for wedding season only because the rest of the year conflicts with more seminars, conferences, and colloquia. The quantum-information-theory conference of the year takes place in January. We wanted to visit Australia in late summer, and Germany in autumn, for conferences. A quantum-thermodynamics conference takes place early in the spring, and the academic year ends in May. Happy is the June bride; happier is the June bride who isn’t preparing a talk.
7) An MIT chaplain will marry you. Who else would sanctify the union of a physicist and a computer scientist?
8) You’ll acquire more in-laws than you bargained for. Biological parents more than suffice for most spouses. My husband has to contend with academic in-laws, as my PhD supervisor is called my “academic father.”
9) Your wedding can double as a conference. Had our wedding taken place in person, collaborations would have flourished during the cocktail hour. Papers would have followed; their acknowledgements sections would have nodded at the wedding; and I’d have requested copies of all manuscripts for our records—which might have included our wedding album.
10) You’ll have trouble identifying a honeymoon destination where she won’t be tempted to give a seminar. I thought that my then-fiancé would enjoy Vienna, but it boasts a quantum institute. So do Innsbruck and Delft. A colleague-friend works in Budapest, and I owe Berlin a professional visit. The list grew—or, rather, our options shrank. But he turned out not to mind my giving a seminar. The pandemic then cancelled our trip, so we’ll stay abroad for a week after some postpandemic European conference (hint hint).
11) Your wedding will feature on the blog of Caltech’s Institute for Quantum Information and Matter. Never mind The New York Times. Where else would you expect to find a quantum information physicist? I feel fortunate to have found someone with whom I wouldn’t rather be anywhere else.
1“I know that if Nicole picked him to stand by her side, he must be a FEYNMAN and not a BOZON.”
Mr. Mole is trapped indoors, alone. Spring is awakening outside, but he’s confined to his burrow. Birds are twittering, and rabbits are chattering, but he has only himself for company.
Spring—crocuses, daffodils, and hyacinths budding; leaves unfurling; and birds warbling—burst upon Cambridge, Massachusetts last month. The city’s shutdown vied with the season’s vivaciousness. I relieved the tension by rereading The Wind in the Willows, which I’ve read every spring since 2017.
Project Gutenberg offers free access to Kenneth Grahame’s 1908 novel. He wrote the book for children, but never mind that. Many masterpieces of literature happen to have been written for children.
One line in the novel demanded, last year, that I memorize it. On page one, Mole is cleaning his house beneath the Earth’s surface. He’s been dusting and whitewashing for hours when the spring calls to him. Life is pulsating on the ground and in the air above him, and he can’t resist joining the party. Mole throws down his cleaning supplies and tunnels upward through the soil: “he scraped and scratched and scrabbled and scrooged, and then he scrooged again and scrabbled and scratched and scraped.”
The quotation appealed to me not only because of its alliteration and chiasmus. Mole’s journey reminded me of research.
Over four years ago, other collaborators and I uncovered a thermodynamics problem, as did two other groups at the same time. Thermodynamicists often consider small systems that interact with large environments, like a magnolia flower releasing its perfume into the air. The two systems—magnolia flower and air—exchange things, such as energy and scent particles. The total amount of energy in the flower and the air remains constant, as does the total number of perfume particles. So we call the energy and the perfume-particle number conservedquantities.
We represent quantum conserved quantities with matrices and . We nearly always assume that, in this thermodynamic problem, those matrices commute with each other: . Almost no one mentions this assumption; we make it without realizing. Eliminating this assumption invalidates a derivation of the state reached by the small system after a long time. But why assume that the matrices commute? Noncommutation typifies quantum physics and underlies quantum error correction and quantum cryptography.
What if the little system exchanges with the large system thermodynamic quantities represented by matrices that don’t commute with each other?
Colleagues and I began answering this question, four years ago. The small system, we argued, thermalizes to near a quantum state that contains noncommuting matrices. We termed that state, , the non-Abelian thermal state. The ’s represent conserved quantities, and the ’s resemble temperatures. The real number ensures that, if you measure any property of the state, you’ll obtain some outcome. Our arguments relied on abstract mathematics, resource theories, and more quantum information theory.
Over the past four years, noncommuting conserved quantities have propagated across quantum-information-theoretic thermodynamics.1 Watching the idea take root has been exhilarating, but the quantum information theory didn’t satisfy me. I wanted to see a real physical system thermalize to near the non-Abelian thermal state.
