maxwell demon experiment

Maxwell's demon experiment could be made real without breaking physics

A thought experiment called Maxwell’s demon, long hypothesised to break the laws of physics, could be made using simple electronic devices at macroscopic scales – without upsetting the laws of thermodynamics

By Leah Crane

10 May 2022

Illustration of a computer motherboard with microchips, transistors and semiconductors.

An illustration of a computer motherboard with transistors

EDUARD MUZHEVSKYI / SCIENCE PHOTO LIBRARY

A 155-year-old thought experiment, long thought to break the laws of thermodynamics, could be made real on a large scale.

Maxwell’s demon is a thought experiment first proposed by Scottish mathematician James Clerk Maxwell in 1867. He imagined a tiny demon controlling a door between two gas-filled chambers of a box. By carefully opening and closing the door, the demon allows only fast-moving gas particles into one chamber and slow-moving ones into the other. Because the speed of its particles determines a gas’s temperature ,…

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A Reboot of the Maxwell’s Demon Thought Experiment—in Real Life

Antoine Naert has reconstructed a prop straight out of the Industrial Revolution. But don’t expect steam-powered engines or decorative brass gears. Naert, a physicist at ENS Lyon in France, and his team have a unique take : Their contraption consists of a soundproofed glass container placed on a vibrating platform that shakes some 300 steel beads inside it like a quiet maraca.

The device looks less steampunk and more 21st-century science fair project. But make no mistake—it’s Naert’s reinterpretation of a 19th-century thought experiment known as Maxwell’s Demon. To look for violations of the second law of thermodynamics, in 1867 Scottish physicist James Clerk Maxwell proposed the concept of a “demon” that could interact with microscopic particles. In a nutshell, the second law states that without the input of additional energy, heat always flows from a hot region toward a cold one. Imagining a demon that could disrupt the flow forces physicists to contemplate what the second law actually means.

“We can get inside the working of the second law by such thought experiments,” says Naert. “And we built it for real.”

Maxwell’s contemporaries had uncovered the second law while investigating the most efficient way to convert heat—from burning coal, for example—into the motion of pistons and turbines. But this principle turns out to have far broader implications than for just steam engines. It’s why ice always melts in a drink at room temperature, but the drink never turns into ice. Put another way, the second law indicates that certain processes in nature can proceed only in one direction. Such irreversible processes distinguish the past from present—what physicists call “the arrow of time.” “This is really the principle that describes why we get old,” says Naert.

The second law “is so much of our experience,” says physicist Harvey Leff, a professor emeritus at California State Polytechnic University, Pomona. “We observe it when we put our hand near a fire and are like, ‘Oops, that’s hot,’ and move our hand away.” Any exceptions, like a refrigerator where heat flows from cold to hot, requires a power source.

But what does it mean for something to be hot or cold? To answer that question for steam and other gases, 19th-century physicists developed the concept of temperature to describe the average speed of numerous particles bouncing around in random directions—some faster, some slower. (Later, they would discover the particles were atoms and molecules.) Hotter temperatures mean particles moving at a faster average speed.

The goal of a steam engine is to convert the chaotic motion of hot water vapor into motion in a defined direction, such as the vertical motion of a piston. To create orderly motion, the second law says that you need to keep the gas in two regions at different temperatures. If the gas were all at the same temperature, the vapor particles would move in random directions, and that random motion would not push the piston in a specific direction.

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Maxwell sought potential violations of the second law, as scientists do when someone proposes that a principle should apply to all of nature. Specifically, he tried to devise a theoretical engine that exploited gas at a single temperature. In 1867, he presented a thought experiment, commonly depicted as a box with two compartments containing the same gas at a single temperature. The gas molecules bounce around at a range of speeds following a specific distribution.

But what if a little demon resided at the partition between the two compartments and sorted the molecules according to speed? One compartment would end up with faster molecules on average, corresponding to a hotter temperature, while the other compartment would contain slower molecules on average, corresponding to a colder temperature. The demon would have created two regions of different temperatures. Heat would again flow from hot to cold, making it possible to generate motion in a particular direction, which someone could use to move a piston.

Thus, the demon seems to have created an engine from a gas at one temperature, violating the second law. “Maxwell’s Demon was one of the greatest threats to the second law,” says physicist Nicole Yunger Halpern of the University of Maryland.

For more than a century, physicists have contemplated the thought experiment, and in recent years have even converted it into real-life machines. Naert’s device emulates Maxwell’s Demon, except instead of randomly bouncing molecules, he uses randomly bouncing steel beads. The beads shake around the container to hit a rotating blade in different directions. Naert has engineered it such that the blade turns a dynamo to generate an electric current—but only when the blade rotates in a particular direction. He can then use that current to turn a motor, for example. Like Maxwell’s Demon, the contraption turns chaotic motion—analogous to heat—into orderly motion.

It’s surprising that the device could generate orderly motion at all, according to Yunger Halpern, because the machine is so large. Most real-world constructions of Maxwell’s Demon use microscopic particles such as atoms or electrons .

To be clear, Naert’s device does not violate the second law of thermodynamics, nor does Maxwell’s Demon. Physicists, pondering the demon over decades, have delivered multiple explanations for why it doesn’t. One is that in order to sort the beads, the demon has to be cooler than the rest of the gas, says Naert. Thus, the container of gas particles is not a single temperature, which contradicts the premise of the thought experiment. In the case of Naert’s device, the rapidly bouncing steel beads are at one temperature, whereas the electronic component that converts the beads’ motion into the rotation of a blade is another temperature.

So why recreate Maxwell’s Demon? Physicists have used the thought experiment to explore common concepts in wildly different contexts. For example, in the 20th century, it led physicists to discover the physical nature of information. In order for the demon to sort molecules by speed, it needs some way of knowing the particles’ speed. The demon would need to store that knowledge and erase that information. From these ideas, physicists figured out that information isn’t just some abstract concept that we humans harness to communicate. It’s the physical state of some object, like representing the voltage across a transistor as a bit of information—a key concept now fundamental to the study of computing.

In addition, the second law of thermodynamics signifies the statistical nature of the universe. Its building blocks are not stars, planets, humans, or bacteria—they’re the atoms and molecules that make us up. You can think of the atoms in the universe as a deck of cards, constantly being shuffled and reshuffled. By the end of the reshuffling, the deck will have no semblance of order. But instead of dealing with a deck of 52 cards, the universe has a deck on the order of 10 82 atoms.

Or if you want to be more manageable, consider the 10 24 molecules in a cup of coffee. If you drop a sugar cube into that coffee, those sugar molecules have so many more ways of redistributing themselves throughout the coffee than staying in cube form. Or consider someone who releases perfume in a room. That perfume will rush to fill the space. This illustrates the concept of entropy, often described as “disorder.” The most likely arrangement of atoms has the highest entropy. A deck of cards sorted according to the four suits has lower entropy, for example, than one that is not. Similarly, dissolved sugar molecules cannot re-cube, and the perfume cannot rush back into the vial, without some external intervention requiring energy.

