Category Archives: Real science

Spotlight on research performed by teachers and students as part of our outreach efforts.

Let gravity do its work

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One day, early this spring, I found myself in a hotel elevator with three other people. The cohort consisted of two theoretical physicists, one computer scientist, and what appeared to be a normal person. I pressed the elevator’s 4 button, as my husband (the computer scientist) and I were staying on the hotel’s fourth floor. The button refused to light up.

“That happened last time,” the normal person remarked. He was staying on the fourth floor, too.

The other theoretical physicist pressed the 3 button.

“Should we press the 5 button,” the normal person continued, “and let gravity do its work?”

I took a moment to realize that he was suggesting we ascend to the fifth floor and then induce the elevator to fall under gravity’s influence to the fourth. We were reaching floor three, so I exchanged a “have a good evening” with the other physicist, who left. The door shut, and we began to ascend.

“As it happens,” I remarked, “he’s an expert on gravity.” The other physicist was Herman Verlinde, a professor at Princeton.

Such is a side effect of visiting the Simons Center for Geometry and Physics. The Simons Center graces the Stony Brook University campus, which was awash in daffodils and magnolia blossoms last month. The Simons Center derives its name from hedge-fund manager Jim Simons (who passed away during the writing of this article). He achieved landmark physics and math research before earning his fortune on Wall Street as a quant. Simons supported his early loves by funding the Simons Center and other scientific initiatives. The center reminded me of the Perimeter Institute for Theoretical Physics, down to the café’s linen napkins, so I felt at home.

I was participating in the Simons Center workshop “Entanglement, thermalization, and holography.” It united researchers from quantum information and computation, black-hole physics and string theory, quantum thermodynamics and many-body physics, and nuclear physics. We were to share our fields’ approaches to problems centered on thermalization, entanglement, quantum simulation, and the like. I presented about the eigenstate thermalization hypothesis, which elucidates how many-particle quantum systems thermalize. The hypothesis fails, I argued, if a system’s dynamics conserve quantities (analogous to energy and particle number) that can’t be measured simultaneously. Herman Verlinde discussed the ER=EPR conjecture.

My PhD advisor, John Preskill, blogged about ER=EPR almost exactly eleven years ago. Read his blog post for a detailed introduction. Briefly, ER=EPR posits an equivalence between wormholes and entanglement. 

The ER stands for Einstein–Rosen, as in Einstein–Rosen bridge. Sean Carroll provided the punchiest explanation I’ve heard of Einstein–Rosen bridges. He served as the scientific advisor for the 2011 film Thor. Sean suggested that the film feature a wormhole, a connection between two black holes. The filmmakers replied that wormholes were passé. So Sean suggested that the film feature an Einstein–Rosen bridge. “What’s an Einstein–Rosen bridge?” the filmmakers asked. “A wormhole.” So Thor features an Einstein–Rosen bridge.

EPR stands for Einstein–Podolsky–Rosen. The three authors published a quantum paradox in 1935. Their EPR paper galvanized the community’s understanding of entanglement.

ER=EPR is a conjecture that entanglement is closely related to wormholes. As Herman said during his talk, “You probably need entanglement to realize a wormhole.” Or any two maximally entangled particles are connected by a wormhole. The idea crystallized in a paper by Juan Maldacena and Lenny Susskind. They drew on work by Mark Van Raamsdonk (who masterminded the workshop behind this Quantum Frontiers post) and Brian Swingle (who’s appeared in further posts).

Herman presented four pieces of evidence for the conjecture, as you can hear in the video of his talk. One piece emerges from the AdS/CFT duality, a parallel between certain space-times (called anti–de Sitter, or AdS, spaces) and quantum theories that have a certain symmetry (called conformal field theories, or CFTs). A CFT, being quantum, can contain entanglement. One entangled state is called the thermofield double. Suppose that a quantum system is in a thermofield double and you discard half the system. The remaining half looks thermal—we can attribute a temperature to it. Evidence indicates that, if a CFT has a temperature, then it parallels an AdS space that contains a black hole. So entanglement appears connected to black holes via thermality and temperature.

Despite the evidence—and despite the eleven years since John’s publication of his blog post—ER=EPR remains a conjecture. Herman remarked, “It’s more like a slogan than anything else.” His talk’s abstract contains more hedging than a suburban yard. I appreciated the conscientiousness, a college acquaintance having once observed that I spoke carefully even over sandwiches with a friend.

A “source of uneasiness” about ER=EPR, to Herman, is measurability. We can’t check whether a quantum state is entangled via any single measurement. We have to prepare many identical copies of the state, measure the copies, and process the outcome statistics. In contrast, we seem able to conclude that a space-time is connected without measuring multiple copies of the space-time. We can check that a hotel’s first floor is connected to its fourth, for instance, by riding in an elevator once.

Or by riding an elevator to the fifth floor and descending by one story. My husband, the normal person, and I took the stairs instead of falling. The hotel fixed the elevator within a day or two, but who knows when we’ll fix on the truth value of ER=EPR?

With thanks to the conference organizers for their invitation, to the Simons Center for its hospitality, to Jim Simons for his generosity, and to the normal person for inspiration.

The rain in Portugal

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My husband taught me how to pronounce the name of the city where I’d be presenting a talk late last July: Aveiro, Portugal. Having studied Spanish, I pronounced the name as Ah-VEH-roh, with a v partway to a hard b. But my husband had studied Portuguese, so he recommended Ah-VAI-roo. 

His accuracy impressed me when I heard the name pronounced by the organizer of the conference I was participating in—Theory of Quantum Computation, or TQC. Lídia del Rio grew up in Portugal and studied at the University of Aveiro, so I bow to her in matters of Portuguese pronunciation. I bow to her also for organizing one of the world’s largest annual quantum-computation conferences (with substantial help—fellow quantum physicist Nuriya Nurgalieva shared the burden). But Lídia cofounded Quantum, a journal that’s risen from a Gedankenexperiment to a go-to venue in six years. So she gives the impression of being able to manage anything.

Aveiro architecture

Watching Lídia open TQC gave me pause. I met her in 2013, the summer before beginning my PhD at Caltech. She was pursuing her PhD at ETH Zürich, which I was visiting. Lídia took me dancing at an Argentine-tango studio one evening. Now, she’d invited me to speak at an international conference that she was coordinating.

Lídia and me in Zürich as PhD students
Lídia opening TQC

Not only Lídia gave me pause; so did the three other invited speakers. Every one of them, I’d met when each of us was a grad student or a postdoc. 

Richard Küng described classical shadows, a technique for extracting information about quantum states via measurements. Suppose we wish to infer about diverse properties of a quantum state \rho (about diverse observables’ expectation values). We have to measure many copies of \rho—some number n of copies. The community expected n to grow exponentially with the system’s size—for instance, with the number of qubits in a quantum computer’s register. We can get away with far fewer, Richard and collaborators showed, by randomizing our measurements. 