Michael and Amir joined the mission to propose an experiment. We kept nosing toward a solution, then dislodging a rock that would shower dirt on us and block our path. But we scrabbled onward.
Imagine a line of ions trapped by lasers. Each ion contains the physical manifestation of a qubit—a quantum two-level system, the basic unit of quantum information. You can think of a qubit as having a quantum analogue of angular momentum, called spin. The spin has three components, one per direction of space. These spin components are represented by matrices , , and that don’t commute with each other.
A couple of qubits can form the small system, analogous to the magnolia flower. The rest of the qubits form the large system, analogous to the air. I constructed a Hamiltonian—a matrix that dictates how the qubits evolve—that transfers quanta of all the spin’s components between the small system and the large. (Experts: The Heisenberg Hamiltonian transfers quanta of all the spin components between two qubits while conserving .)
The Hamiltonian led to our first scrape: I constructed an integrable Hamiltonian, by accident. Integrable Hamiltonians can’t thermalize systems. A system thermalizes by losing information about its initial conditions, evolving to a state with an exponential form, such as . We clawed at the dirt and uncovered a solution: My Hamiltonian coupled together nearest-neighbor qubits. If the Hamiltonian coupled also next-nearest-neighbor qubits, or if the ions formed a 2D or 3D array, the Hamiltonian would be nonintegrable.
We had to scratch at every stage—while formulating the setup, preparation procedure, evolution, measurement, and prediction. But we managed; Physical Review E published our paper last month. We showed how a quantum system can evolve to the non-Abelian thermal state. Trapped ions, ultracold atoms, and quantum dots can realize our experimental proposal. We imported noncommuting conserved quantities in thermodynamics from quantum information theory to condensed matter and atomic, molecular, and optical physics.
As Grahame wrote, the Mole kept “working busily with his little paws and muttering to himself, ‘Up we go! Up we go!’ till at last, pop! his snout came out into the sunlight and he found himself rolling in the warm grass of a great meadow.”
On my right sat a quantum engineer. She was facing a melanoma specialist who works at a medical school. Leftward of us sat a networks expert, a flatworm enthusiast, and a condensed-matter theorist.
Farther down sat a woman who slices up mouse brains.
Welcome to “Coherent Spins in Biology,” a conference that took place at the University of California, Los Angeles (UCLA) this past December. Two southern Californians organized the workshop: Clarice Aiello heads UCLA’s Quantum Biology Tech lab. Thorsten Ritz, of the University of California, Irvine, cofounded a branch of quantum biology.
Quantum biology served as the conference’s backdrop. According to conventional wisdom, quantum phenomena can’t influence biology significantly: Biological systems have high temperatures, many particles, and fluids. Quantum phenomena, such as entanglement (a relationship that quantum particles can share), die quickly under such conditions.
Yet perhaps some survive. Quantum biologists search for biological systems that might use quantum resources. Then, they model and measure the uses and resources. Three settings (at least) have held out promise during the past few decades: avian navigation, photosynthesis, and olfaction. You can read about them in this book, cowritten by a conference participant for the general public. I’ll give you a taste (or a possibly quantum smell?) by sketching the avian-navigation proposal, developed by Thorsten and colleagues.
Birds migrate southward during the autumn and northward during the spring. How do they know where to fly? At least partially by sensing the Earth’s magnetic field, which leads compass needles to point northward. How do birds sense the field?
Possibly with a protein called “cryptochrome.” A photon (a particle of light) could knock an electron out of part of the protein and into another part. Each part would have one electron that lacked a partner. The electrons would share entanglement. One electron would interact with the Earth’s magnetic field differently than its partner, because its surroundings would differ. (Experts: The electrons would form a radical pair. One electron would neighbor different atoms than the other, so the electron would experience a different local magnetic field. The discrepancy would change the relative phase between the electrons’ spins.) The discrepancy could affect the rate at which the chemical system could undergo certain reactions. Which reactions occur could snowball into large and larger effects, eventually signaling the brain about where the bird should fly.
Quantum mechanics and life rank amongst the universe’s mysteries. How could a young researcher resist the combination? A postdoc warned me away, one lunchtime at the start of my PhD. Quantum biology had enjoyed attention several years earlier, he said, but noise the obscured experimental data. Controversy marred the field.