Ultimately, the second law of thermodynamics says that energy moves around in nature to increase entropy. “If you ask what physics is, you might just say it is the study of energy,” says Leff. “What’s happening as far as I can see is that energy keeps redistributing itself.”

However, as people invent new technology, it’s not always clear how the second law applies. For example, seemingly straightforward concepts like temperature get complicated. Naert’s steel beads are at room temperature in the conventional sense, defined according to the average speed of their constituent molecules. This is the same temperature that you might associate with how it would feel to touch the bead. But Naert has identified another property of his system, which he interprets as a different type of temperature, defined not by the speed of its constituent molecules, but that of the glass beads bouncing around. It’s mathematically analogous to conventional temperature, as both involve the speed of discrete particles, but has no relation to whether you will burn or cool your hand when touching it. Naert plans to work with theorists to better understand what this type of temperature means, along with measuring and understanding the role of entropy in his device.

In addition, physicists have had to revisit the second law as researchers build smaller and smaller devices, such as quantum engines —made of a few atoms. They want to know, for example, whether the second law limits these quantum engines in the same way as conventional macroscopic engines, says Yunger Hapern.

Naert’s personal motivation to build this machine was intellectual curiosity, but he thinks that studying the second law in macroscopic contexts could potentially lead to more efficient machines for harvesting energy from ocean waves, for example, as it illustrates the conversion of chaotic macroscopic motion into orderly motion that could be used to charge a battery or move a turbine. In addition, he sees his device as a teaching tool. “This is incredibly close to the original idea from the 19th century,” he says. But because he uses beads instead of molecules, “you can see everything because it’s in centimeters.” With his new device, Naert has invited Maxwell’s Demon to confuse and enlighten us at a new scale.

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Maxwell’s demon , hypothetical intelligent being (or a functionally equivalent device) capable of detecting and reacting to the motions of individual molecules. It was imagined by James Clerk Maxwell in 1871, to illustrate the possibility of violating the second law of thermodynamics . Essentially, this law states that heat does not naturally flow from a cool body to a warmer; work must be expended to make it do so. Maxwell envisioned two vessels containing gas at equal temperatures and joined by a small hole. The hole could be opened or closed at will by “a being” to allow individual molecules of gas to pass through. By passing only fast-moving molecules from vessel A to vessel B and only slow-moving ones from B to A, the demon would bring about an effective flow from A to B of molecular kinetic energy . This excess energy in B would be usable to perform work ( e.g., by generating steam), and the system could be a working perpetual motion machine. By allowing all molecules to pass only from A to B, an even more readily useful difference in pressure would be created between the two vessels. About 1950 the French physicist Léon Brillouin exorcised the demon by demonstrating that the decrease in entropy resulting from the demon’s actions would be exceeded by the increase in entropy in choosing between the fast and slow molecules.

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Maxwell’s Demon: A Uniquely Quantum Effect in Erasing Information Discovered

By Trinity College Dublin October 27, 2020

A bit of information can be encoded in the position of a particle (left or right). A demon can erase a classical bit (blue) by raising one side until the particle is definitely on the right. A quantum particle (red) can also tunnel under the barrier, which generates more heat. Credit: Professor Goold, Trinity College Dublin

Researchers from Trinity have discovered a uniquely quantum effect in erasing information that may have significant implications for the design of quantum computing chips. Their surprising discovery brings back to life the paradoxical “Maxwell’s demon,” which has tormented physicists for over 150 years.

The thermodynamics of computation was brought to the fore in 1961 when Rolf Landauer, then at IBM, discovered a relationship between the dissipation of heat and logically irreversible operations. Landauer is known for the mantra “Information is Physical,” which reminds us that information is not abstract and is encoded on physical hardware.

The “bit” is the currency of information (it can be either 0 or 1) and Landauer discovered that when a bit is erased there is a minimum amount of heat released. This is known as Landauer’s bound and is the definitive link between information theory and thermodynamics.

Professor John Goold’s QuSys group at Trinity is analyzing this topic with quantum computing in mind, where a quantum bit (a qubit, which can be 0 and 1 at the same time) is erased.

In just-published work in the journal, Physical Review Letters , the group discovered that the quantum nature of the information to be erased can lead to large deviations in the heat dissipation, which is not present in conventional bit erasure.

Thermodynamics and Maxwell’s demon

One hundred years previous to Landauer’s discovery people like Viennese scientist, Ludwig Boltzmann, and Scottish physicist, James Clerk Maxwell, were formulating the kinetic theory of gases, reviving an old idea of the ancient Greeks by thinking about matter being made of atoms and deriving macroscopic thermodynamics from microscopic dynamics.

Professor Goold says:

“Statistical mechanics tells us that things like pressure and temperature, and even the laws of thermodynamics themselves, can be understood by the average behavior of the atomic constituents of matter. The second law of thermodynamics concerns something called entropy which, in a nutshell, is a measure of the disorder in a process. The second law tells us that in the absence of external intervention, all processes in the universe tend, on average, to increase their entropy and reach a state known as thermal equilibrium.

“It tells us that, when mixed, two gases at different temperatures will reach a new state of equilibrium at the average temperature of the two. It is the ultimate law in the sense that every dynamical system is subject to it. There is no escape: all things will reach equilibrium, even you!”

However, the founding fathers of statistical mechanics were trying to pick holes in the second law right from the beginning of the kinetic theory. Consider again the example of a gas in equilibrium: Maxwell imagined a hypothetical “neat-fingered” being with the ability to track and sort particles in a gas based on their speed.

Maxwell’s demon, as the being became known, could quickly open and shut a trap door in a box containing gas, and let hot particles through to one side of the box but restrict cold ones to the other. This scenario seems to contradict the second law of thermodynamics as the overall entropy appears to decrease and perhaps physics’ most famous paradox was born.

But what about Landauer’s discovery about the heat-dissipated cost of erasing information? Well, it took another 20 years until that was fully appreciated, the paradox solved, and Maxwell’s demon finally exorcised.

Landauer’s work inspired Charlie Bennett – also at IBM – to investigate the idea of reversible computing. In 1982 Bennett argued that the demon must have a memory, and that it is not the measurement but the erasure of the information in the demon’s memory which is the act that restores the second law in the paradox. And, as a result, computation thermodynamics was born.

New findings

Now, 40 years on, this is where the new work led by Professor Goold’s group comes to the fore, with the spotlight on quantum computation thermodynamics.

In the recent paper, published with collaborator Harry Miller at the University of Manchester and two postdoctoral fellows in the QuSys Group at Trinity, Mark Mitchison and Giacomo Guarnieri, the team studied very carefully an experimentally realistic erasure process that allows for quantum superposition (the qubit can be in state 0 and 1 at the same time).