Richard postdocked at Caltech while I was a grad student there. Two properties of his stand out in my memory: his describing, during group meetings, the math he’d been exploring and the Austrian accent in which he described that math.

Did this restaurant’s owners realize that quantum physicists were descending on their city? I have no idea.

Also while I was a grad student, Daniel Stilck França visited Caltech. Daniel’s TQC talk conveyed skepticism about whether near-term quantum computers can beat classical computers in optimization problems. Near-term quantum computers are NISQ (noisy, intermediate-scale quantum) devices. Daniel studied how noise (particularly, local depolarizing noise) propagates through NISQ circuits. Imagine a quantum computer suffering from a 1% noise error. The quantum computer loses its advantage over classical competitors after 10 layers of gates, Daniel concluded. Nor does he expect error mitigation—a bandaid en route to the sutures of quantum error correction—to help much.

I’d coauthored a paper with the fourth invited speaker, Adam Bene Watts. He was a PhD student at MIT, and I was a postdoc. At the time, he resembled the 20th-century entanglement guru John Bell. Adam still resembles Bell, but he’s moved to Canada.

Adam speaking at TQC
From a 2021 Quantum Frontiers post of mine. I was tickled to see that TQC’s organizers used the photo from my 2021 post as Adam’s speaker photo.

Adam distinguished what we can compute using simple quantum circuits but not using simple classical ones. His results fall under the heading of complexity theory, about which one can rarely prove anything. Complexity theorists cling to their jobs by assuming conjectures widely expected to be true. Atop the assumptions, or conditions, they construct “conditional” proofs. Adam proved unconditional claims in complexity theory, thanks to the simplicity of the circuits he compared.

In my estimation, the talks conveyed cautious optimism: according to Adam, we can prove modest claims unconditionally in complexity theory. According to Richard, we can spare ourselves trials while measuring certain properties of quantum systems. Even Daniel’s talk inspired more optimism than he intended: a few years ago, the community couldn’t predict how noisy short-depth quantum circuits could perform. So his defeatism, rooted in evidence, marks an advance.

Aveiro nurtures optimism, I expect most visitors would agree. Sunshine drenches the city, and the canals sparkle—literally sparkle, as though devised by Elsa at a higher temperature than usual. Fresh fruit seems to wend its way into every meal.1 Art nouveau flowers scale the architecture, and fanciful designs pattern the tiled sidewalks.

What’s more, quantum information theorists of my generation were making good. Three riveted me in their talks, and another co-orchestrated one of the world’s largest quantum-computation gatherings. To think that she’d taken me dancing years before ascending to the global stage.

My husband and I made do, during our visit, by cobbling together our Spanish, his Portuguese, and occasional English. Could I hold a conversation with the Portuguese I gleaned? As adroitly as a NISQ circuit could beat a classical computer. But perhaps we’ll return to Portugal, and experimentalists are doubling down on quantum error correction. I remain cautiously optimistic.

1As do eggs, I was intrigued to discover. Enjoyed a hardboiled egg at breakfast? Have a fried egg on your hamburger at lunch. And another on your steak at dinner. And candied egg yolks for dessert.

This article takes its title from a book by former US Poet Laureate Billy Collins. The title alludes to a song in the musical My Fair Lady, “The Rain in Spain.” The song has grown so famous that I don’t think twice upon hearing the name. “The rain in Portugal” did lead me to think twice—and so did TQC.

With thanks to Lídia and Nuriya for their hospitality. You can submit to TQC2024 here.

What geckos have to do with quantum computing

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When my brother and I were little, we sometimes played video games on weekend mornings, before our parents woke up. We owned a 3DO console, which ran the game Gex. Gex is named after its main character, a gecko. Stepping into Gex’s shoes—or toe pads—a player can clamber up walls and across ceilings. 

I learned this month how geckos clamber, at the 125th Statistical Mechanics Conference at Rutgers University. (For those unfamiliar with the field: statistical mechanics is a sibling of thermodynamics, the study of energy.) Joel Lebowitz, a legendary mathematical physicist and nonagenarian, has organized the conference for decades. This iteration included a talk by Kanupriya (Kanu) Sinha, an assistant professor at the University of Arizona. 

Kanu studies open quantum systems, or quantum systems that interact with environments. She often studies a particle that can be polarized. Such a particle carries an electric charge, which can be distributed unevenly across the particle. Examples include a water molecule. As encoded in its chemical symbol, H2O, a water molecule consists of two hydrogen atoms and one oxygen atom. The oxygen attracts the molecule’s electrons more strongly than the hydrogen atoms do. So the molecule’s oxygen end carries a negative charge, and the hydrogen ends carry positive charges.1

The red area represents the oxygen, and the gray areas represent the hydrogen atoms. Image from the American Chemical Society.

When certain quantum particles are polarized, we can control their positions using lasers. After all, a laser consists of light—an electromagnetic field—and electric fields influence electrically charged particles’ movements. This control enables optical tweezers—laser beams that can place certain polarizable atoms wherever an experimentalist wishes. Such atoms can form a quantum computer, as John Preskill wrote in a blog post on Quantum Frontiers earlier this month.

Instead of placing polarizable atoms in an array that will perform a quantum computation, you can place the atoms in an outline of the Eiffel Tower. Image from Antoine Browaeys’s lab.

A tweezered atom’s environment consists not only of a laser, but also everything else around, including dust particles. Undesirable interactions with the environment deplete an atom of its quantum properties. Quantum information stored in the atom leaks into the environment, threatening a quantum computer’s integrity. Hence the need for researchers such as Kanu, who study open quantum systems.

Kanu illustrated the importance of polarizable particles in environments, in her talk, through geckos. A gecko’s toe pads contain tiny hairs that polarize temporarily. The electric charges therein can be attracted to electric charges in a wall. We call this attraction the van der Waals force. So Gex can clamber around for a reason related to why certain atoms suit quantum computing.

Kanu explaining how geckos stick.

Winter break offers prime opportunities for kicking back with one’s siblings. Even if you don’t play Gex (and I doubt whether you do), behind your game of choice may lie more physics than expected.

1Water molecules are polarized permanently, whereas Kanu studies particles that polarize temporarily.

The power of awe

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Mid-afternoon, one Saturday late in September, I forgot where I was. I forgot that I was visiting Seattle for the second time; I forgot that I’d just finished co-organizing a workshop partially about nuclear physics for the first time. I’d arrived at a crowded doorway in the Chihuly Garden and Glass museum, and a froth of blue was towering above the onlookers in front of me. Glass tentacles, ranging from ultramarine through turquoise to clear, extended from the froth. Golden conch shells, starfish, and mollusks rode the waves below. The vision drove everything else from my mind for an instant.

