Category: Analysis

• What does a quantum Bayes’s rule look like?

  Bayes’s rule is one of the most fundamental principles in probability and statistics. It allows us to update our beliefs in the face of new evidence. In its simplest form, the rule tells us how to revise the probability of a hypothesis once new data becomes available.

  A standard way to teach it involves drawing coloured balls from a pouch: you start with some expectation (e.g. “there’s a 20% chance I’ll draw a blue ball”), then you update your belief depending on what you observe (“I’ve drawn a red ball, so the actual chance of drawing a blue ball is 10%”). While this example seems simple, the rule carries considerable weight: physicists and mathematicians have described it as the most consistent way to handle uncertainty in science, and it’s a central part of logic, decision theory, and indeed nearly every field of applied science.

  There are two well-known ways of arriving at Bayes’s rule. One is the axiomatic route, which treats probability as a set of logical rules and shows that Bayesian updating is the only way to preserve consistency. The other is variational, which demands that updates should stay as close as possible to prior beliefs while remaining consistent with new data. This latter view is known as the principle of minimum change. It captures the intuition that learning should be conservative: we shouldn’t alter our beliefs more than is necessary. This principle explains why Bayesian methods have become so effective in practical statistical inference: because they balance a respect for new data with loyalty to old information.

  A natural question arises here: can Bayes’s rule be extended into the quantum world?

  Quantum theory can be thought of as a noncommutative extension of probability theory. While there are good reasons to expect there should be a quantum analogue of Bayes’s rule, the field has for a long time struggled to identify a unique and universally accepted version. Instead, there are several competing proposals. One of them stands out: the Petz transpose map. This is a mathematical transformation that appears in many areas of quantum information theory, particularly in quantum error correction and statistical sufficiency. Some scholars have even argued that it’s the “correct” quantum Bayes’s rule. Still, the situation remains unsettled.

  In probability, the joint distribution is like a big table that lists the chances of every possible pair of events happening together. If you roll a die and flip a coin, the joint distribution specifies the probability of getting “heads and a 3”, “tails and a 5”, and so on. In this big table, you can also zoom out and just look at one part. For example, if you only care about the die, you can add up over all coin results to get the probability of each die face. Or if you only care about the coin, you can add up over all die results to get the probability of heads or tails. These zoomed-out views are called marginals.

  The classical Bayes’s rule doesn’t just update the zoomed-out views but the whole table — i.e. the entire joint distribution — so the connection between the two events also remains consistent with the new evidence.

  In the quantum version, the joint distribution isn’t a table of numbers but a mathematical object that records how the input and output of a quantum process are related. The point of the new study is that if you want a true quantum Bayes’s rule, you need to update that whole object, not just one part of it.

  A new study by Ge Bai, Francesco Buscemi, and Valerio Scarani in Physical Review Letters has taken just this step. In particular, they’ve presented a quantum version of the principle of minimum change by showing that when the measure of change is chosen to be quantum fidelity — a widely used measure of similarity between states — this optimisation leads to a unique solution. Equally remarkably, this solution coincided with the Petz transpose map in many important cases. As a result, the researchers have built a strong bridge between classical Bayesian updating, the minimum change principle, and a central tool of quantum information.

  The motivation for this new work isn’t only philosophical. If we’re to generalise Bayes’s rule to include quantum mechanics as well, we need to do so in a way that respects the structural constraints of quantum theory without breaking away from its classical roots.

  The researchers began by recalling how the minimum change principle works in classical probability. Instead of updating only a single marginal distribution, the principle works at the level of the joint input-output distribution. Updating then becomes an optimisation problem, i.e. finding the subsequent distribution that’s consistent with the new evidence but minimally different from the evidence from before.

  In ordinary probability, we talk about stochastic processes. These are rules that tell us how an input is turned into an output, with certain probabilities. For example if you put a coin into a vending machine, there might be a 90% chance you get a chips packet and a 10% chance you get nothing. This rule describes a stochastic process. This process can also be described with a joint distribution.

  In quantum physics, however, it’s tricky. The inputs and outputs aren’t just numbers or events but quantum states, which are described by wavefunctions or density matrices. This makes the maths much more complex. The resulting stochastic processes also become sequences of events called completely positive trace-preserving (CPTP) maps.

  A CPTP map is the most general kind of physical evolution allowed: it takes a quantum state and transforms it into another quantum state. And in the course of doing so, it needs to follow two rules: it shouldn’t yield any negative probabilities and it should ensure the total probability adds up to 1. That is, your chance of getting a chips packet shouldn’t be –90% nor should it be 90% plus a 20% chance of getting nothing.

  These complications mean that, while the joint distribution in classical Bayesian updating is a simple table, the one in quantum theory is more sophisticated. It uses two mathematical tools in particular. One is purification, a way to embed a mixed quantum state into a larger ‘pure’ state so that mathematicians can keep track of correlations. The other is Choi operators, a standard way of representing a CPTP map as a big matrix that encodes all possible input-output behaviour at once.

  Together, these tools play the role of the joint distribution in the quantum setting: they record the whole picture of how inputs and outputs are related.

  Now, how do you compare two processes, i.e. the actual forward process (input → output) and the guessed reverse process (output → input)?