I ate lunch with that postdoc in 2013. Interest in quantum biology is reviving, as evidenced in the conference. Two reasons suggested themselves: new technologies and new research avenues. For example, Thorsten described the disabling and deletion of genes that code for cryptochrome. Such studies require years’ more work but might illuminate whether cryptochrome affects navigation.
The keynote speaker, Harvard’s Misha Lukin, illustrated new technologies and new research avenues. Misha’s lab has diamonds that contain quantum defects, which serve as artificial atoms. The defects sense tiny magnetic fields and temperatures. Misha’s group applies these quantum sensors to biology problems.
For example, different cells in an embryo divide at different times. Imagine reversing the order in which the cells divide. Would the reversal harm the organism? You could find out by manipulating the temperatures in different parts of the embryo: Temperature controls the rate at which cells divide.
Misha’s team injected nanoscale diamonds into a worm embryo. (See this paper for a related study.) The diamonds reported the temperature at various points in the worm. This information guided experimentalists who heated the embryo with lasers.
The manipulated embryos grew into fairly normal adults. But their cells, and their descendants’ cells, cycled through the stages of life slowly. This study exemplified, to me, one of the most meaningful opportunities for quantum physicists interested in biology: to develop technologies and analyses that can answer biology questions.
I mentioned, in an earlier blog post, another avenue emerging in quantum biology: Physicist Matthew Fisher proposed a mechanism by which entanglement might enhance coordinated neuron firing. My collaborator Elizabeth Crosson and I analyzed how the molecules in Matthew’s proposal—Posner clusters—could process quantum information. The field of Posner quantum biology had a population of about two, when Elizabeth and I entered, and I wondered whether anyone would join us.
The conference helped resolve my uncertainty. Three speakers (including me) presented work based on Matthew’s; two other participants were tilling the Posner soil; and another speaker mentioned Matthew’s proposal. The other two Posner talks related data from three experiments. The experimentalists haven’t finished their papers, so I won’t share details. But stay tuned.
Clarice and Thorsten’s conference reminded me of a conference I’d participated in at the end of my PhD: Last month, I moonlighted as a quantum biologist. In 2017, I moonlighted as a quantum-gravity theorist. Two years earlier, I’d been dreaming about black holes and space-time. At UCLA, I was finishing the first paper I’ve coauthored with biophysicists. What a toolkit quantum information theory and thermodynamics provide, that it can unite such disparate fields.
The contrast—on top of what I learned at UCLA—filled my mind for weeks. And reminded me of the description of asexual reproduction that we heard from the conference’s flatworm enthusiast. According to Western Michigan University’s Wendy Beane, a flatworm “glues its butt down, pops its head off, and grows a new one. Y’know. As one does.”
I hope I never flinch from popping my head off and growing a new one—on my quantum-information-thermodynamics spine—whenever new science calls for figuring out.
With thanks to Clarice, Thorsten, and UCLA for their invitation and hospitality.
A year ago, the “I’m a little teapot” song kept playing in my head.
I was finishing a collaboration with David Limmer, a theoretical chemist at the University of California Berkeley. David studies quantum and classical systems far from equilibrium, including how these systems exchange energy and information with their environments. Example systems include photoisomers.
Your average-Joe photoisomer spends much of its life in equilibrium, exchanging heat with room-temperature surroundings. The molecule has the shape above, called the cis configuration. Imagine shining a laser or sunlight on the photoisomer. The molecule can absorb a photon, or particle of light, gaining energy. The energized switch has the opportunity to switch: One chemical group can rotate downward. The molecule will occupy its trans configuration.
The molecule now has more energy than it had while equilibrium, albeit less energy than it had right after absorbing the photon. The molecule can remain in this condition for a decent amount of time. (Experts: The molecule occupies a metastable state.) That is, the molecule can store sunlight. For that reason, experimentalists at Harvard and MIT attached photoisomers to graphene nanotubules, improving the nanotubules’ storage of solar fuel.
With what probability does a photoisomer switch upon absorbing a photon? This question has resisted easy answering, because photoisomers prove difficult to model: They’re small, quantum, and far from equilibrium. People have progressed by making assumptions, but such assumptions can lack justifications or violate physical principles. David wanted to derive a simple, general bound—of the sort in which thermodynamicists specialize—on a photoisomer’s switching probability.
