An equation fit for a novel

Archana Kamal was hunting for an apartment in Cambridge, Massachusetts. She was moving MIT, to work as a postdoc in physics. The first apartment she toured had housed John Updike, during his undergraduate career at Harvard. No other apartment could compete; Archana signed the lease.

The apartment occupied the basement of a red-brick building covered in vines. The rooms spanned no more than 350 square feet. Yet her window opened onto the neighbors’ garden, whose leaves she tracked across the seasons. And Archana cohabited with history.

Apartment photos

She’s now studying the universe’s history, as an assistant professor of physics at the University of Massachusetts Lowell. The cosmic microwave background (CMB) pervades the universe. The CMB consists of electromagnetic radiation, or light. Light has particle-like properties and wavelike properties. The wavelike properties include wavelength, the distance between successive peaks. Long-wavelength light includes red light, infrared light, and radio waves. Short-wavelength light includes blue light, ultraviolet light, and X-rays. Light of one wavelength and light of another wavelength are said to belong to different modes.

Wavelength

Does the CMB have nonclassical properties, impossible to predict with classical physics but (perhaps) predictable with quantum theory? The CMB does according to the theory of inflation. According to the theory, during a short time interval after the Big Bang, the universe expanded very quickly: Spacetime stretched. Inflation explains features of our universe, though we don’t know what mechanism would have effected the expansion.

According to inflation, around the Big Bang time, all the light in the universe crowded together. The photons (particles of light) interacted, entangling (developing strong quantum correlations). Spacetime then expanded, and the photons separated. But they might retain entanglement.

Detecting that putative entanglement poses challenges. For instance, the particles that you’d need to measure could produce a signal too weak to observe. Cosmologists have been scratching their heads about how to observe nonclassicality in the CMB. One team—Nishant Agarwal at UMass Lowell and Sarah Shandera at Pennsylvania State University—turned to Archana for help.

A sky full of stars

Archana studies the theory of open quantum systems, quantum systems that interact with their environments. She thinks most about systems such as superconducting qubits, tiny circuits with which labs are building quantum computers. But the visible universe constitutes an open quantum system.

We can see only part of the universe—or, rather, only part of what we believe is the whole universe. Why? We can see only stuff that’s emitted light that has reached us, and light has had only so long to travel. But the visible universe interacts (we believe) with stuff we haven’t seen. For instance, according to the theory of inflation, that rapid expansion stretched some light modes’ wavelengths. Those wavelengths grew longer than the visible universe. We can’t see those modes’ peak-to-peak variations or otherwise observe the modes, often called “frozen.” But the frozen modes act as an environment that exchanges information and energy with the visible universe.

We describe an open quantum system’s evolution with a quantum master equation, which I blogged about four-and-a-half years ago. Archana and collaborators constructed a quantum master equation for the visible universe. The frozen modes, they found, retain memories of the visible universe. (Experts: the bath is non-Markovian.) Next, they need to solve the equation. Then, they’ll try to use their solution to identify quantum observables that could reveal nonclassicality in the CMB.

Frozen modes

Frozen modes

Archana’s project caught my fancy for two reasons. First, when I visited her in October, I was collaborating on a related project. My coauthors and I were concocting a scheme for detecting nonclassical correlations in many-particle systems by measuring large-scale properties. Our paper debuted last month. It might—with thought and a dash of craziness—be applied to detect nonclassicality in the CMB. Archana’s explanation improved my understanding of our scheme’s potential. 

Second, Archana and collaborators formulated a quantum master equation for the visible universe. A quantum master equation for the visible universe. The phrase sounded romantic to me.1 It merited a coauthor who’d seized on an apartment lived in by a Pulitzer Prize-winning novelist. 

Archana’s cosmology and Updike stories reminded me of one reason why I appreciate living in the Boston area: History envelops us here. Last month, while walking to a grocery, I found a sign that marks the building in which the poet e. e. cummings was born. My walking partner then generously tolerated a recitation of cummings’s “anyone lived in a pretty how town.” History enriches our lives—and some of it might contain entanglement.