Much had been weighing on my mind that week. The previous day had marked the end of a workshop hosted by the Inqubator for Quantum Simulation (IQuS, pronounced eye-KWISS) at the University of Washington. I’d co-organized the workshop with IQuS member Niklas Mueller, NIST physicist Alexey Gorshkov, and nuclear theorist Raju Venugopalanan (although Niklas deserves most of the credit). We’d entitled the workshop “Thermalization, from Cold Atoms to Hot Quantum Chromodynamics.” Quantum chromodynamics describes the strong force that binds together a nucleus’s constituents, so I call the workshop “Journey to the Center of the Atom” to myself. 

We aimed to unite researchers studying thermal properties of quantum many-body systems from disparate perspectives. Theorists and experimentalists came; and quantum information scientists and nuclear physicists; and quantum thermodynamicists and many-body physicists; and atomic, molecular, and optical physicists. Everyone cared about entanglement, equilibration, and what else happens when many quantum particles crowd together and interact. 

We quantum physicists crowded together and interacted from morning till evening. We presented findings to each other, questioned each other, coagulated in the hallways, drank tea together, and cobbled together possible projects. The week electrified us like a chilly ocean wave but also wearied me like an undertow. Other work called for attention, and I’d be presenting four more talks at four more workshops and campus visits over the next three weeks. The day after the workshop, I worked in my hotel half the morning and then locked away my laptop. I needed refreshment, and little refreshes like art.

Strongly interacting physicists

Chihuly Garden and Glass, in downtown Seattle, succeeded beyond my dreams: the museum drew me into somebody else’s dreams. Dale Chihuly grew up in Washington state during the mid-twentieth century. He studied interior design and sculpture before winning a Fulbright Fellowship to learn glass-blowing techniques in Murano, Italy. After that, Chihuly transformed the world. I’ve encountered glass sculptures of his in Pittsburgh; Florida; Boston; Jerusalem; Washington, DC; and now Seattle—and his reach dwarfs my travels. 

Chihuly chandelier at the Renwick Gallery in Washington, DC

After the first few encounters, I began recognizing sculptures as Chihuly’s before checking their name plates. Every work by his team reflects his style. Tentacles, bulbs, gourds, spheres, and bowls evidence what I never expected glass to do but what, having now seen it, I’m glad it does.

This sentiment struck home a couple of galleries beyond the Seaforms. The exhibit Mille Fiori drew inspiration from the garden cultivated by Chihuly’s mother. The name means A Thousand Flowers, although I spied fewer flowers than what resembled grass, toadstools, and palm fronds. Visitors feel like grasshoppers amongst the red, green, and purple stalks that dwarfed some of us. The narrator of Jules Vernes’s Journey to the Center of the Earth must have felt similarly, encountering mastodons and dinosaurs underground. I encircled the garden before registering how much my mind had lightened. Responsibilities and cares felt miles away—or, to a grasshopper, backyards away. Wonder does wonders.

Mille Fiori

Near the end of the path around the museum, a theater plays documentaries about Chihuly’s projects. The documentaries include interviews with the artist, and several quotes reminded me of the science I’d been trained to seek out: “I really wanted to take glass to its glorious height,” Chihuly said, “you know, really make something special.” “Things—pieces got bigger, pieces got taller, pieces got wider.” He felt driven to push art forms as large as the glass would permit his team. Similarly, my PhD advisor John Preskill encouraged me to “think big.” What physics is worth doing—what would create an impact?

How did a boy from Tacoma, Washington impact not only fellow blown-glass artists—not only artists—not only an exhibition here and there in his home country—but experiences across the globe, including that of a physicist one weekend in September?

One idea from the IQuS workshop caught my eye. Some particle colliders accelerate heavy ions to high energies and then smash the ions together. Examples include lead and gold ions studied at CERN in Geneva. After a collision, the matter expands and cools. Nuclear physicists don’t understand how the matter cools; models predict cooling times longer than those observed. This mismatch has persisted across decades of experiments. The post-collision matter evades attempts at computer simulation; it’s literally a hot mess. Can recent advances in many-body physics help?

The exhibit Persian Ceiling at Chihuly Garden and Glass. Doesn’t it look like it could double as an artist’s rendering of a heavy-ion collision?

Martin Savage, the director of IQuS, hopes so. He hopes that IQuS will impact nuclear physics across the globe. Every university and its uncle boasts a quantum institute nowadays, but IQuS seems to me to have carved out a niche for itself. IQuS has grown up in the bosom of the Institute for Nuclear Theory at the University of Washington, which has guided nuclear theory for decades. IQuS is smashing that history together with the future of quantum simulators. IQuS doesn’t strike me as just another glass bowl in the kitchen of quantum science. A bowl worthy of Chihuly? I don’t know, but I’d like to hope so.

I left Chihuly Garden and Glass with respect for the past week and energy for the week ahead. Whether you find it in physics or in glass or in both—or in plunging into a dormant Icelandic volcano in search of the Earth’s core—I recommend the occasional dose of awe.

Participants in the final week of the workshop

With thanks to Martin Savage, IQuS, and the University of Washington for their hospitality.

Astrobiology meets quantum computation?

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The origin of life appears to share little with quantum computation, apart from the difficulty of achieving it and its potential for clickbait. Yet similar notions of complexity have recently garnered attention in both fields. Each topic’s researchers expect only special systems to generate high values of such complexity, or complexity at high rates: organisms, in one community, and quantum computers (and perhaps black holes), in the other. 

Each community appears fairly unaware of its counterpart. This article is intended to introduce the two. Below, I review assembly theory from origin-of-life studies, followed by quantum complexity. I’ll then compare and contrast the two concepts. Finally, I’ll suggest that origin-of-life scientists can quantize assembly theory using quantum complexity. The idea is a bit crazy, but, well, so what?

Assembly theory in origin-of-life studies

Imagine discovering evidence of extraterrestrial life. How could you tell that you’d found it? You’d have detected a bunch of matter—a bunch of particles, perhaps molecules. What about those particles could evidence life?

This question motivated Sara Imari Walker and Lee Cronin to develop assembly theory. (Most of my assembly-theory knowledge comes from Sara, about whom I wrote this blog post years ago and with whom I share a mentor.) Assembly theory governs physical objects, from proteins to self-driving cars. 

Imagine assembling a protein from its constituent atoms. First, you’d bind two atoms together. Then, you might bind another two atoms together. Eventually, you’d bind two pairs together. Your sequence of steps would form an algorithm for assembling the protein. Many algorithms can generate the same protein. One algorithm has the least number of steps. That number is called the protein’s assembly number.

Different natural processes tend to create objects that have different assembly numbers. Stars form low-assembly-number objects by fusing two hydrogen atoms together into helium. Similarly, random processes have high probabilities of forming low-assembly-number objects. For example, geological upheavals can bring a shard of iron near a lodestone. The iron will stick to the magnetized stone, forming a two-component object.

My laptop has an enormous assembly number. Why can such an object exist? Because of information, Sara and Lee emphasize. Human beings amassed information about materials science, Boolean logic, the principles of engineering, and more. That information—which exists only because organisms exists—helped engender my laptop.