Professor Goold explains:

“In reality, computers function well away from Landauer’s bound for heat dissipation because they are not perfect systems. However, it is still important to think about the bound because as the miniaturization of computing components continues, that bound becomes ever closer, and it is becoming more relevant for quantum computing machines. What is amazing is that with technology these days you can really study erasure approaching that limit.

“We asked: ‘What difference does this distinctly quantum feature make for the erasure protocol?’ And the answer was something we did not expect. We found that even in an ideal erasure protocol – due to quantum superposition – you get very rare events that dissipate heat far greater than the Landauer limit.

“In the paper we prove mathematically that these events exist and are a uniquely quantum feature. This is a highly unusual finding that could be really important for heat management on future quantum chips – although there is much more work to be done, in particular in analyzing faster operations and the thermodynamics of other gate implementations.

“Even in 2020, Maxwell’s demon continues to pose fundamental questions about the laws of nature.”

Reference: “Quantum Fluctuations Hinder Finite-Time Information Erasure near the Landauer Limit” by Harry J. D. Miller, Giacomo Guarnieri, Mark T. Mitchison and John Goold, 15 October 2020, Physical Review Letters . DOI: 10.1103/PhysRevLett.125.160602

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1 comment on "maxwell’s demon: a uniquely quantum effect in erasing information discovered".

I totally clicked on this article to inspect those colorful titty models

Maxwell’s demon and the hunt for alien life

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Timo Hannay is founder of the education data-analytics company SchoolDash, based in London.

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DIC micrograph of a section through a moss plant leaf, showing the chloroplasts (round, green) within the cells (hexagonal).

Chloroplasts inside moss cells. These organelles conduct photosynthesis, a process that relies on quantum effects. Credit: John Durham/SPL

The Demon in the Machine: How Hidden Webs of Information Are Finally Solving the Mystery of Life Paul Davies Allen Lane (2019)

Biology, long the domain of qualitative theories and experimental subjects that refuse to do the same thing twice, is now thoroughly data-driven. Propelled by the twentieth-century revolutions in molecular biology and computing, its emphasis has shifted from observing and describing to sequencing and calculating. In the process, biology has increasingly become like physics — a development that has caught the attention of quite a few physicists.

One such boundary-transcending thinker is the cosmologist and writer Paul Davies. His latest book, The Demon in the Machine , presents a case that information is central not just to doing biology, but to understanding life itself. He follows in esteemed footsteps. In 1943, the Austrian physicist Erwin Schrödinger delivered a landmark series of public lectures at Trinity College Dublin. Published the following year as What Is Life? , it explained many principles of molecular genetics — a decade before the structure of DNA was discovered (see P. Ball Nature 560 , 548–550; 2018 ).

As a quantum theorist, Schrödinger was particularly struck by the observation that atoms, although profoundly unpredictable, can form highly ordered systems. Furthermore, those systems persist for long periods and even replicate, thus seeming to evade the second law of thermodynamics, which states that total entropy, or disorder, can only increase.

This classic account serves as Davies’s starting point. As a cosmologist, however, his principal question arises from a consideration not of the irreducibly small, but of the incomparably large. If life exists elsewhere in the Universe, Davies wonders, how can we recognize it? Searches for signs of liquid water, organic chemistry or certain atmospheric gases (such as oxygen, carbon dioxide or methane) make sense given the characteristics of the one ecosystem we know, but to accept these as the essence of life seems to him (and me) desperately narrow-minded.

Davies claims that life’s defining characteristics are better understood in terms of information. This is not as absurd as it may seem. Energy is abstract, yet we have little trouble accepting it as a causal factor. Indeed, energy and information are closely related through entropy.

Davies explains this connection by referring to Maxwell’s demon. Victorian physicist James Clerk Maxwell’s celebrated thought experiment features a hypothetical miniature beast perching at an aperture between two containers of gas, where it allows only certain molecules to pass, depending on their kinetic energy. The demon can thus create a temperature gradient between the containers: a reduction in overall entropy, apparently breaking the second law of thermodynamics. The resolution to this paradox seems to lie in the fact that the demon must gather information about the properties of each molecule, and for this it requires a recording device, such as a brain or a miniature notebook. When its storage space eventually runs out, the information must be deleted, a process that necessarily produces an increase in total entropy.

From this perspective, living systems can be seen as composed of countless such ‘demons’ (proteins and other cellular machinery) that maintain local order by pumping disorder (often in the form of heat) into their surroundings. Davies adroitly brings Schrödinger’s account up to date by way of Claude Shannon’s information theory, Turing machines (universal computers), von Neumann machines (self-replicating universal constructors), molecular biology, epigenetics, information-integration theories of consciousness and quantum biology (which concerns quantum effects in processes from photosynthesis to insect coloration and bird navigation).

Such disparate threads might seem like unpromising material from which to weave a coherent narrative. But Davies does so admirably, with only occasional forays into areas that feel slightly out of place. One such is the brief account of his work on cancer, which he sees less as an example of broken cellular machinery and more as a regression to an earlier evolutionary state, when single-celled organisms responded to adverse conditions by replicating.

What practical difference does it make to see life as informational? We don’t yet know, but can speculate. For one thing, if the essential characteristics of life are entropic, extraterrestrial searches based on chemistry could be misguided. It might be more useful to look for phenomena such as ‘anti-accretion’ — in which matter is regularly transferred from a planet’s surface into space. Earth has experienced this since the 1950s, when the one-way traffic in asteroids and meteorites plunging into the globe was finally counteracted by the launch of the first artificial satellites. Arguably, such situations are not merely consistent with the presence of life, but almost impossible to explain in any other way.

Moreover, a definition of life that depends on its informational characteristics rather than its carbon-based substrate could force a reappraisal of our attitudes towards artificial systems embodied in computers. We are already beginning to treat these as companions; might we eventually come to see them as living creatures rather than mere imitations? With apologies to Charles Darwin, there is grandeur in this view of life.

As well as having eclectic interests, Davies is iconoclastic and opinionated. Although certainly no believer in a vital force distinct from physics or chemistry, he has little time for reductionism, believing that life cannot be fully explained in terms of lower-level laws (such as the second law of thermodynamics), even in principle. In a final nod to Schrödinger — who believed that a proper understanding of life might reveal “other laws of physics hitherto unknown” — Davies closes by arguing that biology might yet contain deep lessons for physics. This is highly speculative and, in my (biologist’s) view, probably wrong. But this is not a criticism. On the contrary, if only more of us were wrong in such thought-provoking ways, we might more readily uncover the truth.

Nature 565 , 427-428 (2019)

doi: https://doi.org/10.1038/d41586-019-00215-9

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Maxwell’s demon—a historical review.