 

1It might sound like gobbledygook to you, if I’ve botched my explanations of the terminology.

With thanks to Archana and the UMass Lowell Department of Physics and Applied Physics for their hospitality and seminar invitation.

The paper that begged for a theme song

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.

A photoisomer is a molecular switch. These switches appear across nature and technologies. We have photoisomers in our eyes, and experimentalists have used photoisomers to boost solar-fuel storage. A photoisomer has two functional groups, or collections of bonded atoms, attached to a central axis. 

Photoisomer

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.

Switch

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.

Teapot 1

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’ve blogged, many times, 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.

Teapot 2

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.

Teapot 3

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.

Teapot 4

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.

The quantum steampunker by Massachusetts Bay

Every spring, a portal opens between Waltham, Massachusetts and another universe. 

The other universe has a Watch City dual to Waltham, known for its watch factories. The cities throw a festival to which explorers, inventors, and tourists flock. Top hats, goggles, leather vests, bustles, and lace-up boots dot the crowds. You can find pet octopodes, human-machine hybrids, and devices for bending space and time. Steam powers everything.

Watch City

Watch City Steampunk Festival

So I learned thanks to Maxim Olshanyi, a professor of physics at the University of Massachusetts Boston. He hosted my colloquium, “Quantum steampunk: Quantum information meets thermodynamics,” earlier this month. Maxim, I discovered, has more steampunk experience than I. He digs up century-old designs for radios, builds the radios, and improves upon the designs. He exhibits his creations at the Watch City Steampunk Festival.

Maxim photo

Maxim Olshanyi

I never would have guessed that Maxim moonlights with steampunkers. But his hobby makes sense: Maxim has transformed our understanding of quantum integrability.

Integrability is to thermalization as Watch City is to Waltham. A bowl of baked beans thermalizes when taken outside in Boston in October: Heat dissipates into the air. After half-an-hour, large-scale properties bear little imprint of their initial conditions: The beans could have begun at 112ºF or 99º or 120º. Either way, the beans have cooled.

Integrable systems avoid thermalizing; more of their late-time properties reflect early times. Why? We can understand through an example, an integrable system whose particles don’t interact with each other (whose particles are noninteracting fermions). The dynamics conserve the particles’ momenta. Consider growing the system by adding particles. The number of conserved quantities grows as the system size. The conserved quantities retain memories of the initial conditions.

Imagine preparing an integrable system, analogously to preparing a bowl of baked beans, and letting it sit for a long time. Will the system equilibrate, or settle down to, a state predictable with a simple rule? We might expect not. Obeying the same simple rule would cause different integrable systems to come to resemble each other. Integrable systems seem unlikely to homogenize, since each system retains much information about its initial conditions.

Boston baked beans

Maxim and collaborators exploded this expectation. Integrable systems do relax to simple equilibrium states, which the physicists called the generalized Gibbs ensemble (GGE). Josiah Willard Gibbs cofounded statistical mechanics during the 1800s. He predicted the state to which nonintegrable systems, like baked beans in autumnal Boston, equilibrate. Gibbs’s theory governs classical systems, like baked beans, as does the GGE theory. But also quantum systems equilibrate to the GGE, and Gibbs’s conclusions translate into quantum theory with few adjustments. So I’ll explain in quantum terms.

Consider quantum baked beans that exchange heat with a temperature-T environment. Let \hat{H} denote the system’s Hamiltonian, which basically represents the beans’ energy. The beans equilibrate to a quantum Gibbs state, e^{ - \hat{H} / ( k_{\rm B} T ) } / Z. The k_{\rm B} denotes Boltzmann’s constant, a fundamental constant of nature. The partition function Z enables the quantum state to obey probability theory (normalizes the state).