If any object has a high enough assembly number, Sara and Lee posit, that object evidences life. Absent life, natural processes have too low a probability of randomly throwing together molecules into the shape of a computer. How high is “high enough”? Approximately fifteen, experiments by Lee’s group suggest. (Why do those experiments point to the number fifteen? Sara’s group is working on a theory for predicting the number.)

In summary, assembly number quantifies complexity in origin-of-life studies, according to Sara and Lee. The researchers propose that only living beings create high-assembly-number objects.

Quantum complexity in quantum computation

Quantum complexity defines a stage in the equilibration of many-particle quantum systems. Consider a clump of N quantum particles isolated from its environment. The clump will be in a pure quantum state | \psi(0) \rangle at a time t = 0. The particles will interact, evolving the clump’s state as a function  | \psi(t) \rangle. 

Quantum many-body equilibration is more complicated than the equilibration undergone by your afternoon pick-me-up as it cools.

The interactions will equilibrate the clump internally. One stage of equilibration centers on local observables O. They’ll come to have expectation values \langle \psi(t) | O | \psi(t) \rangle approximately equal to thermal expectation values {\rm Tr} ( O \, \rho_{\rm th} ), for a thermal state \rho_{\rm th} of the clump. During another stage of equilibration, the particles correlate through many-body entanglement. 

The longest known stage centers on the quantum complexity of | \psi(t) \rangle. The quantum complexity is the minimal number of basic operations needed to prepare | \psi(t) \rangle from a simple initial state. We can define “basic operations” in many ways. Examples include quantum logic gates that act on two particles. Another example is an evolution for one time step under a Hamiltonian that couples together at most k particles, for some k independent of N. Similarly, we can define “a simple initial state” in many ways. We could count as simple only the N-fold tensor product | 0 \rangle^{\otimes N} of our favorite single-particle state | 0 \rangle. Or we could call any N-fold tensor product simple, or any state that contains at-most-two-body entanglement, and so on. These choices don’t affect the quantum complexity’s qualitative behavior, according to string theorists Adam Brown and Lenny Susskind.

How quickly can the quantum complexity of | \psi(t) \rangle grow? Fast growth stems from many-body interactions, long-range interactions, and random coherent evolutions. (Random unitary circuits exemplify random coherent evolutions: each gate is chosen according to the Haar measure, which we can view roughly as uniformly random.) At most, quantum complexity can grow linearly in time. Random unitary circuits achieve this rate. Black holes may; they scramble information quickly. The greatest possible complexity of any N-particle state scales exponentially in N, according to a counting argument. 

A highly complex state | \psi(t) \rangle looks simple from one perspective and complicated from another. Human scientists can easily measure only local observables O. Such observables’ expectation values \langle \psi(t) | O | \psi(t) \rangle  tend to look thermal in highly complex states, \langle \psi(t) | O | \psi(t) \rangle \approx {\rm Tr} ( O \, \rho_{\rm th} ), as implied above. The thermal state has the greatest von Neumann entropy, - {\rm Tr} ( \rho \log \rho), of any quantum state \rho that obeys the same linear constraints as | \psi(t) \rangle (such as having the same energy expectation value). Probed through simple, local observables O, highly complex states look highly entropic—highly random—similarly to a flipped coin.

Yet complex states differ from flipped coins significantly, as revealed by subtler analyses. An example underlies the quantum-supremacy experiment published by Google’s quantum-computing group in 2018. Experimentalists initialized 53 qubits (quantum two-level systems) in a tensor product. The state underwent many gates, which prepared a highly complex state. Then, the experimentalists measured the z-component \sigma_z of each qubit’s spin, randomly obtaining a -1 or a 1. One trial yielded a 53-bit string. The experimentalists repeated this process many times, using the same gates in each trial. From all the trials’ bit strings, the group inferred the probability p(s) of obtaining a given string s in the next trial. The distribution \{ p(s) \} resembles the uniformly random distribution…but differs from it subtly, as revealed by a cross-entropy analysis. Classical computers can’t easily generate \{ p(s) \}; hence the Google group’s claiming to have achieved quantum supremacy/advantage. Quantum complexity differs from simple randomness, that difference is difficult to detect, and the difference can evidence quantum computers’ power.

A fridge that holds one of Google’s quantum computers.

Comparison and contrast

Assembly number and quantum complexity resemble each other as follows:

  1. Each function quantifies the fewest basic operations needed to prepare something.
  2. Only special systems (organisms) can generate high assembly numbers, according to Sara and Lee. Similarly, only special systems (such as quantum computers and perhaps black holes) can generate high complexity quickly, quantum physicists expect.
  3. Assembly number may distinguish products of life from products of abiotic systems. Similarly, quantum complexity helps distinguish quantum computers’ computational power from classical computers’.
  4. High-assembly-number objects are highly structured (think of my laptop). Similarly, high-complexity quantum states are highly structured in the sense of having much many-body entanglement.
  5. Organisms generate high assembly numbers, using information. Similarly, using information, organisms have created quantum computers, which can generate quantum complexity quickly.

Assembly number and quantum complexity differ as follows:

  1. Classical objects have assembly numbers, whereas quantum states have quantum complexities.
  2. In the absence of life, random natural processes have low probabilities of producing high-assembly-number objects. That is, randomness appears to keep assembly numbers low. In contrast, randomness can help quantum complexity grow quickly.
  3. Highly complex quantum states look very random, according to simple, local probes. High-assembly-number objects do not.
  4. Only organisms generate high assembly numbers, according to Sara and Lee. In contrast, abiotic black holes may generate quantum complexity quickly.

Another feature shared by assembly-number studies and quantum computation merits its own paragraph: the importance of robustness. Suppose that multiple copies of a high-assembly-number (or moderate-assembly-number) object exist. Not only does my laptop exist, for example, but so do many other laptops. To Sara, such multiplicity signals the existence of some stable mechanism for creating that object. The multiplicity may provide extra evidence for life (including life that’s discovered manufacturing), as opposed to an unlikely sequence of random forces. Similarly, quantum computing—the preparation of highly complex states—requires stability. Decoherence threatens quantum states, necessitating quantum error correction. Quantum error correction differs from Sara’s stable production mechanism, but both evidence the importance of robustness to their respective fields.

A modest proposal

One can generalize assembly number to quantum states, using quantum complexity. Imagine finding a clump of atoms while searching for extraterrestrial life. The atoms need not have formed molecules, so the clump can have a low classical assembly number. However, the clump can be in a highly complex quantum state. We could detect the state’s complexity only (as far as I know) using many copies of the state, so imagine finding many clumps of atoms. Preparing highly complex quantum states requires special conditions, such as a quantum computer. The clump might therefore evidence organisms who’ve discovered quantum physics. Using quantum complexity, one might extend the assembly number to identify quantum states that may evidence life. However, quantum complexity, or a high rate of complexity generation, alone may not evidence life—for example, if achievable by black holes. Fortunately, a black hole seems unlikely to generate many identical copies of a highly complex quantum state. So we seem to have a low probability of mistakenly attributing a highly complex quantum state, sourced by a black hole, to organisms (atop our low probability of detecting any complex quantum state prepared by anyone other than us).