He had a hunch as to how he could derive such a bound. I’veblogged,manytimes,about thermodynamic resource theories. Thermodynamic resource theories are simple models, developed in quantum information theory, for exchanges of heat, particles, information, and more. These models involve few assumptions: the conservation of energy, quantum theory, and, to some extent, the existence of a large environment (Markovianity). With such a model, David suspected, he might derive his bound.
I knew nothing about photoisomers when I met David, but I knew about thermodynamic resource theories. I’d contributed to their development, to the theorems that have piled up in the resource-theory corner of quantum information theory. Then, the corner had given me claustrophobia. Those theorems felt so formal, abstract, and idealized. Formal, abstract theory has drawn me ever since I started studying physics in college. But did resource theories model physical reality? Could they impact science beyond our corner of quantum information theory? Did resource theories matter?
I called for connecting thermodynamic resource theories to physical reality four years ago, in a paper that begins with an embarrassing story about me. Resource theorists began designing experiments whose results should agree with our theorems. Theorists also tried to improve the accuracy with which resource theories model experimentalists’ limitations. See David’s and my paper for a list of these achievements. They delighted me, as a step toward the broadening of resource theories’ usefulness.
Like any first step, this step pointed toward opportunities. Experiments designed to test our theorems essentially test quantum mechanics. Scientists have tested quantum mechanics for decades; we needn’t test it much more. Such experimental proposals can push experimentalists to hone their abilities, but I hoped that the community could accomplish more. We should be able to apply resource theories to answer questions cultivated in other fields, such as condensed matter and chemistry. We should be useful to scientists outside our corner of quantum information.
David’s idea lit me up like photons on a solar-fuel-storage device. He taught me about photoisomers, I taught him about resource theories, and we derived his bound. Our proof relies on the “second laws of thermodynamics.” These abstract resource-theory results generalize the second law of thermodynamics, which helps us understand why time flows in only one direction. We checked our bound against numerical simulations (experts: of Lindbladian evolution). Our bound is fairly tight if the photoisomer has a low probability of absorbing a photon, as in the Harvard-MIT experiment.
Experts: We also quantified the photoisomer’s coherences relative to the energy eigenbasis. Coherences can’t boost the switching probability, we concluded. But, en route to this conclusion, we found that the molecule is a natural realization of a quantum clock. Our quantum-clock modeling extends to general dissipative Landau-Zener transitions, prevalent across condensed matter and chemistry.
As I worked on our paper one day, a jingle unfolded in my head. I recognized the tune first: “I’m a little teapot.” I hadn’t sung that much since kindergarten, I realized. Lyrics suggested themselves:
I’m a little isomer with two hands.
Here is my cis pose; here is my trans.
Stand me in the sunlight; watch me spin.
I’ll keep solar energy in!
The song lodged itself in my head for weeks. But if you have to pay an earworm to collaborate with David, do.
Imagine that you earned tenure and your field’s acclaim decades ago. Perhaps you received a Nobel Prize. Perhaps you’re directing an institute for science that you helped invent. Do you still do science? Does mentoring youngsters, advising the government, raising funds, disentangling logistics, presenting keynote addresses at conferences, chairing committees, and hosting visitors dominate the time you dedicate to science? Or do you dabble, attend seminars, and read, following progress without spearheading it?
People have asked whether my colleagues do science when weighed down with laurels. The end of August illustrates my answer.
At the end of August, I participated in the eighth Conference on Quantum Information and Quantum Control (CQIQC) at Toronto’s Fields Institute. CQIQC bestows laurels called “the John Stewart Bell Prize” on quantum-information scientists. John Stewart Bell revolutionized our understanding of entanglement, strong correlations that quantum particles can share and that power quantum computing. Aephraim Steinberg, vice-chair of the selection committee, bestowed this year’s award. The award, he emphasized, recognizes achievements accrued during the past six years. This year’s co-winners have been leading quantum information theory for decades. But the past six years earned the winners their prize.
Peter Zoller co-helms IQOQI in Innsbruck. (You can probably guess what the acronym stands for. Hint: The name contains “Quantum” and “Institute.”) Ignacio Cirac is a director of the Max Planck Institute of Quantum Optics near Munich. Both winners presented recent work about quantum many-body physics at the conference. You can watch videos of their talks here.
Peter discussed how a lab in Austria and a lab across the world can check whether they’ve prepared the same quantum state. One lab might have trapped ions, while the other has ultracold atoms. The experimentalists might not know which states they’ve prepared, and the experimentalists might have prepared the states at different times. Create multiple copies of the states, Peter recommended, measure the copies randomly, and play mathematical tricks to calculate correlations.