Maxim and friends modeled their generalized Gibbs ensemble on the Gibbs state. Let \hat{I}_m denote a quantum integrable system’s m^{\rm th} conserved quantity. This system equilibrates to e^{ - \sum_m \lambda_m \hat{I}_m } / Z_{\rm GGE}. The Z_{\rm GGE} normalizes the state. The intensive parameters \lambda_m’s serve analogously to temperature and depend on the conserved quantities’ values. Maxim and friends predicted this state using information theory formalized by Ed Jaynes. Inventing the GGE, they unlocked a slew of predictions about integrable quantum systems. 

Olchanyi__radioboard_comp_2015_picture

A radio built by Maxim. According to him, “The invention was to replace a diode with a diode bridge, in a crystal radio, thus gaining a factor of two in the output power.”

I define quantum steampunk as the intersection of quantum theory, especially quantum information theory, with thermodynamics, and the application of this intersection across science. Maxim has used information theory to cofound a branch of quantum statistical mechanics. Little wonder that he exhibits homemade radios at the Watch City Steampunk Festival. He also holds a license to drive steam engines and used to have my postdoc position. I appreciate having older cousins to look up to. Here’s hoping that I become half the quantum steampunker that I found by Massachusetts Bay.

With thanks to Maxim and the rest of the University of Massachusetts Boston Department of Physics for their hospitality.

The next Watch City Steampunk Festival takes place on May 9, 2020. Contact me if you’d attend a quantum-steampunk meetup!

Quantum conflict resolution

If only my coauthors and I had quarreled.

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 x. If you measure the position immediately afterward, you’ll obtain x 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.

Tug-of-war

How should we quantify the disagreement of two quantum observables, \hat{A} and \hat{B}? 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,

\Delta \hat{A} \, \Delta \hat{B} \geq \frac{1}{i \hbar} \langle [\hat{A}, \hat{B}] \rangle.

Imagine preparing a quantum state | \psi \rangle and measuring \hat{A}, then repeating this protocol in many trials. Each trial has some probability p_a of yielding the outcome a. Different trials will yield different a’s. We quantify the spread in a values with the standard deviation \Delta \hat{A} = \sqrt{ \langle \psi | \hat{A}^2 | \psi \rangle - \langle \psi | \hat{A} | \psi \rangle^2 }. We define \Delta \hat{B} analogously. \hbar denotes Planck’s constant, a number that characterizes our universe as the electron’s mass does. 

[\hat{A}, \hat{B}] denotes the observables’ commutator. The numbers that we use in daily life commute: 7 \times 5 = 5 \times 7. Quantum numbers, or operators, represent \hat{A} and \hat{B}. Operators don’t necessarily commute. The commutator [\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A} represents how little \hat{A} and \hat{B} resemble 7 and 5. 

Robertson’s uncertainty relation means, “If you can predict an \hat{A} measurement’s outcome precisely, you can’t predict a \hat{B} measurement’s outcome precisely, and vice versa. The uncertainties must multiply to at least some number. The number depends on how much \hat{A} fails to commute with \hat{B}.” The higher an uncertainty bound (the greater the inequality’s right-hand side), the more the operators disagree.

fistfight-cloud

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 H, 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

H (\hat{A}) + H( \hat{B} ) \geq - \log c. \qquad (^*)

The entropy H( \hat{A} ) quantifies the difficulty of predicting the outcome a of an \hat{A} measurement. H( \hat{B} ) is defined analogously. c is called the overlap. It quantifies your ability to predict what happens if you prepare your system with a well-defined \hat{A} value, then measure \hat{B}. For further analysis, check out this paper. Entropic uncertainty relations have blossomed within QI theory over the past few years. 

Blossom

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 \hat{W} represent the poke. Suppose that the system evolves chaotically for a time t afterward, the qubits interacting. Information about the poke spreads through many-body entanglement, or scrambles.

Spin chain

Imagine measuring an observable \hat{V} of a few qubits far from the \hat{W} qubits. A little information about \hat{W} migrates into the \hat{V} qubits. But measuring \hat{V} reveals almost nothing about \hat{W}, because most of the information about \hat{W} has spread across the system. \hat{V} disagrees with \hat{W}, in a sense. Actually, \hat{V} disagrees with \hat{W}(t). The (t) represents the time evolution.