Would I expect a quantum assembly number to greatly improve humanity’s search for extraterrestrial life? I’m no astrobiology expert (NASA videos notwithstanding), but I’d expect probably not. Still, astrobiology requires chemistry, which requires quantum physics. Quantum complexity seems likely to find applications in the assembly-number sphere. Besides, doesn’t juxtaposing the search for extraterrestrial life and the understanding of life’s origins with quantum computing sound like fun? And a sense of fun distinguishes certain living beings from inanimate matter about as straightforwardly as assembly number does.

With thanks to Jim Al-Khalili, Paul Davies, the From Physics to Life collaboration, and UCLA for hosting me at the workshop that spurred this article.

The Book of Mark, Chapter 2

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Late in the summer of 2021, I visited a physics paradise in a physical paradise: the Kavli Institute for Theoretical Physics (KITP). The KITP sits at the edge of the University of California, Santa Barbara like a bougainvillea bush at the edge of a yard. I was eating lunch outside the KITP one afternoon, across the street from the beach. PhD student Arman Babakhani, whom a colleague had just introduced me to, had joined me.

The KITP’s Kohn Hall

What physics was I working on nowadays? Arman wanted to know.

Thermodynamic exchanges. 

The world consists of physical systems exchanging quantities with other systems. When a rose blooms outside the Santa Barbara mission, it exchanges pollen with the surrounding air. The total amount of pollen across the rose-and-air whole remains constant, so we call the amount a conserved quantity. Quantum physicists usually analyze conservation of particles, energy, and magnetization. But quantum systems can conserve quantities that participate in uncertainty relations. Such quantities are called incompatible, because you can’t measure them simultaneously. The x-, y-, and z-components of a qubit’s spin are incompatible.

The Santa Barbara mission…
…and its roses

Exchanging and conserving incompatible quantities, systems can violate thermodynamic expectations. If one system is much larger than the other, we expect the smaller system to thermalize; yet incompatibility invalidates derivations of the thermal state’s form. Incompatibility reduces the thermodynamic entropy produced by exchanges. And incompatibility can raise the average amount entanglement in the pair of systems—the total system.

If the total system conserves incompatible quantities, what happens to the eigenstate thermalization hypothesis (ETH)? Last month’s blog post overviewed the ETH, a framework for understanding how quantum many-particle systems thermalize internally. That post labeled Mark Srednicki, a professor at the KITP, a high priest of the ETH. I want, I told Arman, to ask Mark what happens when you combine the ETH with incompatible conserved quantities.

I’ll do it, Arman said.

Soon after, I found myself in the fishbowl. High up in the KITP, a room filled with cushy seats overlooks the ocean. The circular windows lend the room its nickname. Arrayed on the armchairs and couches were Mark, Arman, Mark’s PhD student Fernando Iniguez, and Mark’s recent PhD student Chaitanya Murthy. The conversation went like this:

Mark was frustrated about not being able to answer the question. I was delighted to have stumped him. Over the next several weeks, the group continued meeting, and we emailed out notes for everyone to criticize. I particulary enjoyed watching Mark and Chaitanya interact. They’d grown so intellectually close throughout Chaitanya’s PhD studies, they reminded me of an old married couple. One of them had to express only half an idea for the other to realize what he’d meant and to continue the thread. Neither had any qualms with challenging the other, yet they trusted each other’s judgment.1

In vintage KITP fashion, we’d nearly completed a project by the time Chaitanya and I left Santa Barbara. Physical Review Letters published our paper this year, and I’m as proud of it as a gardener of the first buds from her garden. Here’s what we found.

Southern California spoiled me for roses.

Incompatible conserved quantities conflict with the ETH and the ETH’s prediction of internal thermalization. Why? For three reasons. First, when inferring thermalization from the ETH, we assume that the Hamiltonian lacks degeneracies (that no energy equals any other). But incompatible conserved quantities force degeneracies on the Hamiltonian.2 

Second, when inferring from the ETH that the system thermalizes, we assume that the system begins in a microcanonical subspace. That’s an eigenspace shared by the conserved quantities (other than the Hamiltonian)—usually, an eigenspace of the total particle number or the total spin’s z-component. But, if incompatible, the conserved quantities share no eigenbasis, so they might not share eigenspaces, so microcanonical subspaces won’t exist in abundance.

Third, let’s focus on a system of N qubits. Say that the Hamiltonian conserves the total spin components S_x, S_y, and S_z. The Hamiltonian obeys the Wigner–Eckart theorem, which sounds more complicated than it is. Suppose that the qubits begin in a state | s_\alpha, \, m \rangle labeled by a spin quantum number s_\alpha and a magnetic spin quantum number m. Let a particle hit the qubits, acting on them with an operator \mathcal{O} . With what probability (amplitude) do the qubits end up with quantum numbers s_{\alpha'} and m'? The answer is \langle s_{\alpha'}, \, m' | \mathcal{O} | s_\alpha, \, m \rangle. The Wigner–Eckart theorem dictates this probability amplitude’s form. 

| s_\alpha, \, m \rangle and | s_{\alpha'}, \, m' \rangle are Hamiltonian eigenstates, thanks to the conservation law. The ETH is an ansatz for the form of \langle s_{\alpha'}, \, m' | \mathcal{O} | s_\alpha, \, m \rangle—of the elements of matrices that represent operators \mathcal{O} relative to the energy eigenbasis. The ETH butts heads with the Wigner–Eckart theorem, which also predicts the matrix element’s form.

The Wigner–Eckart theorem wins, being a theorem—a proved claim. The ETH is, as the H in the acronym relates, only a hypothesis.

If conserved quantities are incompatible, we have to kiss the ETH and its thermalization predictions goodbye. But must we set ourselves adrift entirely? Can we cling to no buoy from physics’s best toolkit for quantum many-body thermalization?

No, and yes, respectively. Our clan proposed a non-Abelian ETH for Hamiltonians that conserve incompatible quantities—or, equivalently, that have non-Abelian symmetries. The non-Abelian ETH depends on s_\alpha and on Clebsch–Gordan coefficients—conversion factors between total-spin eigenstates | s_\alpha, \, m \rangle and product states | s_1, \, m_1 \rangle \otimes | s_2, \, m_2 \rangle.

Using the non-Abelian ETH, we proved that many systems thermalize internally, despite conserving incompatible quantities. Yet the incompatibility complicates the proof enormously, extending it from half a page to several pages. Also, under certain conditions, incompatible quantities may alter thermalization. According to the conventional ETH, time-averaged expectation values \overline{ \langle \mathcal{O} \rangle }_t come to equal thermal expectation values \langle \mathcal{O} \rangle_{\rm th} to within O( N^{-1} ) corrections, as I explained last month. The correction can grow polynomially larger in the system size, to O( N^{-1/2} ), if conserved quantities are incompatible. Our conclusion holds under an assumption that we argue is physically reasonable.