Ignacio expounded upon how to simulate particle physics on a quantum computer formed from ultracold atoms trapped by lasers. For expert readers: Simulate matter fields with fermionic atoms and gauge fields with bosonic atoms. Give the optical lattice the field theory’s symmetries. Translate the field theory’s Lagrangian into Hamiltonian language using Kogut and Susskind’s prescription.
Even before August, I’d collected an arsenal of seasoned scientists who continue to revolutionize their fields. Frank Wilczek shared a physics Nobel Prize for theory undertaken during the 1970s. He and colleagues helped explain matter’s stability: They clarified how close-together quarks (subatomic particles) fail to attract each other, though quarks draw together when far apart. Why stop after cofounding one subfield of physics? Frank spawned another in 2012. He proposed the concept of a time crystal, which is like table salt, except extended across time instead of across space. Experimentalistsrealized a variation on Frank’s prediction in 2018, and time crystals have exploded across the scientific literature.1
Rudy Marcus is 96 years old. He received a chemistry Nobel Prize, for elucidating how electrons hop between molecules during reactions, in 1992. I took a nonequilibrium-statistical-mechanics course from Rudy four years ago. Ever since, whenever I’ve seen him, he’s asked for the news in quantum information theory. Rudy’s research group operates at Caltech, and you won’t find “Emeritus” in the title on his webpage.
My PhD supervisor, John Preskill, received tenure at Caltech for particle-physics research performed before 1990. You might expect the rest of his career to form an afterthought. But he helped establish quantum computing, starting in the mid-1990s. During the past few years, he co-midwifed the subfield of holographic quantum information theory, which concerns black holes, chaos, and the unification of quantum theory with general relativity. Watching a subfield emerge during my PhD left a mark like a tree on a bicyclist (or would have, if such a mark could uplift instead of injure). John hasn’t helped create subfields only by garnering resources and encouraging youngsters. Several papers by John and collaborators—about topological quantum matter,black holes,quantum error correction, and more—have transformed swaths of physics during the past 15 years. Nor does John stamp his name on many papers: Most publications by members of his group don’t list him as a coauthor.
Do my colleagues do science after laurels pile up on them? The answer sounds to me, in many cases, more like a roar than like a “yes.” Much science done by senior scientists inspires no less than the science that established them. Beyond their results, their enthusiasm inspires. Never mind receiving a Bell Prize. Here’s toworking toward deserving a Bell Prize every six years.
With thanks to the Fields Institute, the University of Toronto, Daniel F. V. James, Aephraim Steinberg, and the rest of the conference committee for their invitation and hospitality.
You can find videos of all the conference’s talks here. My talk is shown here.
1To scientists, I recommend thisPhysics Today perspective on time crystals. Few articles have awed and inspired me during the past year as much as this review did.
I was working with Tony Bartolotta, a PhD student in theoretical physics at Caltech, and Jason Pollack, a postdoc in cosmology at the University of British Columbia. They acted as the souls of consideration. We missed out on dozens of opportunities to bicker—about the paper’s focus, who undertook which tasks, which journal to submit to, and more. Bickering would have spiced up the story behind our paper, because the paper concerns disagreement.
Quantum observables can disagree. Observables are measurable properties, such as position and momentum. Suppose that you’ve measured a quantum particle’s position and obtained an outcome . If you measure the position immediately afterward, you’ll obtain again. Suppose that, instead of measuring the position again, you measure the momentum. All the possible outcomes have equal probabilities of obtaining. You can’t predict the outcome.
The particle’s position can have a well-defined value, or the momentum can have a well-defined value, but the observables can’t have well-defined values simultaneously. Furthermore, if you measure the position, you randomize the outcome of a momentum measurement. Position and momentum disagree.
How should we quantify the disagreement of two quantum observables, and ? The question splits physicists into two camps. Pure quantum information (QI) theorists use uncertainty relations, whereas condensed-matter and high-energy physicists prefer out-of-time-ordered correlators. Let’s meet the camps in turn.
Heisenberg intuited an uncertainty relation that Robertson formalized during the 1920s,
Imagine preparing a quantum state and measuring , then repeating this protocol in many trials. Each trial has some probability of yielding the outcome . Different trials will yield different ’s. We quantify the spread in values with the standard deviation . We define analogously. denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does.
denotes the observables’ commutator. The numbers that we use in daily life commute: . Quantum numbers, or operators, represent and . Operators don’t necessarily commute. The commutator represents how little and resemble 7 and 5.