1. Introduction

2. early history.

The law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations—then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation, well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation [ 1 ]. —Arthur Eddington

One of the best established facts in thermodynamics is that it is impossible in a system enclosed in an envelope which permits neither change of volume nor passage of heat, and in which both the temperature and the pressure are everywhere the same, to produce any inequality of temperature or of pressure without the expenditure of work. This is the second law of thermodynamics, and it is undoubtedly true as long as we can deal with bodies only in mass, and have no power of perceiving or handling the separate molecules of which they are made up. But if we conceive a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us. For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics. [ 2 ]

3. Different Demons

While he did not fully solve the puzzle, the tremendous import of Szilard’s 1929 paper is clear: He identified the three central issues related to information-gathering Maxwell’s demons as we understand them today—measurement, information, and entropy—and he established the underpinnings of information theory and its connections with physics. [ 8 ]

4. Measurement, Information, and Erasure

5. molecular ratchets, 6. photonic maxwell’s demon.

As the measurement devices are randomized in this process, the experiment cannot violate the second law of thermodynamics.

7. Information and Quantum Computing

8. other recent work.

“…whereas the original photon was part of an orderly train of photons (the laser beam), the scattered photons go off in random directions. The photons thus become more disordered, and we showed that the corresponding increase in the entropy of the light exactly balanced the entropy reduction of the atoms because they get confined by the one-way gate. Therefore, single-photon cooling works as a Maxwell demon in the very sense envisioned by Leo Szilard in 1929.” [ 31 ]

We believe that we are still several steps away from a complete understanding of the physical nature of information. First, it is necessary to unify the existing theoretical frameworks, and investigate a comprehensive theory for general processes. Second, we need to verify if other phenomena, such as copolymerization and proofreading, can be analyzed within this unified framework. And third, we must return to Maxwell’s original concern about the second law and try to address the basic problems of statistical mechanics, such as the emergence of the macroscopic world and the subjectivity of entropy, in the light of a general theory of information. [ 36 ]

9. Conclusions and a Parable

Acknowledgments, conflicts of interest.

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Rex, A. Maxwell’s Demon—A Historical Review. Entropy 2017 , 19 , 240. https://doi.org/10.3390/e19060240

Rex A. Maxwell’s Demon—A Historical Review. Entropy . 2017; 19(6):240. https://doi.org/10.3390/e19060240

Rex, Andrew. 2017. "Maxwell’s Demon—A Historical Review" Entropy 19, no. 6: 240. https://doi.org/10.3390/e19060240

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Maxwell’s demon in the quantum world

Departamento de Física Atómica, Molecular y Nuclear and GISC, Universidad Complutense de Madrid, 28040-Madrid, Spain

Information and thermodynamics are intimately connected. This idea was first illustrated by Maxwell with his celebrated demon: an intelligent being who uses his knowledge about the position and velocity of the molecules in a gas to transfer heat against a temperature gradient without expenditure of work, beating the second law of thermodynamics. The Szilard engine is a stylized version of the demon, where a yes/no measurement of a classical single-particle system allows one to extract a tiny amount of energy, k T ln 2 , k being the Boltzmann constant, from a thermal reservoir at temperature T . The engine has been around for almost a century now [1] . Along the way it has furnished insight into the foundations of statistical mechanics, become the canonical model for investigations of feedback-controlled systems, and even spurred the creation of a new field: the thermodynamics of computation.

In a paper appearing in Physical Review Letters [2] , Sang Wook Kim at the Pusan National University, Korea, and the University of Tokyo, Japan, along with collaborators from the University of Tokyo analyze a multiparticle quantum version of the Szilard engine, highlighting how the quantum statistics of fermions and bosons can dramatically affect the engine’s performance. The paper helps to clarify the interplay between information and entropy in the quantum world, the thermodynamic consequences of measurement, and the information content of spatially extended multiparticle states.

The original Szilard engine consists of a classical particle in a box attached to a thermal reservoir at temperature T . An external agent inserts a wall in the middle of the box, confining the particle in one half. Next, the agent measures in which half of the box the particle is trapped, and then slowly moves the wall to the opposite side of the box, allowing the particle to perform mechanical work. The motion of the wall is an isothermal expansion and the amount of work performed can easily be calculated, k T ln 2 . In the classical case, the insertion of the wall can be done, ideally, at zero energy cost. Therefore the whole process results in the extraction of a net amount of energy, k T ln 2 , from the thermal bath or a decrease of the entropy by k ln 2 . This is precisely the information gathered by the measurement, in the appropriate units (the information in a yes/no measurement is one bit or ln 2 nats, a unit of information that uses natural instead of base 2 logarithms). The second law of thermodynamics demands that this decrease in the entropy must be compensated by an increase in the entropy of the agent operating the engine. This can occur either in the measurement or in the erasure of the information gathered [1] . Sagawa and Ueda have unified both possibilities in a simple and elegant theoretical framework [3] .

Despite the age of the problem, the analysis of the Szilard engine continues to benefit from new theoretical and experimental developments. The engine has been studied using a new class of powerful results in nonequilibrium statistical mechanics that characterize the fluctuations in the energetics of arbitrary thermodynamic processes: work and fluctuation theorems. In particular, the extraction of work has been related to the time-reversal asymmetry of the engine’s operation [4, 5] . Additionally, the engine has recently been realized in the laboratory—almost one century since Szilard proposed the engine as a gedanken experiment—using a charged Brownian rotor in an electrostatic field controlled by feedback [6] .

In their paper, Kim et al. present a general analysis of the isothermal Szilard engine using an arbitrary number of quantum particles. The quantum Szilard engine exhibits intriguing differences with respect to the classical case: the insertion of the wall cannot be done without expending work [7] , and the measurement generally involves the collapse of the wave function [1] . Although these subtleties have been analyzed to some extent in previous works, the present article extends these results. Most importantly, by considering two or more noninteracting quantum particles, quantum statistics enters the game, bringing in new effects. For an illustration of the operation of this quantum Szilard engine, see Fig. 1 .

Take, for example, the case of two particles: Kim et al. show that the work extracted in one operational cycle is 2 k T f 0 ln f 0 , where f 0 is the probability of finding both particles in the same half of the box. For distinguishable particles, f 0 = 1 / 2 , and one recovers the classical result for a single particle. The reason is that although the measurement carries more information ( 2 bits) than in the one-particle case, two of the possible outcomes (namely, one particle in each half of the box) are not used to extract energy.