  In quantum mechanics, one of the best measures of similarity is fidelity. It’s a number between 0 and 1. 0 means two processes are completely different and 1 means they’re exactly the same.

  In this context, the researchers’ problem statement was this: given a forward process, what reverse process is closest to it?

  To solve this, they looked over all possible reverse processes that obeyed the two rules, then they picked the one that maximised the fidelity, i.e. the CPTP map most similar to the forward process. This is the quantum version of applying the principle of minimum change.

  In the course of this process, the researchers found that in natural conditions, the Petz transpose map emerges as the quantum Bayes’s rule.

  In quantum mechanics, two objects (like matrices) commute if the order in which you apply them doesn’t matter. That is, A then B produces the same outcome as B then A. In physical terms, if two quantum states commute, they behave more like classical probabilities.

  The researchers found that when the CPTP map that takes an input and produces an output, called the forward channel, commutes with the new state, the updating process is nothing but the Petz transpose map.

  This is an important result for many reasons. Perhaps foremost is that it explains why the Petz map has shown up consistently across different parts of quantum information theory. It appears it isn’t just a useful tool but the natural consequence of the principle of minimum change applied in the quantum setting.

  The study also highlighted instances where the Petz transpose map isn’t optimal, specifically when the commutativity condition fails. In these situations, the optimal updating process depends more intricately on the new evidence. This subtlety departs clearly from classical Bayesian logic because in the quantum case, the structure of non-commutativity forces updates to depend non-linearly on the evidence (i.e. the scope of updating can be disproportionate to changes in evidence).

  Finally, the researchers have shown how their framework can recover special cases of practical importance. If some new evidence perfectly agrees with prior expectations, the forward and reverse processes become identical, mirroring the classical situation where Bayes’s rule simply reaffirms existing beliefs. Similarly, in contexts like quantum error correction, the Petz transpose map’s appearance is explained by its status as the optimal minimal-change reverse process.

  But the broader significance of this work lies in the way it unifies different strands of quantum information theory under a single conceptual roof. By proving that the Petz transpose map can be derived from the principle of minimum change, the study has provided a principled justification for its widespread use rather than being restricted to particular contexts. This fact has immediate consequences for quantum computing, where physicists are looking for ways to reverse the effects of noise on fragile quantum states. The Petz transpose map has long been known to do a good job of recovering information from these states after they’ve been affected by noise. Now that physicists know the map embodies the smallest update required to stay consistent with the observed outcomes, they may be able to design new recovery schemes that exploit the structure of minimal change more directly.

  The study may also open doors to extending Bayesian networks into the quantum regime. In classical probability, a Bayesian network provides a structured way to represent cause-effect relationships. By adapting the minimum change framework, scientists may be able to develop ‘quantum Bayesian networks’ where the way one updates their expectations of a particular outcome respects the peculiar constraints of CPTP maps. This could have applications in quantum machine learning and in the study of quantum causal models.

  There are also some open questions as well. For instance, the researchers have noted that if different measures of divergence other than fidelity are used, e.g. the Hilbert-Schmidt distance or quantum relative entropy, the resulting quantum Bayes’s rules may be different. This in turn indicates that there could be multiple valid updating rules, each suited to different contexts. Future research will need to map out these possibilities and determine which ones are most useful for particular applications.

  In all, the study provides both a conceptual advance and a technical tool. Conceptually, it shows how the spirit of Bayesian updating can carry over into the quantum world; technically, it provides a rigorous derivation of when and why the Petz transpose map is the optimal quantum Bayes’s rule. Taken together, the study’s finding strengthens the bridge between classical and quantum reasoning and offers a deeper understanding of how information is updated in a world where uncertainty is baked into reality rather than being due to an observer’s ignorance.

  2025.10.04

• Using 10,000 atoms and 1 to probe the Bohr-Einstein debate

  The double-slit experiment has often been described as the most beautiful demonstration in physics. In one striking image, it shows the strange dual character of matter and light. When particles such as electrons or photons are sent through two narrow slits, the resulting pattern on a screen behind them is not the simple outline of the slits, but a series of alternating bright and dark bands. This pattern looks exactly like the ripples produced by waves on the surface of water when two stones are thrown in together. But when detectors are placed to see which slit each particle passes through, the pattern changes: the wave-like interference disappears and the particles line up as if they had travelled like microscopic bullets.

  This puzzling switch between wave and particle behaviour became the stage for one of the deepest disputes of the 20th century. The two central figures were Albert Einstein and Niels Bohr, each with a different vision of what the double-slit experiment really meant. Their disagreement was not about the results themselves but about how these results should be interpreted, and what they revealed about the nature of reality.

  What does it mean to interpret quantum physics?

  Einstein believed strongly that the purpose of physics was to describe an external reality that exists independently of us. For him, the universe must have clear properties whether or not anyone is looking. In a double-slit experiment, this meant an electron or photon must in fact have taken a definite path, through one slit or the other, before striking the screen. The interference pattern might suggest some deeper process that we don’t yet understand but, to Einstein, it couldn’t mean that the particle lacked a path altogether.