Robertson’s uncertainty relation means, “If you can predict an measurement’s outcome precisely, you can’t predict a measurement’s outcome precisely, and vice versa. The uncertainties must multiply to at least some number. The number depends on how much fails to commute with .” The higher an uncertainty bound (the greater the inequality’s right-hand side), the more the operators disagree.
Heisenberg and Robertson explored operator disagreement during the 1920s. They wouldn’t have seen eye to eye with today’s QI theorists. For instance, QI theorists consider how we can apply quantum phenomena, such as operator disagreement, to information processing. Information processing includes cryptography. Quantum cryptography benefits from operator disagreement: An eavesdropper must observe, or measure, a message. The eavesdropper’s measurement of one observable can “disturb” a disagreeing observable. The message’s sender and intended recipient can detect the disturbance and so detect the eavesdropper.
How efficiently can one perform an information-processing task? The answer usually depends on an entropy , a property of quantum states and of probability distributions.Uncertainty relations cry out for recasting in terms of entropies. So QI theorists have devised entropic uncertainty relations, such as
The entropy quantifies the difficulty of predicting the outcome of an measurement. is defined analogously. is called the overlap. It quantifies your ability to predict what happens if you prepare your system with a well-defined value, then measure . For further analysis, check out this paper. Entropic uncertainty relations have blossomed within QI theory over the past few years.
Pure QI theorists, we’ve seen, quantify operator disagreement with entropic uncertainty relations. Physicists at the intersection of condensed matter and high-energy physics prefer out-of-time-ordered correlators (OTOCs). I’ve blogged about OTOCs so many times, Quantum Frontiers regulars will be able to guess the next two paragraphs.
Consider a quantum many-body system, such as a chain of qubits. Imagine poking one end of the system, such as by flipping the first qubit upside-down. Let the operator represent the poke. Suppose that the system evolves chaotically for a time afterward, the qubits interacting.Information about the poke spreads through many-body entanglement, or scrambles.
Imagine measuring an observable of a few qubits far from the qubits. A little information about migrates into the qubits. But measuring reveals almost nothing about , because most of the information about has spread across the system. disagrees with , in a sense. Actually, disagrees with . The represents the time evolution.
The OTOC’s smallness reflects how much disagrees with at any instant . At early times , the operators agree, and the OTOC . At late times, the operators disagree loads, and the OTOC .
Different camps of physicists, we’ve seen, quantify operator disagreement with different measures: Today’s pure QI theorists use entropic uncertainty relations. Condensed-matter and high-energy physicists use OTOCs. Trust physicists to disagree about what “quantum operator disagreement” means.
I want peace on Earth. I conjectured, in 2016 or so, that one could reconcile the two notions of quantum operator disagreement. One must be able to prove an entropic uncertainty relation for scrambling, wouldn’t you think?
You might try substituting for the in Ineq. , and for the . You’d expect the uncertainty bound to tighten—the inequality’s right-hand side to grow—when the system scrambles. Scrambling—the condensed-matter and high-energy-physics notion of disagreement—would coincide with a high uncertainty bound—the pure-QI-theory notion of disagreement. The two notions of operator disagreement would agree. But the bound I’ve described doesn’t reflect scrambling. Nor do similar bounds that I tried constructing. I banged my head against the problem for about a year.
The sky brightened when Jason and Tony developed an interest in the conjecture. Their energy and conversation enabled us to prove an entropic uncertainty relation for scrambling, published this month.1 We tested the relation in computer simulations of a qubit chain. Our bound tightens when the system scrambles, as expected: The uncertainty relation reflects the same operator disagreement as the OTOC. We reconciled two notions of quantum operator disagreement.
As Quantum Frontiers regulars will anticipate, our uncertainty relation involves weak measurements and quasiprobability distributions: I’ve been studying their roles in scrambling over the pastthreeyears,with colleagues for whose collaborations I have the utmost gratitude. I’m grateful to have collaborated with Tony and Jason. Harmony helps when you’re tackling (quantum operator) disagreement—even if squabbling would spice up your paper’s backstory.