The OTOC’s smallness reflects how much \hat{W}(t) disagrees with \hat{V} at any instant t. At early times t \gtrsim 0, the operators agree, and the OTOC \approx 1. At late times, the operators disagree loads, and the OTOC \approx 0.

Dove

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 \hat{W}(t) for the \hat{A} in Ineq. {(^*)}, and \hat{V} for the \hat{B}. 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.

Handshake

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 past three years, 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.

 

1Thanks to Communications Physics for publishing the paper. For pedagogical formatting, read the arXiv version. 

What distinguishes quantum thermodynamics from quantum statistical mechanics?

Yoram Alhassid asked the question at the end of my Yale Quantum Institute colloquium last February. I knew two facts about Yoram: (1) He belongs to Yale’s theoretical-physics faculty. (2) His PhD thesis’s title—“On the Information Theoretic Approach to Nuclear Reactions”—ranks among my three favorites.1 

Over the past few months, I’ve grown to know Yoram better. He had reason to ask about quantum statistical mechanics, because his research stands up to its ears in the field. If forced to synopsize quantum statistical mechanics in five words, I’d say, “study of many-particle quantum systems.” Examples include gases of ultracold atoms. If given another five words, I’d add, “Calculate and use partition functions.” A partition function is a measure of the number of states, or configurations, accessible to the system. Calculate a system’s partition function, and you can calculate the system’s average energy, the average number of particles in the system, how the system responds to magnetic fields, etc.

Line in the sand

My colloquium concerned quantum thermodynamics, which I’ve blogged about many times. So I should have been able to distinguish quantum thermodynamics from its neighbors. But the answer I gave Yoram didn’t satisfy me. I mulled over the exchange for a few weeks, then emailed Yoram a 502-word essay. The exercise grew my appreciation for the question and my understanding of my field. 

An adaptation of the email appears below. The adaptation should suit readers who’ve majored in physics, but don’t worry if you haven’t. Bits of what distinguishes quantum thermodynamics from quantum statistical mechanics should come across to everyone—as should, I hope, the value of question-and-answer sessions:

One distinction is a return to the operational approach of 19th-century thermodynamics. Thermodynamicists such as Sadi Carnot wanted to know how effectively engines could operate. Their practical questions led to fundamental insights, such as the Carnot bound on an engine’s efficiency. Similarly, quantum thermodynamicists often ask, “How can this state serve as a resource in thermodynamic tasks?” This approach helps us identify what distinguishes quantum theory from classical mechanics.

For example, quantum thermodynamicists found an advantage in charging batteries via nonlocal operations. Another example is the “MBL-mobile” that I designed with collaborators. Many-body localization (MBL), we found, can enhance an engine’s reliability and scalability. 

Asking, “How can this state serve as a resource?” leads quantum thermodynamicists to design quantum engines, ratchets, batteries, etc. We analyze how these devices can outperform classical analogues, identifying which aspects of quantum theory power the outperformance. This question and these tasks contrast with the questions and tasks of many non-quantum-thermodynamicists who use statistical mechanics. They often calculate response functions and (e.g., ground-state) properties of Hamiltonians.

These goals of characterizing what nonclassicality is and what it can achieve in thermodynamic contexts resemble upshots of quantum computing and cryptography. As a 21st-century quantum information scientist, I understand what makes quantum theory quantum partially by understanding which problems quantum computers can solve efficiently and classical computers can’t. Similarly, I understand what makes quantum theory quantum partially by understanding how much more work you can extract from a singlet \frac{1}{ \sqrt{2} } ( | 0 1 \rangle - |1 0 \rangle ) (a maximally entangled state of two qubits) than from a product state in which the reduced states have the same forms as in the singlet, \frac{1}{2} ( | 0 \rangle \langle 0 | + | 1 \rangle \langle 1 | ).