  Based on this idea, Einstein argued that quantum mechanics (as formulated in the 1920s) couldn’t be the full story. The strange idea that a particle had no definite position until measured, or that its path depended on the presence of a detector, was unacceptable to him. He felt that there must be hidden details that explained the apparently random outcomes. These details would restore determinism and make physics once again a science that described what happens, not just what is observed.

  Bohr, however, argued that Einstein’s demand for definite paths misunderstood what quantum mechanics was telling us. Bohr’s central idea was called complementarity. According to this principle, particles like electrons or photons can show both wave-like and particle-like behaviour, but never both at the same time. Which behaviour appears depends entirely on how an experiment is arranged.

  In the double-slit experiment, if the apparatus is set up to measure which slit the particle passes through, the outcome will display particle-like behaviour and the interference pattern will vanish. If the apparatus is set up without path detectors, the outcome will display wave-like interference. For Bohr, the two descriptions are not contradictions but complementary views of the same reality, each valid only within its experimental context.

  Specifically, Bohr insisted that physics doesn’t reveal a world of objects with definite properties existing independently of measurement. Instead, physics provides a framework for predicting the outcomes of experiments. The act of measurement is inseparable from the phenomenon itself. Asking what “really happened” to the particle when no one was watching was, for Bohr, a meaningless question.

  Thus, while Einstein demanded hidden details to restore certainty, Bohr argued that uncertainty was built into nature itself. The double-slit experiment, for Bohr, showed that the universe at its smallest scales does not conform to classical ideas of definite paths and objective reality.

  Disentangling entanglement

  The disagreement between Einstein and Bohr was not simply about technical details but a clash of philosophies. Einstein’s view was rooted in the classical tradition: the world exists in a definite state and science should describe that state. Quantum mechanics, he thought, was useful but incomplete, like a map missing a part of the territory.

  Bohr’s view was more radical. He believed that the limits revealed by the double-slit experiment were not shortcomings of the theory but truths about the universe. For him, the experiment demonstrated that the old categories of waves and particles, causes and paths, couldn’t be applied without qualification. Science had to adapt its concepts to match what experiments revealed, even if that meant abandoning the idea of an observer-independent reality.

  Though the two men never reached agreement, their debate has continued to inspire generations of physicists and philosophers. The double-slit experiment remains the clearest demonstration of the puzzle they argued over. Do particles truly have no definite properties until measured, as Bohr claimed? Or are we simply missing hidden elements that would complete the picture, as Einstein insisted?

  A new study in Physical Review Letters has taken the double-slit spirit into the realm of single atoms and scattered photons. And rather than ask whether an electron goes through one slit or another, it has asked whether scattered light carries “which-way” information about an atom. By focusing on the coherence or incoherence of scattered light, the researchers — from the Massachusetts Institute of Technology — have effectively reopened the old debate in a modern setting.

  The researchers trapped rubidium atoms held in an optical lattice, a regular grid of light that traps atoms in well-defined positions, like pieces on a chessboard. By carefully preparing these atoms in a particular state, each lattice site contained exactly one atom in its lowest energy state. The lattice could then be suddenly switched off, letting the atoms expand as localised wavepackets (i.e. wave-like packets of energy). A short pulse of laser light was directed at these atoms. The photons it emitted were scattered off the atoms and collected by a detector.

  By checking whether the scattered light was coherent (with a steady, predictable phase) or incoherent (with a random phase), the scientists could tell if the photons carried hints of the motion of the atom that scattered them.

  The main finding was that even a single atom scattered light that was only partly coherent. In other words, the scattered light wasn’t completely wave-like: one part of it showed a clear phase pattern, another part looked random. The randomness came from the fact that the scattering process linked, or entangled, the photon with the atom’s movement. This was because each time a photon was scattered off, the atom recoiled just a little, and that recoil left behind a faint clue about which atom had scattered the photon. This in turn meant that if the scientists looked close enough, they could work out where the photon came from in theory.

  A tribute to rubidium

  To study this effect, the team compared three cases. First, they observed atoms still held tightly in the optical lattice. In this case, scattering could create sidebands — frequency shifts in the scattered light — that reflected changes in the atom’s motion. These sidebands represented incoherent scattering. Second, they looked at atoms immediately after switching off the lattice, before the expanding wavepackets had spread out. Third, they examined atoms after a longer expansion in free space, when the wavepackets had grown even wider.

  In all three cases, the ratio of coherent to incoherent light could be described by a simple mathematical term called the Debye-Waller factor. This factor depends only on the spatial spread of the wavepacket. As the atoms expanded in space, the Debye-Waller factor decreased, meaning more and more of the scattered light became incoherent. Eventually, after long enough expansion, essentially all the scattered light was incoherent.

  Experiments with two different atomic species supported this picture. With lithium-7 atoms, which are very light, the wavepackets expanded quickly, so the transition from partial coherence to full incoherence was rapid. With the much heavier dysprosium-162 atoms, the expansion was slower, allowing the researchers to track the change in more detail. In both cases, the results agreed with theoretical predictions.

  An especially striking observation was that the presence or absence of the trap made no difference to the basic coherence properties. The same mix of coherent and incoherent scattering appeared whether the atoms were confined in the lattice or expanding in free space. This showed that sidebands and trapping states were not the fundamental source of incoherence. Instead, what mattered was the partial entanglement between the light and the atoms.