More interesting effects occur when both particles are bosons or fermions, especially close to zero temperature where the quantum nature of the particles is more significant. For fermions near zero temperature, the Pauli exclusion principle forces the two fermions to be found in separate halves of the box, thus f 0 ≈ 0 and the work extracted is much smaller than in the classical case. On the other hand, at zero temperature, bosons like to clump together. Kim et al. have shown that in this case, f 0 = 1 / 3 , and the work extracted is larger than in the classical case. Consequently, bosons more efficiently extract work from measurement, whereas fermions can be completely inefficient. These differences with respect to the classical case remain even when the thermal energy k T is far above the energy of the ground state.

While the field of quantum information is well developed, we are still a long way from a full understanding of the interplay between information and thermodynamics in quantum systems. Although the paper of Kim et al. does not make explicit use of quantum coherence, it is a first step in this pursuit. Experimental realization of a quantum Szilard engine is still some way off, although potential candidates include trapped cold atoms in the bosonic case and quantum dots from semiconductor heterostructures for the fermionic version.

H. S. Leff and A. F. Rex, Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics, Bristol, 2003)[ Amazon ][ WorldCat ]
S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys. Rev. Lett. 106 , 070401 (2011)
T. Sagawa and M. Ueda, Phys. Rev. Lett. 102 , 250602 (2009)
R. Kawai, J. M. R. Parrondo, and C. Van den Broeck, Phys. Rev. Lett. 98 , 080602 (2007)
J. M. Horowitz and S. Vaikuntanathan, Phys. Rev. E 82 , 061120 (2010)
S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nature Phys. 6 , 988 (2010)
J. Gea-Banacloche and H. S. Leff, Fluct. Noise Lett. 5 , C39 (2005)

About the Authors

Juan M.R. Parrondo is a Professor at Universidad Complutense de Madrid, Spain. His research focuses on foundations of statistical mechanics and fluctuations and stochastic processes applied to statistical physics, condensed matter, and biophysics. He obtained his Ph.D. at the Universidad Complutense in 1992.

Jordan M. Horowitz is currently a postdoctoral researcher at Universidad Complutense de Madrid. He received his B.A. in physics and mathematics at Columbia University in New York in 2005 and his Ph.D. in physics from the University of Maryland in 2010. His research interests are the thermodynamics and statistical mechanics of far-from-equilibrium systems and the thermodynamic consequences of feedback and information processing.

Quantum Szilard Engine

Sang Wook Kim, Takahiro Sagawa, Simone De Liberato, and Masahito Ueda

Phys. Rev. Lett. 106 , 070401 (2011)

Published February 14, 2011

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How maxwell’s demon continues to startle scientists.

By Jonathan O’Callaghan
April 29, 2021

Reprinted with permission from Quanta Magazine’s Abstractions blog.

T he universe bets on disorder. Imagine, for example, dropping a thimbleful of red dye into a swimming pool. All of those dye molecules are going to slowly spread throughout the water.

Physicists quantify this tendency to spread by counting the number of possible ways the dye molecules can be arranged. There’s one possible state where the molecules are crowded into the thimble. There’s another where, say, the molecules settle in a tidy clump at the pool’s bottom. But there are uncountable billions of permutations where the molecules spread out in different ways throughout the water. If the universe chooses from all the possible states at random, you can bet that it’s going to end up with one of the vast set of disordered possibilities.

Seen in this way, the inexorable rise in entropy, or disorder, as quantified by the second law of thermodynamics, takes on an almost mathematical certainty. So of course physicists are constantly trying to break it.

One almost did. A thought experiment devised by the Scottish physicist James Clerk Maxwell in 1867 stumped scientists for 115 years. And even after a solution was found, physicists have continued to use “Maxwell’s demon” to push the laws of the universe to their limits.

In the thought experiment, Maxwell imagined splitting a room full of gas into two compartments by erecting a wall with a small door. Like all gases, this one is made of individual particles. The average speed of the particles corresponds to the temperature of the gas—faster is hotter. But at any given time, some particles will be moving more slowly than others.

What if, suggested Maxwell, a tiny imaginary creature—a demon, as it was later called —sat at the door. Every time it saw a fast-moving particle approaching from the left-hand side, it opened the door and let it into the right-hand compartment. And every time a slow-moving particle approached from the right, the demon let it into the left-hand compartment.

After a while, the left-hand compartment would be full of slow, cold particles, and the right-hand compartment would grow hot. This isolated system would seem to grow more orderly, not less, because two distinguishable compartments have more order than two identical compartments. Maxwell had created a system that appeared to defy the rise of entropy, and thus the laws of the universe.

“He tried to prove a system where the entropy would decrease,” said Laia Delgado Callico , a physicist at King’s College London. “It’s a paradox.”

Two advances would be crucial to solving Maxwell’s demon. The first was by the American mathematician Claude Shannon, regarded as the founder of information theory. In 1948, Shannon showed that the information content of a message could be quantified with what he called the information entropy. “In the 19th century, no one knew about information,” said Takahiro Sagawa , a physicist at the University of Tokyo. “The modern understanding of Maxwell’s demon was established by Shannon’s work.”

The second vital piece of the puzzle was the principle of erasure. In 1961, the German American physicist Rolf Landauer showed that any logically irreversible computation, such as the erasing of information from a memory, would result in a minimal nonzero amount of work converted into heat dumped into the environment, and a corresponding rise in entropy. Landauer’s erasure principle provided a tantalizing link between information and thermodynamics. “Information is physical,” he later proclaimed .

In 1982, the American physicist Charles Bennett put the pieces of the puzzle together . He realized that Maxwell’s demon was at core an information-processing machine: It needed to record and store information about individual particles in order to decide when to open and close the door. Periodically it would need to erase this information. According to Landauer’s erasure principle, the rise in entropy from the erasure would more than compensate for the decrease in entropy caused by the sorting of the particles. “You need to pay,” said Gonzalo Manzano , a physicist at the Institute for Quantum Optics and Quantum Information in Vienna. The demon’s need to make room for more information inexorably led to a net increase in disorder.

Then in the 21st century, with the thought experiment solved, the real experiments began. “The most important development is we can now realize Maxwell’s demon in laboratories,” said Sagawa.

In 2007 scientists used a light-powered gate to demonstrate the idea of Maxwell’s demon in action; in 2010, another team devised a way to use the energy produced by the demon’s information to coax a bead uphill; and in 2016 scientists applied the idea of Maxwell’s demon to two compartments containing not gas, but light.

“We switched the roles of matter and light,” said Vlatko Vedral , a physicist at the University of Oxford and one of the study’s co-authors. The researchers were ultimately able to charge a very small battery.

Others wondered if there might be less demanding ways to use information to extract useful work from a similar system. And research published in February 2021 in Physical Review Letters seems to have found a way to do so. The work makes the demon into a gambler.