So incompatible conserved quantities do alter the ETH, yet another thermodynamic expectation. Physicist Jae Dong Noh began checking the non-Abelian ETH numerically, and more testing is underway. And I’m looking forward to returning to the KITP this fall. Tales do say that paradise is a garden.

View through my office window at the KITP

1Not that married people always trust each other’s judgment.

2The reason is Schur’s lemma, a group-theoretic result. Appendix A of this paper explains the details.

The Book of Mark

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Mark Srednicki doesn’t look like a high priest. He’s a professor of physics at the University of California, Santa Barbara (UCSB); and you’ll sooner find him in khakis than in sacred vestments. Humor suits his round face better than channeling divine wrath would; and I’ve never heard him speak in tongues—although, when an idea excites him, his hands rise to shoulder height of their own accord, as though halfway toward a priestly blessing. Mark belongs less on a ziggurat than in front of a chalkboard. Nevertheless, he called himself a high priest.

Specifically, Mark jokingly called himself a high priest of the eigenstate thermalization hypothesis, a framework for understanding how quantum many-body systems thermalize internally. The eigenstate thermalization hypothesis has an unfortunate number of syllables, so I’ll call it the ETH. The ETH illuminates closed quantum many-body systems, such as a clump of N ultracold atoms. The clump can begin in a pure product state | \psi(0) \rangle, then evolve under a chaotic1 Hamiltonian H. The time-t state | \psi(t) \rangle will remain pure; its von Neumann entropy will always vanish. Yet entropy grows according to the second law of thermodynamics. Breaking the second law amounts almost to a enacting a miracle, according to physicists. Does the clump of atoms deserve consideration for sainthood?

No—although the clump’s state remains pure, a small subsystem’s state does not. A subsystem consists of, for example, a few atoms. They’ll entangle with the other atoms, which serve as an effective environment. The entanglement will mix the few atoms’ state, whose von Neumann entropy will grow.

The ETH predicts this growth. The ETH is an ansatz about H and an operator O—say, an observable of the few-atom subsystem. We can represent O as a matrix relative to the energy eigenbasis. The matrix elements have a certain structure, if O and H satisfy the ETH. Suppose that the operators do and that H lacks degeneracies—that no two energy eigenvalues equal each other. We can prove that O thermalizes: Imagine measuring the expectation value \langle \psi(t) | O | \psi(t) \rangle at each of many instants t. Averaging over instants produces the time-averaged expectation value \overline{ \langle O \rangle_t }. 

Another average is the thermal average—the expectation value of O in the appropriate thermal state. If H conserves just itself,2 the appropriate thermal state is the canonical state, \rho_{\rm can} := e^{-\beta H}/ Z. The average energy \langle \psi(0) | H | \psi(0) \rangle defines the inverse temperature \beta, and Z normalizes the state. Hence the thermal average is \langle O \rangle_{\rm th}  :=  {\rm Tr} ( O \rho_{\rm can} ). 

The time average approximately equals the thermal average, according to the ETH: \overline{ \langle O \rangle_t }  =  \langle O \rangle_{\rm th} + O \big( N^{-1} \big). The correction is small in the total number N of atoms. Through the lens of O, the atoms thermalize internally. Local observables tend to satisfy the ETH, and we can easily observe only local observables. We therefore usually observe thermalization, consistently with the second law of thermodynamics.

I agree that Mark Srednicki deserves the title high priest of the ETH. He and Joshua Deutsch independently dreamed up the ETH in 1994 and 1991. Since numericists reexamined it in 2008, studies and applications of the ETH have exploded like a desert religion. Yet Mark had never encountered the question I posed about it in 2021. Next month’s blog post will share the good news about that question.

1Nonintegrable.

2Apart from trivial quantities, such as projectors onto eigenspaces of H.

Let the great world spin

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I first heard the song “Fireflies,” by Owl City, shortly after my junior year of college. During the refrain, singer Adam Young almost whispers, “I’d like to make myself believe / that planet Earth turns slowly.” Goosebumps prickled along my neck. Yes, I thought, I’ve studied Foucault’s pendulum.

Léon Foucault practiced physics in France during the mid-1800s. During one of his best-known experiments, he hung a pendulum from high up in a building. Imagine drawing a wide circle on the floor, around the pendulum’s bob.1

Pendulum bob and encompassing circle, as viewed from above.

Imagine pulling the bob out to a point above the circle, then releasing the pendulum. The bob will swing back and forth, tracing out a straight line across the circle.

You might expect the bob to keep swinging back and forth along that line, and to do nothing more, forever (or until the pendulum has spent all its energy on pushing air molecules out of its way). After all, the only forces acting on the bob seem to be gravity and the tension in the pendulum’s wire. But the line rotates; its two tips trace out the circle.

How long the tips take to trace the circle depends on your latitude. At the North and South Poles, the tips take one day.

Why does the line rotate? Because the pendulum dangles from a building on the Earth’s surface. As the Earth rotates, so does the building, which pushes the pendulum. You’ve experienced such a pushing if you’ve ridden in a car. Suppose that the car is zipping along at a constant speed, in an unchanging direction, on a smooth road. With your eyes closed, you won’t feel like you’re moving. The only forces you can sense are gravity and the car seat’s preventing you from sinking into the ground (analogous to the wire tension that prevents the pendulum bob from crashing into the floor). If the car turns a bend, it pushes you sidewise in your seat. This push is called a centrifugal force. The pendulum feels a centrifugal force because the Earth’s rotation is an acceleration like the car’s. The pendulum also feels another force—a Coriolis force—because it’s not merely sitting, but moving on the rotating Earth.

We can predict the rotation of Foucault’s pendulum by assuming that the Earth rotates, then calculating the centrifugal and Coriolis forces induced, and then calculating how those forces will influence the pendulum’s motion. The pendulum evidences the Earth’s rotation as nothing else had before debuting in 1851. You can imagine the stir created by the pendulum when Foucault demonstrated it at the Observatoire de Paris and at the Panthéon monument. Copycat pendulums popped up across the world. One ended up next to my college’s physics building, as shown in this video. I reveled in understanding that pendulum’s motion, junior year.

Image from the Smithsonian Institution Archives.

My professor alluded to a grander Foucault pendulum in Paris. It hangs in what sounded like a temple to the Enlightenment—beautiful in form, steeped in history, and rich in scientific significance. I’m a romantic about the Enlightenment; I adore the idea of creating the first large-scale organizational system for knowledge. So I hungered to make a pilgrimage to Paris.