  The team also compared long and short laser pulses. Long pulses could in principle resolve the sidebands while short pulses could not. Yet the fraction of coherent versus incoherent scattering was the same in both cases. This further reinforced the conclusion that coherence was lost not because of frequency shifts but because of entanglement itself.

  The gap between language and quantum mechanics

  In 2024, another group in China also realised the recoiling-slit thought experiment in practice. Researchers from the University of Science and Technology of China trapped a single rubidium atom in an optical tweezer and cooled it to its quantum ground state, thus making the atom act like a movable slit whose recoil could be directly entangled with scattered photons.

  By tightening or loosening the trap, the scientists could pin the atom more firmly in place. When it was held tightly, the atom’s recoil left almost no mark on the photons, which went on to form a clear interference pattern (like the ripples in water). When the atom was loosely held, however, its recoil was easier to notice and the interference pattern faded. This gave the researchers a controllable way to show how a recoiling slit could erase the wave pattern — which is also the issue at the heart of Bohr-Einstein debate.

  Importantly, the researchers also distinguished true quantum effects from classical noise, such as heating of the atom during repeated scattering. Their data showed that the sharpness of the interference pattern wasn’t an artifact of an imperfect apparatus but a direct result of the atom-photon entanglement itself. In this way, they were able to demonstrate the transition from quantum uncertainty to classical disturbance within a single, controllable system. And even at this scale, the Bohr-Einstein debate couldn’t be settled.

  The results pointed to a physical mechanism for how information becomes embedded in light scattered from atoms. In the conventional double-slit experiment, the question was whether a photon’s path could ever be known without destroying the interference pattern. In the new, modern version, the question was whether a scattered photon carried any ‘imprint’ of the atom’s motion. The MIT team’s measurements showed that it did.

  The Debye-Waller factor — the measure of how much of the scattered light is still coherent — played an important role in this analysis. When atoms are confined tightly in a lattice, their spatial spread is small and the factor is relatively large, meaning a smaller fraction of the light is incoherent and thus reveals which-way information. But as the atoms are released and their wavepackets spread, the factor drops and with it the coherent fraction of scattered light. Eventually, after free expansion for long enough, essentially all of the scattered light becomes incoherent.

  Further, while the lighter lithium atoms expanded so quickly that the coherence decayed almost at once, the heavier dysprosium atoms expanded more slowly, allowing the researchers to track them in detail. Yet both atomic species followed a common rule: the Debye-Waller factor depended solely on how much the atom became delocalised as a wave, and not by the technical details of the traps or the sidebands. The conclusion here was that the light lost its coherence because the atom’s recoil became entangled with the scattered photon.

  Bohr and the breakaway from classical mechanics

  This finding adds substance to the Bohr-Einstein debate. In one sense, Einstein’s intuition has been vindicated: every scattering event leaves behind faint traces of which atom interacted with the light. This recoil information is physically real and, at least in principle, accessible. But Bohr’s point also emerges clearly: that no amount of experimental cleverness can undo the trade-off set by quantum mechanics. The ratio of coherent to incoherent light is dictated not by human knowledge or ignorance but by implicit uncertainties in the spread of the atomic wavepacket itself.

  Together with the MIT results, the second experiment showed that both Einstein’s and Bohr’s insights remain relevant: every scattering leaves behind a real, measurable recoil — yet the amount of interference lost is dictated by the unavoidable quantum uncertainties of the system. When a photon scatters off an atom, the atom must recoil a little bit to conserve momentum. That recoil in principle carries which-way information because it marks the atom as the source of the scattered photon. But whether that information is accessible depends on how sharply the atom’s momentum (and position) can be defined.

  According to the Heisenberg uncertainty principle, the atom can’t simultaneously have both a precisely known position and momentum. In these experiments, the key measure was how delocalised the atom’s wavepacket was in space. If the atom was tightly trapped, its position uncertainty would be small, so its momentum uncertainty would be large. The recoil from a photon is then ‘blurred’ by that momentum spread, meaning the photon doesn’t clearly encode which-way information. Ultimately, interference is preserved.

  By recasting the debate in the language of scattered photons and expanding wavepackets, the MIT experiment has thus moved the double-slit spirit into new terrain. It shows that quantum mechanics doesn’t simply suggest fuzziness in the abstract but enforces it in how matter and light are allowed to share information. The loss of coherence isn’t a flaw in the experimental technique or a sign of missing details, as Einstein might’ve claimed, but the very mechanism by which the microscopic world keeps both Einstein’s and Bohr’s insights in tension. The double-slit experiment, even in a highly sophisticated avatar, continues to reinforce the notion that the universe resists any single-sided description.

  (The researchers leading the two studies are Wolfgang Ketterle and Pan Jianwei, respectively a Nobel laureate and a rockstar in the field of quantum information likely to win a Nobel Prize soon.)

  Featured image created with ChatGPT.

  2025.10.01

• Curiosity as a public good

  India has won 22 Ig Nobel prizes to date. These awards, given annually at Harvard University by the magazine Annals of Improbable Research, honour studies that “first make people laugh, and then make them think” — a description that can suggest the prizes are little more than jokes whereas the research they reward is genuine.