The team, led by Manzano, wondered if there was a way to implement something like Maxwell’s demon but without the information requirements. They imagined a two-compartment system with a door, as before. But in this case, the door would open and close on its own. Sometimes particles would randomly separate themselves into hotter and colder compartments. The demon could only watch this process and decide when to turn the system off. In theory this process could create a small temperature imbalance, and therefore a useful heat engine, if the demon was smart about when to end the experiment and lock any temperature imbalance in place, much as a smart gambler on a hot streak knows when to leave the table. “You can either play all night on the roulette table, or you can stop if you win $100,” said Édgar Roldán , a physicist at the International Center for Theoretical Physics in Italy who was a co-author on the study. “We’re saying we don’t need such a complicated device as Maxwell’s demon to extract work in the second law. We can be more relaxed.” The researchers then implemented such a gambling demon in a nanoelectronic device, to show it was possible.

Ideas like this could prove useful in designing more efficient thermal systems, like refrigerators, or even in developing more advanced computer chips, which may be approaching a fundamental limit dictated by Landauer’s principle.

For the time being, though, our laws of the universe are safe, even when placed under the greatest scrutiny. What has changed is our understanding of information in the universe, and with it our appreciation of Maxwell’s demon, first a troublesome paradox, and now an invaluable concept—one that has helped to illuminate the remarkable link between the physical world and information.

Jonathan O’Callaghan is a freelance space and science journalist based in London. He writes regularly for a number of publications including The New York Times , Scientific American , New Scientist, Forbes, and Wired . You can read more of his work or get in contact at jonathanocallaghan.com , or find him on Twitter @Astro_Jonny .

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February 28, 2011

How Maxwell's Demon Cools a Gas to Microkelvin Temperatures [Animation]

Physicists have brought a 19th-century thought experiment to life

By Davide Castelvecchi

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By Davide Castelvecchi In his article " Demons, Entropy and the Quest for Absolute Zero ," physicist Mark G. Raizen describes how to cool a rarefied gas down to temperatures of just millionths of a degree above absolute zero. The starting point is to take a gas that has already been cooled to one one-hundredth of a kelvin (using a device called an atomic coilgun, also described in the text) and place it in a magnetic trap. Then the serious freeze can begin, using the new technique of single-photon cooling. Single-photon cooling exploits a one-way gate inspired by a 19th-century thought experiment by James Clerk Maxwell. The great Scottish physicist theorized the existence of a "demon" that seemed able to concentrate the atoms of a gas into a smaller volume without raising their temperature, thus reducing their entropy. That feat seemed to violate the second law of thermodynamics, according to which entropy can never decrease. (What's the catch? Read the article to find out.) Raizen's idea is that a one-way gate can help cool a gas in two steps: First let the gate concentrate atoms into a smaller volume (but without raising their temperature), then allow them to expand to the original volume (which brings their temperature down). Click on "START" to see how it works.

What Is Maxwell’s Demon?

Isolated systems and the second law of thermodynamics, maxwell’s demon – the loophole, can an apparatus like this really exist.

Maxwell’s demon is a hypothetical entity that physicist James Clerk Maxwell conjured in one of his thought experiments around 1871. The thought experiment consisted of an apparatus that would extract work from an isolated system, despite it existing in an equilibrium, at a single uniform temperature.

Basically, Maxwell’s notional entity is a sort of deus ex machina that contradicts or has cleverly devised a way around what is the most fundamental and indisputable law of the universe: the 2 nd law of thermodynamics. Naturally, the notion of extracting work or energy from seemingly nothing baffled his colleagues — surely this could mean the end of tirelessly feeding coal to a ravenous steam engine? A free lunch!

Well, not really. To understand how it works, we must first understand what the law entails and why discovering a loophole beckons a riot.

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Maxwell’s Demon Paradox

What is maxwell’s demon paradox.

Imagine a little creature that can sort warm air from cool air without using any energy, breaking the rules we know as the laws of physics. This creature isn’t real, but a thought experiment known as Maxwell’s Demon. The paradox lies in the idea that this imaginary creature can sort molecules in a way that seems to reduce chaos without any cost, challenging our ideas of what’s possible and what isn’t in physics.

Now, let’s break it down a bit further: If you have a room divided in two, and on one side you have fast-moving, energetic air molecules (hot air) and on the other side you have slow-moving, less energetic air molecules (cool air), normally, the air would mix until the whole room is a uniform temperature. The demon, however, can magically sort these molecules without any effort, keeping the hot air on one side and the cool air on the other. This seems to go against the idea that without an input of energy, the disorder or ‘entropy’ in a system shouldn’t decrease – it should either stay the same or get messier.

Key Arguments

The second law of thermodynamics is like the rule that says spilled milk will never un-spill itself; the mess can only get bigger, or if you’re lucky, stay the same.
Maxwell’s Demon goes against this rule by acting like it can un-spill the milk – or separate the mixed air molecules – without a mop or any effort at all.
If this Demon could actually do what it proposes, it would be like having a water wheel that spins forever, which goes against everything we understand about how things stop and start because of friction and other forces.
Many smart people have pointed out that the demon must use some form of energy to sort the air, like the energy needed to turn on a light to see the difference between the fast and slow molecules.
Alternatively, if the demon uses its memory to remember which air molecules are fast or slow, eventually it would need to delete that memory to make room for new information. This “forgetting” is believed to use energy, so there’s no cheating the second law after all.

Answer or Resolution

The twist in the tale came when scientists realized that even imaginary creatures doing imaginary tasks need energy when the tasks involve information. Measuring molecule speeds and remembering them is using information, just like when you learn things in school. Eventually, if you learn new things, you might forget some old stuff to make space in your brain – and that process isn’t free. It uses up energy in your brain, which in the world of physics means you’re still not breaking any rules.

Major Criticism

Though a lot of smart people agree on the information-energy connection that resolves the paradox, there are still those who scratch their heads about what this demon means on a quantum scale – that’s the super small world of atoms and particles. Here, things don’t always act the way you’d expect. Critics also ponder if we’re missing something about how information and chaos intertwine. Even with all the discussion, Maxwell’s Demon remains a brain-bender that makes both scientists and philosophers wonder about the unseen corners of physics.

Practical Applications

Information Theory: Maxwell’s Demon has sharpened our brains on how knowledge and chaos relate, helping us craft the essential ideas used in computer science and phone networks.
Computing: The solutions to the demon’s sorting tricks have inspired ways to compute things with less power, promising a future where gadgets don’t drain batteries so quickly.
Quantum Mechanics: This little thought creature has also pricked the minds of those studying the tango between particles that are lightyears apart yet strangely connected, pushing us toward inventions that compute in ways we can hardly imagine.

These practical uses emphasize that ideas, even those spun from pure imagination, can shape the edge of technological revolutions.

Why is it Important?

Maxwell’s Demon isn’t just a fun bit of brain gymnastics for scientists – it reminds us that the world is full of strange twists and the rules aren’t always as clear-cut as they seem. It encourages us to think critically about what we take for granted and inspires us to keep searching for deeper truths, even in everyday life.