I made the pilgrimage this spring. I was attending a quantum-chaos workshop at the Institut Pascal, an interdisciplinary institute in a suburb of Paris. One quiet Saturday morning, I rode a train into the city center. The city houses a former priory—a gorgeous, 11th-century, white-stone affair of the sort for which I envy European cities. For over 200 years, the former priory has housed the Musée des Arts et Métiers, a museum of industry and technology. In the priory’s chapel hangs Foucault’s pendulum.2

A pendulum of Foucault’s own—the one he exhibited at the Panthéon—used to hang in the chapel. That pendulum broke in 2010; but still, the pendulum swinging today is all but a holy relic of scientific history. Foucault’s pendulum! Demonstrating that the Earth rotates! And in a jewel of a setting—flooded with light from stained-glass windows and surrounded by Gothic arches below a painted ceiling. I flitted around the little chapel like a pollen-happy bee for maybe 15 minutes, watching the pendulum swing, looking at other artifacts of Foucault’s, wending my way around the carved columns.

Almost alone. A handful of visitors trickled in and out. They contrasted with my visit, the previous weekend, to the Louvre. There, I’d witnessed a Disney World–esque line of tourists waiting for a glimpse of the Mona Lisa, camera phones held high. Nobody was queueing up in the musée’s chapel. But this was Foucault’s pendulum! Demonstrating that the Earth rotates!

I confess to capitalizing on the lack of visitors to take a photo with Foucault’s pendulum and Foucault’s Pendulum, though.

Shortly before I’d left for Paris, a librarian friend had recommended Umberto Eco’s novel Foucault’s Pendulum. It occupied me during many a train ride to or from the center of Paris.

The rest of the museum could model in an advertisement for steampunk. I found automata, models of the steam engines that triggered the Industrial Revolution, and a phonograph of Thomas Edison’s. The gadgets, many formed from brass and dark wood, contrast with the priory’s light-toned majesty. Yet the priory shares its elegance with the inventions, many of which gleam and curve in decorative flutes. 

The grand finale at the Musée des Arts et Métiers.

I tore myself away from the Musée des Arts et Métiers after several hours. I returned home a week later and heard the song “Fireflies” again not long afterward. The goosebumps returned worse. Thanks to Foucault, I can make myself believe that planet Earth turns.

With thanks to Kristina Lynch for tolerating my many, many, many questions throughout her classical-mechanics course.

This story’s title refers to a translation of Goethe’s Faust. In the translation, the demon Mephistopheles tells the title character, “You let the great world spin and riot; / we’ll nest contented in our quiet” (to within punctuational and other minor errors, as I no longer have the text with me). A prize-winning 2009 novel is called Let the Great World Spin; I’ve long wondered whether Faust inspired its title.

1Why isn’t the bottom of the pendulum called the alice?

2After visiting the musée, I learned that my classical-mechanics professor had been referring to the Foucault pendulum that hangs in the Panthéon, rather than to the pendulum in the musée. The musée still contains the pendulum used by Foucault in 1851, whereas the Panthéon has only a copy, so I’m content. Still, I wouldn’t mind making a pilgrimage to the Panthéon. Let me know if more thermodynamic workshops take place in Paris!

Memories of things past

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My best friend—who’s held the title of best friend since kindergarten—calls me the keeper of her childhood memories. I recall which toys we played with, the first time I visited her house,1 and which beverages our classmates drank during snack time in kindergarten.2 She wouldn’t be surprised to learn that the first workshop I’ve co-organized centered on memory.

Memory—and the loss of memory—stars in thermodynamics. As an example, take what my husband will probably do this evening: bake tomorrow’s breakfast. I don’t know whether he’ll bake fruit-and-oat cookies, banana muffins, pear muffins, or pumpkin muffins. Whichever he chooses, his baking will create a scent. That scent will waft across the apartment, seep into air vents, and escape into the corridor—will disperse into the environment. By tomorrow evening, nobody will be able to tell by sniffing what my husband will have baked. 

That is, the kitchen’s environment lacks a memory. This lack contributes to our experience of time’s arrow: We sense that time passes partially by smelling less and less of breakfast. Physicists call memoryless systems and processes Markovian.

A snippet from Quantum Steampunk: The Physics of Yesterday’s Tomorrow

Our kitchen’s environment is Markovian because it’s large and particles churn through it randomly. But not all environments share these characteristics. Metaphorically speaking, a dispersed memory of breakfast may recollect, return to a kitchen, and influence the following week’s baking. For instance, imagine an atom in a quantum computer, rather than a kitchen in an apartment. A few other atoms may form our atom’s environment. Quantum information may leak from our atom into that environment, swish around in the environment for a time, and then return to haunt our atom. We’d call the atom’s evolution and environment non-Markovian.

I had the good fortune to co-organize a workshop about non-Markovianity—about memory—this February. The workshop took place at the Banff International Research Station, abbreviated BIRS, which you pronounce like the plural of what you say when shivering outdoors in Canada. BIRS operates in the Banff Centre for Arts and Creativity, high in the Rocky Mountains. The Banff Centre could accompany a dictionary entry for pristine, to my mind. The air feels crisp, the trees on nearby peaks stand out against the snow like evergreen fringes on white velvet, and the buildings balance a rustic-mountain-lodge style with the avant-garde. 

The workshop balanced styles, too, but skewed toward the theoretical and abstract. We learned about why the world behaves classically in our everyday experiences; about information-theoretic measures of the distances between quantum states; and how to simulate, on quantum computers, chemical systems that interact with environments. One talk, though, brought our theory back down to (the snow-dusted) Earth.

Gabriela Schlau-Cohen runs a chemistry lab at MIT. She wants to understand how plants transport energy. Energy arrives at a plant from the sun in the form of light. The light hits a pigment-and-protein complex. If the plant is lucky, the light transforms into a particle-like packet of energy called an exciton. The exciton traverses the receptor complex, then other complexes. Eventually, the exciton finds a spot where it can enable processes such as leaf growth. 

A high fraction of the impinging photons—85%—transform into excitons. How do plants convert and transport energy as efficiently as they do?

Gabriela’s group aims to find out—not by testing natural light-harvesting complexes, but by building complexes themselves. The experimentalists mimic the complex’s protein using DNA. You can fold DNA into almost any shape you want, by choosing the DNA’s base pairs (basic units) adroitly and by using “staples” formed from more DNA scraps. The sculpted molecules are called DNA origami.

Gabriela’s group engineers different DNA structures, analogous to complexes’ proteins, to have different properties. For instance, the experimentalists engineer rigid structures and flexible structures. Then, the group assesses how energy moves through each structure. Each structure forms an environment that influences excitons’ behaviors, similarly to how a memory-containing environment influences an atom.

Courtesy of Gabriela Schlau-Cohen

The Banff environment influenced me, stirring up memories like powder displaced by a skier on the slopes above us. I first participated in a BIRS workshop as a PhD student, and then I returned as a postdoc. Now, I was co-organizing a workshop to which I brought a PhD student of my own. Time flows, as we’re reminded while walking down the mountain from the Banff Centre into town: A cemetery borders part of the path. Time flows, but we belong to that thermodynamically remarkable class of systems that retain memories…memories and a few other treasures that resist change, such as friendships held since kindergarten.