  Many of the Indian wins are in the sciences and they highlight an oft unacknowledged truth: even if the country hasn’t produced a Nobel laureate in science since C.V. Raman in 1930, Indian labs continue to generate knowledge of consequence by pursuing questions that appear odd at first sight. In 2004, for example, IIT Kanpur researchers won an Ig Nobel prize for studying why people spill coffee when they walk. They analysed oscillations and resonance in liquid-filled containers, thus expanding the principles of fluid dynamics into daily life.

  Eleven years later, another team won a prize for measuring the friction coefficients of banana skins, showing why people who step on them are likely to fall. In 2019, doctors in Chennai were feted for documenting how cockroaches can survive inside human skulls, a subject of study drawn from real instances where medical workers had to respond to such challenges in emergency rooms. In 2022, biologists examined how scorpion stings are treated in rural India and compared traditional remedies against science-based pharmacology. More recently, researchers were honoured for describing the role of nasal hair in filtering air and pathogens.

  The wins thus demonstrate core scientific virtues as well as reflect the particular conditions in which research often happens in India. Most of the work also wasn’t supported by lavish grants nor was it published in élite journals with high citation counts. Instead, the work emerged from scientists choosing to follow curiosity rather than institutional incentives. In this sense, the Ig Nobel prizes are less a distraction and more an index of how ‘serious’ science might actually begin.

  Of course it’s also important to acknowledge that India’s research landscape is crowded with work of indifferent quality. A large share of papers are produced to satisfy promotion requirements, with little attention to design or originality, and many find their way into predatory journals where peer review is nonexistent or a joke. Such publications seldom advance knowledge, whether in curiosity-driven or application-oriented paradigms, and they dilute the credibility of the system as a whole.

  Then again whimsy isn’t foreign to the Nobel Prizes themselves, which are generally quite sombre. For example, in 2016, the chemistry prize was awarded to researchers who designed molecular rotors and elevators constructed from just a handful of atoms. The achievement was profound but it also carried the air of play. The prize-giving committee compared the laureates’ work to the invention of the electric motor in the 1830s, noting that even if practical applications (may or may not) come later, the first step remains the act of imagining, not unlike a child. If the Nobel Committee can reward such imaginative departures, India’s Ig Nobel prize wins should be seen as more evidence that playful research is a legitimate part of the scientific enterprise.

  The larger question is whether curiosity-driven research has a place in national science policy. Some experts have argued that in a country like India, with pressing social and economic needs and allegedly insufficient funding to support research, scientists must focus on topics that’re immediately useful: better crops, cheaper drugs, new energy sources, etc. But this is too narrow a view. Science doesn’t have to be useful in the short term to be valuable. The history of discovery is filled with examples that seemed obscure at the time but later transformed technology and society, including X-rays, lasers, and the structure of DNA. Equally importantly, the finitude of resources to which science administrators and lawmakers have often appealed is likely a red herring set up to make excuses for diverting funds away from scientific research.

  What arguments against the ‘next LHC’ say about funding Big Physics

  Measuring why banana skins are slippery didn’t solve a crisis but it advanced scientists’ understanding of biomechanics. Analysing why coffee spills while walking generated models in fluid mechanics that researchers could apply to a range of fluid systems. Together with documenting cockroaches inside skulls and studying scorpion sting therapies, none of this research was wasteful or should be seen that way but more importantly the freedom to pursue such questions is vital. If nothing else, winning a Nobel Prize can’t be engineered by restricting scientists to specific questions. They prizes often go to scientists who are well connected, work in well-funded laboratories, and who publish in highly visible journals — yet bias and visibility explain only part of the pattern. Doing good science depends on an openness to ideas that its exponents can’t be expected to plan in advance.

  This is a broader reason the Ig Nobel prizes are really reminders that curiosity remains alive among Indian scientists, even in a system that often discourages it. They also reveal what we stand to lose when research freedom is curtailed. The point isn’t that every odd question will lead to a breakthrough but that no one can predict in advance which questions will. We don’t know what we don’t know and the only way to find out is to explore.

  India’s 22 Ig Nobel wins in this sense are indicators of a culture of inquiry that deserves more institutional support. If the country wants to achieve scientific recognition of the highest order — the Indian government has in fact been aspiring to “science superpower” status — it must learn to value curiosity as a public good. What may appear whimsical today could prove indispensable tomorrow.

  2025.09.20

• Dispelling Maxwell’s demon

  Maxwell’s demon is one of the most famous thought experiments in the history of physics, a puzzle first posed in the 1860s that continues to shape scientific debates to this day. I’ve struggled to make sense of it for years. Last week I had some time and decided to hunker down and figure it out, and I think I succeeded. The following post describes the fruits of my efforts.

  At first sight, the Maxwell’s demon paradox seems odd because it presents a supernatural creature tampering with molecules of gas. But if you pare down the imagery and focus on the technological backdrop of the time of James Clerk Maxwell, who proposed it, a profoundly insightful probe of the second law of thermodynamics comes into view.

  The thought experiment asks a simple question: if you had a way to measure and control molecules with perfect precision and at no cost, will you able to make heat flow backwards, as if in an engine?