For the average person, the Demon’s tale is about looking beyond what seems impossible and finding solutions and answers in unexpected places. Every time we swipe our phones, send a message, or use electricity, we’re touching a little piece of the vast puzzle that Maxwell’s Demon has a hand in. It’s science touching life, making us wonder and strive for better, smarter technology.

In the end, James Clerk Maxwell’s imaginative demon isn’t just scribbles in an old science book; it’s a lasting riddle that connects the dots between physics, philosophy, and technology. We’ve used it to deepen our understanding of the fundamental concepts of entropy, energy, and information, and as inspiration for technological innovations like energy-efficient computing and potential quantum tech advances. Maxwell’s Demon exemplifies the power of a good scientific puzzle: to enlighten, to challenge, and to provoke enduring curiosity in the mechanics of our world.

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Experimental demonstration of a Maxwell's demon quantum battery in a superconducting noisy intermediate-scale quantum processor

Jiale yu et al., phys. rev. a 109 , 062614 – published 20 june 2024.

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INTRODUCTION
EXPERIMENTAL APPROACH
DEMONIC EFFECTS
QB WORKING CONDITIONS
INFORMATION-TO-WORK CONVERSION
CONCLUSION AND OUTLOOKS
ACKNOWLEDGMENTS

Entering the era of post-quantum supremacy has given one the ability to precisely control noisy intermediate-scale quantum (NISQ) processors with multiqubits and extract valuable quantum many-body correlation resources for many distinct quantum applications. We here construct quantum many-body thermalized states on a 62-qubit superconducting quantum processor and use them to demonstrate the principle of Maxwell's demon. We further demonstrate the direct effect caused by Maxwell's demon on the charging process of a quantum battery (QB). We depicted the nonequilibrium transportation in our QB through measuring the dynamics of the Shannon entropy to explore its working conditions. Finally, we evaluate the information-to-work conversion by varying the readout fidelity to verify the validity of the Sagawa-Ueda equality within the NISQ processor environment and evaluate the qubit-environment interaction such as the measurement backaction. Our experiment suggests that the superconducting NISQ processor with appropriate error mitigation methods will be an ideal platform for studying quantum information thermodynamics through quantum many-body simulations.

Received 9 February 2024
Revised 2 May 2024
Accepted 24 May 2024

DOI: https://doi.org/10.1103/PhysRevA.109.062614

Physics Subject Headings (PhySH)

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Physical Systems

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Vol. 109, Iss. 6 — June 2024

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(a) Schematic illustration of the Maxwell's demon thought experiment where the demon extracts work and controls heat transfer direction [ 38 ] through information of the microscopic state in gas molecules. (b) The application of Maxwell's demon in the QB scenario. The Maxwell's demon should choose a reasonable time when charging the battery. (c) Schematic diagram of Maxwell's demon with the 2D superconducting qubit processor to move the excitation gradually from subsystem B to A. The bottom layer of Maxwell's demon includes erase, write, logic operation, and result classification, where erasing the last information with an irreversible logic operation will increase the entropy cost [ 40 ]. The processing method of measurement outcomes by Maxwell's demon determines that the entropy cost in our feedback protocol is still bounded by the Landauer limit.

The energy transfer process from subsystem B to subsystem A of size 3 × 4 each round. (a) The initial state of the total system where two qubits were excited. (b) The σ z measurement probability distribution using 3 million results of each qubit after evolution. (c)–(h) Plots of the probability distributions after round 2, 3, 4 without or with the action of Maxwell's demon, respectively. In (c)–(e) we use an i swap gate to connect the subsection without the feedback protocol, while in (f)–(h) we use Maxwell's demon qubits that are measured with only the | 0 〉 state for target qubit A and | 1 〉 state for target qubit B being retained. (g) The temperature in different rounds of each subsystem.

Dynamics of the Shannon entropy rate related to maximum entropy η Sh in the Maxwell's demon QB for subsystem 2 × 2 A and 3 × 4 A in (a) and (b) respectively. The averaged values in (a) after 50 ns are 0.744 and 0.508 for r = 2 and r = 3 , respectively, while those in (b) are 0.888 and 0.809, respectively. The maximum peak values of experimental data are noted with corresponding colors. Also shown with solid lines are the results from the numerical simulation. Full details are given in the Appendixes.

Verification for the second law of thermodynamics and Sagawa-Ueda equality in a single-excitation quantum many-body environment. (a) Modified second law verification where the blue hollow square, dark green circle, and gray diamond dots show the relationship between mutual information 〈 I 〉 , 〈 Δ F − W A 〉 , and conversion efficiency η = 〈 Δ F − W A 〉 ε / ( β 〈 I 〉 ) vs the measurement error of target qubit B. The blue dashed line shows the Shannon entropy of target qubit B, S ( t evol ) ∼ 0.613 ln 2 , which is the upper bound of 〈 Δ F − W A 〉 . (b) The Sagawa-Ueda equality where the dark blue, green, and purple circles plot 〈 e β ( Δ F − W A ) 〉 , 〈 e β ( Δ F − W A ) − I 〉 , and 〈 e β ( Δ F − W A + 〈 H ̂ q − e n v 〉 ) − I 〉 , respectively.

(a) Layout of the qubits in experimental area of the quantum processor where the numbers under qubit in the nonexperimental area are frequency in GHz. Red dashed line marks the proximity and subproximity qubits of experimental area, which are biased to 5.0 GHz. Qubits outside of the red dashed line were not used in this experiment. (b) The waveforms corresponding to each step in Fig. 1 .

Qubit parameter distributions, including qubit maximum frequency, qubit idle frequency, anharmonicity, dipersive shift, qubit readout drive frequency, resonator line width, readout fidelity, coupling strength between qubit and resonator, qubit energy relaxation time T 1 at the idle/work point, qubit dephasing time T 2 * at the idle point, effective temperature, and distribution of the coupling strengths between neighboring qubits. Each square in the diagrams represents a qubit; the number and color in the square show the value of the corresponding parameter. The square connecting two qubits shows the effective coupling strength between them when these two qubits are tuned to the interaction frequency of 5.13 GHz in subsystem A and 5.2 GHz in subsystem B.

Schematic diagram of our subsystem frequency alignment process. (a) One step of the subsystem B frequency alignment process for qubits U30Q1, U30Q2, U20Q2, U30Q0, U31Q0, where the excited qubit marked with red is U30Q1. (b) Experimental and simulated value of the probabilistic evolution of the qubits in (a). This is the basis for calculating the distance and updating the frequency correction matrix.

Waveform schematic diagram of the each experimental circuit of the Maxwell's demon swap process parameters calibration process. (a) The waveform to determine the exchange operation frequency point between target qubits of the two subsystems. (b) The waveform to determine the shift of working point on target qubits after continues readout caused by AC Stark effect. (c) The waveform of the two-qubit gate XEB test for i swap gate composed of target qubits.