Video games have been a part of my life for about as long as I can remember. From Paperboy and The Last Ninja on the Commodore 64 when I was barely old enough to operate a keyboard, to Mario Kart 8 and Zelda on the Nintendo Switch, as a postdoc at Caltech, working on quantum computing and condensed matter physics. Up until recently, I have kept my two lives separate: my love of video games and my career in quantum physics.
The realization that I could combine quantum physics with games came during an entertaining discussion with my current supervisor, Gil Refael. Gil and I were brainstorming approaches to develop a quantum version of Tetris. Instead of stopping and laughing it off, or even keeping the idea on the horizon, Gil suggested that we talk to Spyridon (Spiros) Michalakis for some guidance.
This is not the story of Quantum Tetris (yet), but rather the story of how we made a quantum version of a much older, and possibly more universally known game. This is a new game that Spiros and myself have been testing at elementary schools.
And so I am super excited to be able to finally present to you: Quantum TiqTaqToe! As of right now, the app is available both for Android devices and iPhone/iPad:
Previous quantum games
Gil and I knew that Spiros had been involved in prior quantum games (most notably qCraft and Quantum Chess), so he seemed like the perfect contact point. He was conveniently located on the same campus, and even in the same department. But more importantly, he was curious about the idea and eager to talk.
After introducing the idea of Quantum Tetris, Spiros came up with an alternative approach. Seeing as this was going to be my first attempt at creating a video game, not to mention building a game from the ground up with quantum physics, he proposed to put me in touch with Chris Cantwell and help him improve the AI for Quantum Chess.
I thought long and hard about this proposition. Like five seconds. It was an amazing opportunity. I would get to look under the hood of a working and incredibly sophisticated video game, unlike any game ever made: the only game in the world I knew of that was truly based on quantum physics. And I would be solving a critical problem that I would have to deal with eventually, by adapting a conventional, classical rules-based game AI for quantum.
Fun and Games
My first focus was to jump on Quantum Chess full-force, with the aim of helping Chris implement a new AI player for the game. After evaluating some possible chess-playing AI engines, including state-of-the-art players based off of Google’s AlphaZero, we landed on Stockfish as our best candidate for integration. The AI is currently hot-swappable though, so users can try to develop their own!
While some of the work for implementing the AI could be done directly using Chris’s C++ implementation of Quantum Chess, other aspects of the work required me to learn the program he had used to develop the user interface. That program is called Unity. Unity is a free game development program that I would highly recommend trying out and playing around with.
This experience was essential to the birth of Quantum TiqTaqToe. In my quest to understand Unity and Quantum Games, I set out to implement a “simple” game to get a handle on how all the different game components worked together. Having a game based on quantum mechanics is one thing; making sure it is fun to play requires an entirely different skill set.
Classic Tic-Tac-Toe is a game in which two players, called X and O, take turns in placing their symbols on a 3×3 grid. The first player to get 3 of their symbols in a line (diagonally, vertically or horizontally) wins. The game goes as far back as ancient Egypt, and evidence of the game has been found on roof tiles dating to 1300 BC .
Many variations of the game have existed across many cultures. The first print reference to a game called “tick-tack-toe” was in 1884. In the US the game was renamed “tic-tac-toe” sometime in the 20th century. Here’s a random fun fact: in Dutch, the game is most often referred to as “Butter-Cheese-and-Eggs” . In 1952, computer scientist Alexander S. Douglas at the University of Cambridge turned it into one of the first computer games, featuring an AI player that could play perfect games against a human opponent.
Combinatorics has determined that whoever plays first will win 91 out of 138 possible board combinations. The second player will win in 44 boards. However, if both players play optimally, looking ahead through all the possible future outcomes, neither player should ever win and the game always ends in a draw, in one of only 3 board combinations.
In Quantum TiqTaqToe, with the current ruleset, we don’t yet know if a winning strategy exists.
I explicitly refer to the current ruleset because we currently limit the amount of quantumness in the game. We want to make sure the game is fun to play and ‘graspable’ for now. In addition, it turns out there already is a game called Quantum TicTacToe, developed by Allan Goff . That version of TicTacToe has similar concepts but has a different set of rules.
A typical game of Quantum TiqTaqToe will look very much like regular Tic-Tac-Toe until one of the players decides to make a quantum move:
At this point, the game board enters into a superposition. The X is in each position with 50/50 chance; in one universe the X is on the left and in the other it is on the right. Neither player knows how things will play out. And the game only gets more interesting from here. The opponent can choose to place their O in a superposition between an empty square and a square occupied by a quantum X.