  Picture a box of air divided into two halves by a partition. In the partition is a very small trapdoor. It has a hinge so it can swing open and shut. Now imagine a microscopic valve operator that can detect the speed of each gas molecule as it approaches the trapdoor, decide whether to open or close the door, and actuate the door accordingly.

  The operator follows two simple rules: let fast molecules through from left to right and let slow molecules through from right to left. The temperature of a system is nothing but the average kinetic energy of its constituent particles. As the operator operates, over time the right side will heat up and the left side will cool down — thus producing a temperature gradient for free. Where there’s a temperature gradient, it’s possible to run a heat engine. (The internal combustion engine in fossil-fuel vehicles is a common example.)

  

  But the possibility that this operator can detect and sort the molecules, thus creating the temperature gradient without consuming some energy of its own, seems to break the second law of thermodynamics. The second law states that the entropy of a closed system increases over time — whereas the operator ensures that the temperature will decrease, violating the law. This was the Maxwell’s demon thought experiment, with the demon as a whimsical stand-in for the operator.

  The paradox was made compelling by the silent assumption that the act of sorting the molecules could have no cost — i.e. that the imagined operator didn’t add energy to the system (the air in the box) but simply allowed molecules that are already in motion to pass one way and not the other. In this sense the operator acted like a valve or a one-way gate. Devices of this kind — including check valves, ratchets, and centrifugal governors — were already familiar in the 19th century. And scientists assumed that if they were scaled down to the molecular level, they’d be able to work without friction and thus separate hot and cold particles without drawing more energy to overcome that friction.

  This detail is in fact the fulcrum of the paradox, and the thing that’d kept me all these years from actually understanding what the issue was. Maxwell et al. assumed that it was possible that an entity like this gate could exist: one that, without spending energy to do work (and thus increase entropy), could passively, effortlessly sort the molecules. Overall, the paradox stated that if such a sorting exercise really had no cost, the second law of thermodynamics would be violated.

  The second law had been established only a few decades before Maxwell thought up this paradox. If entropy is taken to be a measure of disorder, the second law states that if a system is left to itself, heat will not spontaneously flow from cold to hot and whatever useful energy it holds will inevitably degrade into the random motion of its constituent particles. The second law is the reason why perpetual motion machines are impossible, why the engines in our cars and bikes can’t be 100% efficient, and why time flows in one specific direction (from past to future).

  Yet Maxwell’s imagined operator seemed to be able to make heat flow backwards, sifting molecules so that order increases spontaneously. For many decades, this possibility challenged what physicists thought they knew about physics. While some brushed it off as a curiosity, others contended that the demon itself must expend some energy to operate the door and that this expense would restore the balance. However, Maxwell had been careful when he conceived the thought experiment: he specified that the trapdoor was small and moved without friction, so it could in principle operate in a negligible way. The real puzzle lay elsewhere.

  In 1929, the Hungarian physicist Leó Szilard sharpened the problem by boiling it down to a single-particle machine. This so-called Szilard engine imagined one gas molecule in a box with a partition that could be inserted or removed. By observing on which side the molecule lay and then allowing it to push a piston, the operator could apparently extract work from a single particle at uniform temperature. Szilard showed that the key step was not the movement of the piston but the acquisition of information: knowing where the particle was. That is, Szilard reframed the paradox to be not about the molecules being sorted but about an observer making a measurement.

  (Aside: Szilard was played by Máté Haumann in the 2023 film Oppenheimer.)

  

  The next clue to cracking the puzzle came in the mid-20th century from the growing field of information theory. In 1961, the German-American physicist Rolf Landauer proposed a principle that connected information and entropy directly. Landauer’s principle states that while it’s possible in principle to acquire information in a reversible way — i.e. to be able to acquire it as well as lose it — erasing information from a device with memory has a non-zero thermodynamic cost that can’t be avoided. That is, the act of resetting a memory register of one bit to a standard state generates a small amount of entropy (proportional to Boltzmann’s constant multiplied by the logarithm of two).

  The American information theorist Charles H. Bennett later built on Landauer’s principle and argued that Maxwell’s demon could gather information and act on it — but in order to continue indefinitely, it’d have to erase or overwrite its memory. And that this act of resetting would generate exactly the entropy needed to compensate for the apparent decrease, ultimately preserving the second law of thermodynamics.

  Taken together, Maxwell’s demon was defeated not by the mechanics of the trapdoor but by the thermodynamic cost of processing information. Specifically, the decrease in entropy as a result of the molecules being sorted by their speed is compensated for by the increase in entropy due to the operator’s rewriting or erasure of information about the molecules’ speed. Thus a paradox that’d begun as a challenge to thermodynamics ended up enriching it — by showing information could be physical. It also revealed to scientists that entropy is disorder in matter and energy as well as is linked to uncertainty and information.

  Over time, Maxwell’s demon also became a fount of insight across multiple branches of physics. In classical thermodynamics, for example, entropy came to represent a measure of the probabilities that the system could exist in different combinations of microscopic states. That is, the probabilities referred to the likelihood that a given set of molecules could be arranged in one way instead of another. In statistical mechanics, Maxwell’s demon gave scientists a concrete way to think about fluctuations. In any small system, random fluctuations can reduce entropy for some time in a small portion. While the demon seemed to exploit these fluctuations, the laws of probability were found to ensure that on average, entropy would increase. So the demon became a metaphor for how selection based on microscopic knowledge could alter outcomes but also why such selection can’t be performed without paying a cost.