As quantum thermodynamics shares its operational approach with quantum information theory, quantum thermodynamicists use mathematical tools developed in quantum information theory. An example consists of generalized entropies. Entropies quantify the optimal efficiency with which we can perform information-processing and thermodynamic tasks, such as data compression and work extraction.

Most statistical-mechanics researchers use just the Shannon and von Neumann entropies, H_{\rm Sh} and H_{\rm vN}, and perhaps the occasional relative entropy. These entropies quantify optimal efficiencies in large-system limits, e.g., as the number of messages compressed approaches infinity and in the thermodynamic limit.

Other entropic quantities have been defined and explored over the past two decades, in quantum and classical information theory. These entropies quantify the optimal efficiencies with which tasks can be performed (i) if the number of systems processed or the number of trials is arbitrary, (ii) if the systems processed share correlations, (iii) in the presence of “quantum side information” (if the system being used as a resource is entangled with another system, to which an agent has access), or (iv) if you can tolerate some probability \varepsilon that you fail to accomplish your task. Instead of limiting ourselves to H_{\rm Sh} and H_{\rm vN}, we use also “\varepsilon-smoothed entropies,” Rényi divergences, hypothesis-testing entropies, conditional entropies, etc.

Another hallmark of quantum thermodynamics is results’ generality and simplicity. Thermodynamics characterizes a system with a few macroscopic observables, such as temperature, volume, and particle number. The simplicity of some quantum thermodynamics served a chemist collaborator and me, as explained in the introduction of https://arxiv.org/abs/1811.06551.

Yoram’s question reminded me of one reason why, as an undergrad, I adored studying physics in a liberal-arts college. I ate dinner and took walks with students majoring in economics, German studies, and Middle Eastern languages. They described their challenges, which I analyzed with the physics mindset that I was acquiring. We then compared our approaches. Encountering other disciplines’ perspectives helped me recognize what tools I was developing as a budding physicist. How can we know our corner of the world without stepping outside it and viewing it as part of a landscape?

Plane

1The title epitomizes clarity and simplicity. And I have trouble resisting anything advertised as “the information-theoretic approach to such-and-such.”

The importance of being open

Barcelona refused to stay indoors this May.

Merchandise spilled outside shops onto the streets, restaurateurs parked diners under trees, and ice-cream cones begged to be eaten on park benches. People thronged the streets, markets filled public squares, and the scents of flowers wafted from vendors’ stalls. I couldn’t blame the city. Its sunshine could have drawn Merlin out of his crystal cave. Insofar as a city lives, Barcelona epitomized a quotation by thermodynamicist Ilya Prigogine: “The main character of any living system is openness.”

Prigogine (1917–2003), who won the Nobel Prize for chemistry, had brought me to Barcelona. I was honored to receive, at the Joint European Thermodynamics Conference (JETC) there, the Ilya Prigogine Prize for a thermodynamics PhD thesis. The JETC convenes and awards the prize biennially; the last conference had taken place in Budapest. Barcelona suited the legacy of a thermodynamicist who illuminated open systems.

IMG_0324

The conference center. Not bad, eh?

Ilya Prigogine began his life in Russia, grew up partially in Germany, settled in Brussels, and worked at American universities. His nobelprize.org biography reveals a mind open to many influences and disciplines: Before entering university, his “interest was more focused on history and archaeology, not to mention music, especially piano.” Yet Prigogine pursued chemistry. 

He helped extend thermodynamics outside equilibrium. Thermodynamics is the study of energy, order, and time’s arrow in terms of large-scale properties, such as temperature, pressure, and volume. Many physicists think that thermodynamics describes only equilibrium. Equilibrium is a state of matter in which (1) large-scale properties remain mostly constant and (2) stuff (matter, energy, electric charge, etc.) doesn’t flow in any particular direction much. Apple pies reach equilibrium upon cooling on a countertop. When I’ve described my research as involving nonequilibrium thermodynamics, some colleagues have asked whether I’ve used an oxymoron. But “nonequilibrium thermodynamics” appears in Prigogine’s Nobel Lecture. 