Et voilà, player O has entangled his fate with his opponent’s. Once the two squares become entangled, the only outcomes are X-O or O-X, each with probability ½. Interestingly, since the game is fully quantum, the phase between the two entangled outcomes can in principle be leveraged to create interesting plays through destructive and constructive interference. The app features a simple tutorial (to be updated) that teaches you these moves and a few others. There are boards that classically result in a draw but are quantumly “winnable”.
A quick note on the quantumness
The squares in TiqTaqToe are all fully quantum. I represent them as qutrits (like qubits, but instead of having states 0 and 1 my qutrits have states 0, 1 and 2), and moves made by the players are unitary operations acting on them. So the game consists of these essential elements:
The squares of the 3×3 grid are turned into qutrits (Empty, X, O). Each move is a unitary gate operation on those qutrits. I’ll leave the details of the math out, but for the case of qubits check out Chris’ detailed writeup on Quantum Chess .
Quantum TiqTaqToe allows you to select two squares in the grid, providing you with the option of creating a superposition or an entangled state. For the sake of simplicity (i.e. keeping the game fun to play and ‘graspable’ for now), no more than 3 squares can be involved in a given entangled state.
I chose to explicitly track sets of qutrits that share a Hilbert space. The entire quantum state of the game combines these sets with classical strings of the form “XEEOXEOXE”, indicating that the first square is an X, the second is Empty, etc.
Victory in the multiverse
So, when does the game end if these quantum states are in play? In Quantum TiqTaqToe, the board collapses to a single classical state as soon as it is full (i.e. every square is non-empty). The resulting state is randomly chosen from all the possible outcomes, with a probability that is equal to the (square of the) wave-function amplitude (basic quantum mechanics). If there is a winner after the collapse, the game ends. Otherwise, the game continues until either there is a winner or until there are no more moves to be made (ending in a draw). On top of this, players get the option to forfeit their move for the opportunity to cause a partial collapse of the state, by using the collapse-mode. Future versions may include other ways of collapse, including one that does not involve rolling dice! 
Can you beat the quantum AI?
Due to quantum physics and the collapse of the state, the game is inherently statistical. So instead of asking: “Can I beat my opponent in a game of Quantum TiqTaqToe?” one should ask “If I play 100 games against my opponent, can I consistently win more than 50 of them?”
You can test your skill against the in-game quantum AI to see if you’ve indeed mastered Quantum TiqTaqToe yet. At the hardest setting, winning even 30% of the time after, say, 20 games may be extraordinary. The implementation of this AI, by the way, would have been a blog-post by itself. For the curious, I can say it is based on the ExpectiMiniMax algorithm. As of the moment of this writing, the hardest AI setting is not available in the app yet. Keep your eyes out for an update soon though!
“Perhaps kids who grow up playing quantum games will acquire a visceral understanding of quantum phenomena that our generation lacks.” – John Preskill, in his recent article .
From the get-go, Quantum TiqTaqToe (and Quantum Chess) have had outreach as a core motivation. Perhaps future quantum engineers and quantum programmers will look back on their youth and remember playing Quantum TiqTaqToe as I remember my Commodore 64 games. I am convinced that these small steps into the realm of Quantum Games are only just the beginning of an entirely new genre of fun and useful games.
In the meantime, we are hard at work implementing an Online mode so you can play with your fellow human friends remotely too. This online mode, plus the option of fighting a strong quantum AI, will be unlockable in-game through a small fee (unless you are an educator who wishes to introduce quantum physics in class through this game; those use cases are fee-free courtesy of IQIM and NSF). Each purchase will go towards supporting the future development of exciting new Quantum TiqTaqToe features, as well as other exciting Quantum Games (Tetris, anyone?)
Just in case you missed it: the app is available both for Android devices and iPhone/iPad right now:
I really hope you enjoy the game, and perhaps use it to get your friends and family excited about quantum physics. Oh, and start practicing! You never know if the online mode will bring along with it a real Quantum TiqTaqToe Tournament down the road 😉
 The origin of this name in Dutch isn’t really certain as far as I know. Alledgedly, it is a left-over from the period in which butter, cheese and eggs were sold at the door (so was milk, but that was done separately since it was sold daily). The salesman had a list with columns for each of these three products, and would jot down a cross or a zero whenever a customer at an address bought or declined a product. Three crosses in a row would earn them praise from the boss.