  

  For information theorists and computer scientists, the demon was an early symbol of the deep ties between computation and thermodynamics. Landauer’s principle showed that erasing information imposes a minimum entropy cost — an insight that matters for how computer hardware should be designed. The principle also influenced debates about reversible computing, where the goal is to design logic gates that don’t ever erase information and thus approach zero energy dissipation. In other words, Maxwell’s demon foreshadowed modern questions about how energy-efficient computing could really be.

  Even beyond physics, the demon has seeped into philosophy, biology, and social thought as a symbol of control and knowledge. In biology, the resemblance between the demon and enzymes that sorts molecules has inspired metaphors about how life maintains order. In economics and social theory, the demon has been used to discuss the limits of surveillance and control. The lesson has been the same in every instance: that information is never free and that the act of using it imposes inescapable energy costs.

  I’m particularly taken by the philosophy that animates the paradox. Maxwell’s demon was introduced as a way to dramatise the tension between the microscopic reversibility of physical laws and the macroscopic irreversibility encoded in the second law of thermodynamics. I found that a few questions in particular — whether the entropy increase due to the use of information is a matter of an observer’s ignorance (i.e. because the observer doesn’t know which particular microstate the system occupies at any given moment), whether information has physical significance, and whether the laws of nature really guarantee the irreversibility we observe — have become touchstones in the philosophy of physics.

  In the mid-20th century, the Szilard engine became the focus of these debates because it refocused the second law from molecular dynamics to the cost of acquiring information. Later figures such as the French physicist Léon Brillouin and the Hungarian-Canadian physicist Dennis Gabor claimed that it’s impossible to measure something without spending energy. Critics however countered that these requirements stipulated the need for specific technologies that would in turn smuggle in some limitations — rather than stipulate the presence of a fundamental principle. That is to say, the debate among philosophers became whether Maxwell’s demon was prevented from breaking the second law by deep and hitherto hidden principles or by engineering challenges.

  This gridlock was broken when physicists observed that even a demon-free machine must leave some physical trace of its interactions with the molecule. That is, any device that sorts particles will end up in different physical states depending on the outcome, and to complete a thermodynamic cycle those states must be reset. Here, the entropy is not due to the informational content but due to the logical structure of memory. Landauer solidified this with his principle that logically irreversible operations such as erasure carry a minimum thermodynamic cost. Bennett extended this by saying that measurements can be made reversibly but not erasure. The philosophical meaning of both these arguments is that entropy increase isn’t just about ignorance but also about parts of information processing being irreversible.

  

  In the quantum domain, the philosophical puzzles became more intense. When an object is measured in quantum mechanics, it isn’t just about an observer updating the information they have about the object — the act of measuring also seems to alter the object’s quantum states. For example, in the Schrödinger’s cat thought experiment, checking whether there’s a cat in the box also causes the cat to default to one of two states: dead or alive. Quantum physicists have recreated Maxwell’s demon in new ways in order to check whether the second law continues to hold. And over the course of many experiments, they’ve concluded that indeed it does.

  The second law didn’t break even when Maxwell’s demon could exploit phenomena that aren’t available in the classical domain, including quantum entanglement, superposition, and tunnelling. This was because, among others, quantum mechanics also has some restrictive rules of its own. For one, some physicists have tried to design “quantum demons” that use quantum entanglement between particles to sort them without expending energy. But these experiments have found that as soon as the demon tries to reset its memory and start again, it must erase the record of what happened before. This step destroys the advantage and the entropy cost returns. The overall result is that even a “quantum demon” gains nothing in the long run.

  For another, the no-cloning theorem states that you can’t make a perfect copy of an unknown quantum state. If the demon could freely copy every quantum particle it measured, it could retain flawless records while still resetting its memory, this avoiding the usual entropy cost. The theorem blocks this strategy by forbidding perfect duplication, ensuring that information can’t be ‘multiplied’ without limit. Similarly, the principle of unitarity implies that a system will always evolve in a way that preserves overall probabilities. As a result, quantum phenomena can’t selectively amplify certain outcomes while discarding others. For the demon, this means it can’t secretly limit the range of possible states the system can occupy into a smaller set where the system has lower entropy, because unitarity guarantees that the full spread of possibilities is preserved across time.

  All these rules together prevent the demon from multiplying or rearranging quantum states in a way that would allow it to beat the second law.

  Then again, these ‘blocks’ that prevent Maxwell’s demon from breaking the second law of thermodynamics in the quantum realm raise a puzzle of their own: is the second law of thermodynamics guaranteed no matter how we interpret quantum mechanics? ‘Interpreting quantum mechanics’ means to interpret what the rules of quantum mechanics say about reality, a topic I covered at length in a recent post. Some interpretations say that when we measure a quantum system, its wavefunction “collapses” to a definite outcome. Others say collapse never happens and that measurement is just entangled with the environment, a process called decoherence. The Maxwell’s demon thought experiment thus forces the question: is the second law of thermodynamics safe in a particular interpretation of quantum mechanics or in all interpretations?