Prigogine photo

Ilya Prigogine

Another Nobel laureate, Lars Onsager, helped extend thermodynamics a little outside equilibrium. He imagined poking a system gently, as by putting a pie on a lukewarm stovetop or a magnet in a weak magnetic field. (Experts: Onsager studied the linear-response regime.) You can read about his work in my blog post “Long live Yale’s cemetery.” Systems poked slightly out of equilibrium tend to return to equilibrium: Equilibrium is stable. Systems flung far from equilibrium, as Prigogine showed, can behave differently. 

A system can stay far from equilibrium by interacting with other systems. Imagine placing an apple pie atop a blistering stove. Heat will flow from the stove through the pie into the air. The pie will stay out of equilibrium due to interactions with what we call a “hot reservoir” (the stove) and a “cold reservoir” (the air). Systems (like pies) that interact with other systems (like stoves and air), we call “open.”

You and I are open: We inhale air, ingest food and drink, expel waste, and radiate heat. Matter and energy flow through us; we remain far from equilibrium. A bumper sticker in my high-school chemistry classroom encapsulated our status: “Old chemists don’t die. They come to equilibrium.” We remain far from equilibrium—alive—because our environment provides food and absorbs heat. If I’m an apple pie, the yogurt that I ate at breakfast serves as my stovetop, and the living room in which I breakfasted serves as the air above the stove. We live because of our interactions with our environments, because we’re open. Hence Prigogine’s claim, “The main character of any living system is openness.”

Apple pie

The author

JETC 2019 fostered openness. The conference sessions spanned length scales and mass scales, from quantum thermodynamics to biophysics to gravitation. One could arrive as an expert in cell membranes and learn about astrophysics.

I remain grateful for the prize-selection committee’s openness. The topics of earlier winning theses include desalination, colloidal suspensions, and falling liquid films. If you tipped those topics into a tube, swirled them around, and capped the tube with a kaleidoscope glass, you might glimpse my thesis’s topic, quantum steampunk. Also, of the nine foregoing Prigogine Prize winners, only one had earned his PhD in the US. I’m grateful for the JETC’s consideration of something completely different.

When Prigogine said, “openness,” he referred to exchanges of energy and mass. Humans can exhibit openness also to ideas. The JETC honored Prigogine’s legacy in more ways than one. Here’s hoping I live up to their example.

IMG_0349

Outside La Sagrada Familia

Thermodynamics of quantum channels

You would hardly think that a quantum channel could have any sort of thermodynamic behavior. We were surprised, too.

How do the laws of thermodynamics apply in the quantum regime? Thanks to novel ideas introduced in the context of quantum information, scientists have been able to develop new ways to characterize the thermodynamic behavior of quantum states. If you’re a Quantum Frontiers regular, you have certainly read about these advances in Nicole’s captivating posts on the subject.

Asking the same question for quantum channels, however, turned out to be more challenging than expected. A quantum channel is a way of representing how an input state can change into an output state according to the laws of quantum mechanics. Let’s picture it as a box with an input state and an output state, like so:

channel-01

A computing gate, the building block of quantum computers, is described by a quantum channel. Or, if Alice sends a photon to Bob over an optical fiber, then the whole process is represented by a quantum channel. Thus, by studying quantum channels directly we can derive statements that are valid regardless of the physical platform used to store and process the quantum information—ion traps, superconducting qubits, photonic qubits, NV centers, etc.

We asked the following question: If I’m given a quantum channel, can I transform it into another, different channel by using something like a miniature heat engine? If so, how much work do I need to spend in order to accomplish this task? The answer is tricky because of a few aspects in which quantum channels are more complicated than quantum states.

In this post, I’ll try to give some intuition behind our results, which were developed with the help of Mario Berta and Fernando Brandão, and which were recently published in Physical Review Letters.