(a) Calibration of time and frequency on target qubit B. The center of pentagram is the minimal value of P 01 , which corresponds to the point with the best energy exchange. This point corresponds to the U21Q3 frequency of 5.1544 GHz with a swap time of 105 ns. (b) Calibration of the waiting time t wait where we plot the frequency of U21Q3 vs t wait after readout. Here the red dashed line is the marker line of 120 ns, while the blue dashed line marks the mean peak frequency value of U21Q3 (5.217 GHz).

Fidelity calibration of the i swap gate constructed by two target qubits. (a) The XEB alpha factor fitting procedure for a two-qubit gate constructed for target qubits. (b) The process of optimizing five kinetic phase factors using the Nelder-Mead method with Pauli XEB error as the objective function. (c) The variation of XEB error with the number of iterations during the optimization of the kinetic phase factors. (d) The iteration process of fitting the two-qubit i swap gate SPB alpha factor.

The expectation of Maxwell's demon observation rounds n obs required for a single trigger of QB in various system size, which are the simulated and experimental values of the second and third rounds respectively.

Postselection rate in first and second readouts of Maxwell's demon QB and background in the double excitation case, and the postselection rate is the ratio of postselection number to the total number of experimental measurements. (a) The number of measurements in the double excitation background experiment is 500 000 for system 2 × 2 , 2 × 3 , and 3 × 3 respectively. The initial excitation qubits of system 2 × 2 are “U31Q0” and “U21Q3,” while initial excitation qubits in system 2 × 3 and 3 × 3 are “U30Q1” and “U21Q3.” (b) In system 3 × 4 , the number of experimental measurements and the number of background measurements are both 3 million, and the initial excitation qubits are “U30Q2” and “U21Q3.”

Experimental data processing flow chart. In the main text, the step of sub-/total system excitation number conservation postselected is used only in the experiment of Figs. 2 and 3 .

Exploration of the Shannon entropy in the Maxwell's demon QB. The number of experimental measurements is 1 million for the system sizes 2 × 2 , 2 × 3 , and 3 × 3 and 3 million for the 3 × 4 system size. The number of simulation points is 500 000 for the 2 × 2 , 2 × 3 , and 3 × 3 system sizes and 1 000 000 for the 3 × 4 case. The initial excitation qubits are “U21Q3” and “U31Q0” in the system 2 × 2 , and “U30Q1” and “U21Q3” for the 2 × 3 and 3 × 3 systems. “U30Q2” and “U21Q3” are excited in the 3 × 4 system. (a)–(h) Simulated and experimental values for Shannon's entropy against the evolution time for the various system sizes. The maximum entropy of system 2 × 2 is 2 and 2.585 for rounds 1 and 2, respectively, 2.585 and 3.907 for the 2 × 3 system, and 3.167 and 5.170 for the 3 × 3 system. Maximum entropy of the 3 × 4 system is 3.585 and 6.044 for rounds 1 and 2, respectively. The red, green, and blue background marks the theoretical upper bounds of maximum entropy for three rounds, respectively, while the red, green, and blue data diamond points in each graph present the entropy at r = 1 , r = 2 , and r = 3 , respectively. The dashed lines of the same color are corresponding simulation values.

KL divergence of system 2 × 2 and system 3 × 4 related to microcanonical ensembles.

Approximate estimation algorithm of Krylov coefficients in our QB. The target qubits are identified by black triangles. (a) Derivation of Krylov coefficients for the system 2 × 2 under single excitation where the green arrow shows the direction of state propagation, while the black numbers are the Krylov coefficients. (b) Krylov coefficients for the single excited 3 × 4 system 3 × 4 . (c) Krylov coefficients for the 2 × 2 system under double excitation. The different events are surrounded by dashed boxes of different colors (six total), while their Krylov coefficients are marked with corresponding colors. (d) The Krylov coefficients for the doubly excited 3 × 4 system. There are 66 different kinds of basic events under the particle number representation. The events enclosed by red dashed boxes are the initial states, and there are three kinds of events: Krylov coefficient of the events enclosed by black dashed boxes contain target qubit A, which is 1; Krylov coefficients of the events enclosed by orange dashed boxes, which is 3; and Krylov coefficients of the events enclosed by purple dashed boxes, which is 4.

Dynamics of ergodicity indicator in our QB where the number of experimental measurements is 1 million (3 million) for the 2 × 2 ( 3 × 4 ) system respectively. The number of simulation points is 0.5 million (1 million) for the 2 × 2 ( 3 × 4 ) systems. (a) Dynamics of the Krylov complexity for the 2 × 2 subsystem A . (b) Dynamics of Krylov complexity for the 3 × 4 subsystem A . (c) E KS δ t in the 2 × 2 subsystem A , where the system is in a nonthermal state (marked with purple color). (d) E KS δ t in the 3 × 4 subsystem A , where the system evolves in two stages: the prethermalized phase (yellow) and the thermalized phase (light orange).

(a) Photon number vs the dispersive shift of the dressed cavity for target qubit B (U21Q3), where χ can be calculated by the coupling strength between the qubit and the resonator g , the frequency difference of qubit and the resonator Δ = | ω q − ω r | , and the anharmonicity η . (b) The photon number n ¯ of every qubit used in the experiment.

Relation between the photon number and measurement error of target qubit B (U21Q3).

Demonic effect in the single-excitation case of system 3 × 4 when t evol = 66 ns. (a) The initial state preparation in the single-excitation case with the initial excitation qubit “U30Q2.” (b) Qubit probability distributions of the two subsystems in the single-excitation case t evol = 66 ns. (c, d) σ z system measurements at r = 2 for system 3 × 4 without or with Maxwell's demon action for 3 million measurements respectively. (f–h) Simulation results corresponding to the first and second rounds of evolution of system 3 × 4 with the same initial setup conditions as (a), (b), and (d). (e) The effective temperature vs rounds of feedback control of each subsystem in size of 3 × 4 in the single excitation case.

Dynamical indicators of Maxwell's demon-type QB in system 2 × 3 and 3 × 3 . In (a) and (b) we plot E KS δ t for the subsystem 2 × 3 A and subsystem 3 × 3 A , respectively. In (c) and (d) we plot the stored work rate occupied in phase space η e of subsystem 2 × 3 A and subsystem 3 × 3 A , respectively.

Evolution of probability and entropy with the number of readout rounds for the 2 × 3 scale superconducting qubit system with excited state transfer from subsystem A to B. (a), (b) Simulation and experimental values of entropy evolution with time for systems 2 × 3 with two subsystems A and B, respectively. The number of experimental measurements and simulation points is 500 000. The initial excitation qubits are “U10Q3” and “U21Q0.”

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