1Plushy versions of Simba and Nala from The Lion King. I remain grateful to her for letting me play at being Nala.

2I’d request milk, another kid would request apple juice, and everyone else would request orange juice.

A (quantum) complex legacy: Part deux

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I didn’t fancy the research suggestion emailed by my PhD advisor.

A 2016 email from John Preskill led to my publishing a paper about quantum complexity in 2022, as I explained in last month’s blog post. But I didn’t explain what I thought of his email upon receiving it.

It didn’t float my boat. (Hence my not publishing on it until 2022.)

The suggestion contained ingredients that ordinarily would have caulked any cruise ship of mine: thermodynamics, black-hole-inspired quantum information, and the concept of resources. John had forwarded a paper drafted by Stanford physicists Adam Brown and Lenny Susskind. They act as grand dukes of the community sussing out what happens to information swallowed by black holes. 

From Rare-Gallery

We’re not sure how black holes work. However, physicists often model a black hole with a clump of particles squeezed close together and so forced to interact with each other strongly. The interactions entangle the particles. The clump’s quantum state—let’s call it | \psi(t) \rangle—grows not only complicated with time (t), but also complex in a technical sense: Imagine taking a fresh clump of particles and preparing it in the state | \psi(t) \rangle via a sequence of basic operations, such as quantum gates performable with a quantum computer. The number of basic operations needed is called the complexity of | \psi(t) \rangle. A black hole’s state has a complexity believed to grow in time—and grow and grow and grow—until plateauing. 

This growth echoes the second law of thermodynamics, which helps us understand why time flows in only one direction. According to the second law, every closed, isolated system’s entropy grows until plateauing.1 Adam and Lenny drew parallels between the second law and complexity’s growth.

The less complex a quantum state is, the better it can serve as a resource in quantum computations. Recall, as we did last month, performing calculations in math class. You needed clean scratch paper on which to write the calculations. So does a quantum computer. “Scratch paper,” to a quantum computer, consists of qubits—basic units of quantum information, realized in, for example, atoms or ions. The scratch paper is “clean” if the qubits are in a simple, unentangled quantum state—a low-complexity state. A state’s greatest possible complexity, minus the actual complexity, we can call the state’s uncomplexity. Uncomplexity—a quantum state’s blankness—serves as a resource in quantum computation.

Manny Knill and Ray Laflamme realized this point in 1998, while quantifying the “power of one clean qubit.” Lenny arrived at a similar conclusion while reasoning about black holes and firewalls. For an introduction to firewalls, see this blog post by John. Suppose that someone—let’s call her Audrey—falls into a black hole. If it contains a firewall, she’ll burn up. But suppose that someone tosses a qubit into the black hole before Audrey falls. The qubit kicks the firewall farther away from the event horizon, so Audrey will remain safe for longer. Also, the qubit increases the uncomplexity of the black hole’s quantum state. Uncomplexity serves as a resource also to Audrey.

A resource is something that’s scarce, valuable, and useful for accomplishing tasks. Different things qualify as resources in different settings. For instance, imagine wanting to communicate quantum information to a friend securely. Entanglement will serve as a resource. How can we quantify and manipulate entanglement? How much entanglement do we need to perform a given communicational or computational task? Quantum scientists answer such questions with a resource theory, a simple information-theoretic model. Theorists have defined resource theories for entanglement, randomness, and more. In many a blog post, I’ve eulogized resource theories for thermodynamic settings. Can anyone define, Adam and Lenny asked, a resource theory for quantum uncomplexity?

Resource thinking pervades our world.

By late 2016, I was a quantum thermodynamicist, I was a resource theorist, and I’d just debuted my first black-hole–inspired quantum information theory. Moreover, I’d coauthored a review about the already-extant resource theory that looked closest to what Adam and Lenny sought. Hence John’s email, I expect. Yet that debut had uncovered reams of questions—questions that, as a budding physicist heady with the discovery of discovery, I could own. Why would I answer a question of someone else’s instead?

So I thanked John, read the paper draft, and pondered it for a few days. Then, I built a research program around my questions and waited for someone else to answer Adam and Lenny.

Three and a half years later, I was still waiting. The notion of uncomplexity as a resource had enchanted the black-hole-information community, so I was preparing a resource-theory talk for a quantum-complexity workshop. The preparations set wheels churning in my mind, and inspiration struck during a long walk.2

After watching my workshop talk, Philippe Faist reached out about collaborating. Philippe is a coauthor, a friend, and a fellow quantum thermodynamicist and resource theorist. Caltech’s influence had sucked him, too, into the black-hole community. We Zoomed throughout the pandemic’s first spring, widening our circle to include Teja Kothakonda, Jonas Haferkamp, and Jens Eisert of Freie University Berlin. Then, Anthony Munson joined from my nascent group in Maryland. Physical Review A published our paper, “Resource theory of quantum uncomplexity,” in January.

The next four paragraphs, I’ve geared toward experts. An agent in the resource theory manipulates a set of n qubits. The agent can attempt to perform any gate U on any two qubits. Noise corrupts every real-world gate implementation, though. Hence the agent effects a gate chosen randomly from near U. Such fuzzy gates are free. The agent can’t append or discard any system for free: Appending even a maximally mixed qubit increases the state’s uncomplexity, as Knill and Laflamme showed. 

Fuzzy gates’ randomness prevents the agent from mapping complex states to uncomplex states for free (with any considerable probability). Complexity only grows or remains constant under fuzzy operations, under appropriate conditions. This growth echoes the second law of thermodynamics. 

We also defined operational tasks—uncomplexity extraction and expenditure analogous to work extraction and expenditure. Then, we bounded the efficiencies with which the agent can perform these tasks. The efficiencies depend on a complexity entropy that we defined—and that’ll star in part trois of this blog-post series.

Now, I want to know what purposes the resource theory of uncomplexity can serve. Can we recast black-hole problems in terms of the resource theory, then leverage resource-theory results to solve the black-hole problem? What about problems in condensed matter? Can our resource theory, which quantifies the difficulty of preparing quantum states, merge with the resource theory of magic, which quantifies that difficulty differently?

Unofficial mascot for fuzzy operations

I don’t regret having declined my PhD advisor’s recommendation six years ago. Doing so led me to explore probability theory and measurement theory, collaborate with two experimental labs, and write ten papers with 21 coauthors whom I esteem. But I take my hat off to Adam and Lenny for their question. And I remain grateful to the advisor who kept my goals and interests in mind while checking his email. I hope to serve Anthony and his fellow advisees as well.

1…en route to obtaining a marriage license. My husband and I married four months after the pandemic throttled government activities. Hours before the relevant office’s calendar filled up, I scored an appointment to obtain our license. Regarding the metro as off-limits, my then-fiancé and I walked from Cambridge, Massachusetts to downtown Boston for our appointment. I thank him for enduring my requests to stop so that I could write notes.

2At least, in the thermodynamic limit—if the system is infinitely large. If the system is finite-size, its entropy grows on average.