First things first, let’s worry about how to study the thermodynamic behavior of miniature systems.

Thermodynamics of small stuff

One of the important ideas that quantum information brought to thermodynamics is the idea of a resource theory. In a resource theory, we declare that there are certain kinds of states that are available for free, and that there are a set of operations that can be carried out for free. In a resource theory of thermodynamics, when we say “for free,” we mean “without expending any thermodynamic work.”

Here, the free states are those in thermal equilibrium at a fixed given temperature, and the free operations are those quantum operations that preserve energy and that introduce no noise into the system (we call those unitary operations). Faced with a task such as transforming one quantum state into another, we may ask whether or not it is possible to do so using the freely available operations. If that is not possible, we may then ask how much thermodynamic work we need to invest, in the form of additional energy at the input, in order to make the transformation possible.

Interestingly, the amount of work needed to go from one state ρ to another state σ might be unrelated to the work required to go back from σ to ρ. Indeed, the freely allowed operations can’t always be reversed; the reverse process usually requires a different sequence of operations, incurring an overhead. There is a mathematical framework to understand these transformations and this reversibility gap, in which generalized entropy measures play a central role. To avoid going down that road, let’s instead consider the macroscopic case in which we have a large number n of independent particles that are all in the same state ρ, a state which we denote by \rho^{\otimes n}. Then something magical happens: This macroscopic state can be reversibly converted to and from another macroscopic state \sigma^{\otimes n}, where all particles are in some other state σ. That is, the work invested in the transformation from \rho^{\otimes n} to \sigma^{\otimes n} can be entirely recovered by performing the reverse transformation:

asympt-interconversion-states-01

If this rings a bell, that is because this is precisely the kind of thermodynamics that you will find in your favorite textbook. There is an optimal, reversible way of transforming any two thermodynamic states into each other, and the optimal work cost of the transformation is the difference of a corresponding quantity known as the thermodynamic potential. Here, the thermodynamic potential is a quantity known as the free energy F(\rho). Therefore, the optimal work cost per copy w of transforming \rho^{\otimes n} into \sigma^{\otimes n} is given by the difference in free energy w = F(\sigma) - F(\rho).

From quantum states to quantum channels

Can we repeat the same story for quantum channels? Suppose that we’re given a channel \mathcal{E}, which we picture as above as a box that transforms an input state into an output state. Using the freely available thermodynamic operations, can we “transform” \mathcal{E} into another channel \mathcal{F}? That is, can we wrap this box with some kind of procedure that uses free thermodynamic operations to pre-process the input and post-process the output, such that the overall new process corresponds (approximately) to the quantum channel \mathcal{F}? We might picture the situation like this:

channel-simulation-E-F-01

Let us first simplify the question by supposing we don’t have a channel \mathcal{E} to start off with. How can we implement the channel \mathcal{F} from scratch, using only free thermodynamic operations and some invested work? That simple question led to pages and pages of calculations, lots of coffee, a few sleepless nights, and then more coffee. After finally overcoming several technical obstacles, we found that in the macroscopic limit of many copies of the channel, the corresponding amount of work per copy is given by the maximum difference of free energy F between the input and output of the channel. We decided to call this quantity the thermodynamic capacity of the channel:

thermodynamic-capacity-def

Intuitively, an implementation of \mathcal{F}^{\otimes n} must be prepared to expend an amount of work corresponding to the worst possible transformation of an input state to its corresponding output state. It’s kind of obvious in retrospect. However, what is nontrivial is that one can find a single implementation that works for all input states.

It turned out that this quantity had already been studied before. An earlier paper by Navascués and García-Pintos had shown that it was exactly this quantity that characterized the amount of work per copy that could be extracted by “consuming” many copies of a process \mathcal{E}^{\otimes n} provided as black boxes.

To our surprise, we realized that Navascués and García-Pintos’s result implied that the transformation of \mathcal{E}^{\otimes n} into \mathcal{F}^{\otimes n} is reversible. There is a simple procedure to convert \mathcal{E}^{\otimes n} into \mathcal{F}^{\otimes n} at a cost per copy that equals T(\mathcal{F}) - T(\mathcal{E}). The procedure consists in first extracting T(\mathcal{E}) work per copy of the first set of channels, and then preparing \mathcal{F}^{\otimes n} from scratch at a work cost of T(\mathcal{F}) per copy:

asympt-conversion-channels-E-to-F-01

Clearly, the reverse transformation yields back all the work invested in the forward transformation, making the transformation reversible. That’s because we could have started with \mathcal{F}’s and finished with \mathcal{E}’s instead of the opposite, and the associated work cost per copy would be T(\mathcal{E}) - T(\mathcal{F}). Thus the transformation is, indeed, reversible:

asympt-interconversion-channels-01

In turn, this implies that in the many-copy regime, quantum channels have a macroscopic thermodynamic behavior. That is, there is a thermodynamic potential—the thermodynamic capacity—that quantifies the minimal work required to transform one macroscopic set of channels into another.

Prospects for the thermodynamic capacity

Resource theories that are reversible are pretty rare. Reversibility is a coveted property because a reversible resource theory is one in which we can easily understand exactly which transformations are possible. Other than the thermodynamic resource theory of states mentioned above, most instances of a resource theory—especially resource theories of channels—typically produce the kind of overheads in the conversion cost that spoil reversibility. So it’s rather exciting when you do find a new reversible resource theory of channels.

Quantum information theorists, especially those working on the theory of quantum communication, care a lot about characterizing the capacity of a channel. This is the maximal amount of information that can be transmitted through a channel. Even though in our case we’re talking about a different kind of capacity—one where we transmit thermodynamic energy and entropy, rather than quantum bits of messages—there are some close parallels between the two settings from which both fields of quantum communication and quantum thermodynamics can profit. Our result draws deep inspiration from the so-called quantum reverse Shannon theorem, an important result in quantum communication that tells us how two parties can communicate using one kind of a channel if they have access to another kind of a channel. On the other hand, the thermodynamic capacity at zero energy is a quantity that was already studied in quantum communication, but it was not clear what that quantity represented concretely. This quantity gained even more importance as it was identified as the entropy of a channel. Now, we see that this quantity has a thermodynamic interpretation. Also, the thermodynamic capacity has a simple definition, it is relatively easy to compute and it is additive—all desirable properties that other measures of capacity of a quantum channel do not necessarily share.

We still have a few rough edges that I hope we can resolve sooner or later. In fact, there is an important caveat that I have avoided mentioning so far—our argument only holds for special kinds of channels, those that do the same thing regardless of when they are applied in time. (Those channels are called time-covariant.) A lot of channels that we’re used to studying have this property, but we think it should be possible to prove a version of our result for any general quantum channel. In fact, we do have another argument that works for all quantum channels, but it uses a slightly different thermodynamic framework which might not be physically well-grounded.

That’s all very nice, I can hear you think, but is this useful for any quantum computing applications? The truth is, we’re still pretty far from founding a new quantum start-up. The levels of heat dissipation in quantum logic elements are still orders of magnitude away from the fundamental limits that we study in the thermodynamic resource theory.

Rather, our result teaches us about the interplay of quantum channels and thermodynamic concepts. We not only have gained useful insight on the structure of quantum channels, but also developed new tools for how to analyze them. These will be useful to study more involved resource theories of channels. And still, in the future when quantum technologies will perhaps approach the thermodynamically reversible limit, it might be good to know how to implement a given quantum channel in such a way that good accuracy is guaranteed for any possible quantum input state, and without any inherent overhead due to the fact that we don’t know what the input state is.

Thermodynamics, a theory developed to study gases and steam engines, has turned out to be relevant from the most obvious to the most unexpected of situations—chemical reactions, electromagnetism, solid state physics, black holes, you name it. Trust the laws of thermodynamics to surprise you again by applying to a setting you’d never imagined them to, like quantum channels.