  

  Landauer’s idea, that erasing information always carries a cost, also applies to quantum information. Even if Maxwell’s demon used qubits instead of bits, it won’t be able to escape the fact that to reuse its memory, it must erase the record, which will generate heat. But then the question becomes more subtle in quantum systems because qubits can be entangled with each other, and their delicate coherence — the special quantum link between quantum states — can be lost when information is processed. This means scientists need to carefully separate two different ideas of entropy: one based on what we as observers don’t know (our ignorance) and another based on what the quantum system itself has physically lost (by losing coherence).

  The lesson is that the second law of thermodynamics doesn’t just guard the flow of energy. In the quantum realm it also governs the flow of information. Entropy increases not only because we lose track of details but also because the very act of erasing and resetting information, whether classical or quantum, forces a cost that no demon can avoid.

  Then again, some philosophers and physicists have resisted the move to information altogether, arguing that ordinary statistical mechanics suffices to resolve the paradox. They’ve argued that any device designed to exploit fluctuations will be subject to its own fluctuations, and thus in aggregate no violation will have occurred. In this view, the second law is self-sufficient and doesn’t need the language of information, memory or knowledge to justify itself. This line of thought is attractive to those wary of anthropomorphising physics even if it also risks trivialising the demon. After all, the demon was designed to expose the gap between microscopic reversibility and macroscopic irreversibility, and simply declaring that “the averages work out” seems to bypass the conceptual tension.

  Thus, the philosophical significance of Maxwell’s demon is that it forces us to clarify the nature of entropy and the second law. Is entropy tied to our knowledge/ignorance of microstates, or is it ontic, tied to the irreversibility of information processing and computation? If Landauer is right, handling information and conserving energy are ‘equally’ fundamental physical concepts. If the statistical purists are right, on the other hand, then information adds nothing to the physics and the demon was never a serious challenge. Quantum theory can further stir both pots by suggesting that entropy is closely linked to the act of measurement, of quantum entanglement, and how quantum systems ‘collapse’ to classical ones by the process of decoherence. The demon debate therefore tests whether information is a physically primitive entity or a knowledge-based tool. Either way, however, Maxwell’s demon endures as a parable.

  Ultimately, what makes Maxwell’s demon a gift that keeps giving is that it works on several levels. On the surface it’s a riddle about sorting molecules between two chambers. Dig a little deeper and it becomes a probe into the meaning of entropy. If you dig even further, it seems to be a bridge between matter and information. As the Schrödinger’s cat thought experiment dramatised the oddness of quantum superposition, Maxwell’s demon dramatised the subtleties of thermodynamics by invoking a fantastical entity. And while Schrödinger’s cat forces us to ask what it means for a macroscopic system to be in two states at once, Maxwell’s demon forces us to ask what it means to know something about a system and whether that knowledge can be used without consequence.

  2025.09.19

• CSIR touts dubious ‘Ayurveda’ product for diabetes

  At 6 am on September 13, the CSIR handle on X.com published the following post about an “anti-diabetic medicine” called either “Daiba 250” or “Diabe 250”, developed at the CSIR-Indian Institute of Integrative Medicine (IIIM):

  Science-led #Ayurveda for #Diabetes Care!@CSIRIIIM's Daiba 250 — an anti-diabetic medicine has been developed with cGMP standards@DrJitendraSingh @moayush @startupindia #AyushmanBharat #StartupIndia #MakeInIndia.#84YearsOfCSIR pic.twitter.com/TfVmIdtlgs

  — CSIR, India (@CSIR_IND) September 13, 2025

  Its “key features”, according to the CSIR, are that it created more than 250 jobs and that Prime Minister Narendra Modi “mentioned the startup” to which it has been licensed in his podcast ‘Mann ki Baat’. What of the clinical credentials of Diabe-250, however?

  Diabe-250 is being marketed on India-based online pharmacies like Tata 1mg as an “Ayurvedic” over-the-counter tablet “for diabetes support/healthy sugar levels”. The listing also claims Diabe-250 is backed by a US patent granted to an Innoveda Biological Solutions Pvt. Ltd. Contrary to the CSIR post calling Diabe-250 “medicine”, some listings also carry the disclaimer that it’s “a dietary nutritional supplement, not for medicinal use”.

  (“Ayurveda” is within double-quotes throughout this post because, like most products like Diabe-250 in the market that are also licensed by the Ministry of AYUSH, there’s no evidence that they’re actually Ayurvedic. They may be, they may not be — and until there’s credible proof, the Ayurvedic identity is just another claim.)

  Second, while e-commerce and brand pages use the spellings “Diabe 250” or “Diabe-250” (without or without the hyphen), the CSIR’s social media posts refer to it as “Daiba 250”. The latter also describe it as an anti-diabetic developed/produced with the CSIR-IIIM in the context of incubation and licensing. These communications don’t constitute clinical evidence but they might be the clearest public basis to link the “Daiba” or “Diabe” spellings with the CSIR.

  Multiple product pages also credit Innoveda Biological Solutions Pvt. Ltd. as a marketer and manufacturer. Corporate registry aggregators corroborate the firm’s existence; its CIN is U24239DL2008PTC178821). Similarly, the claim that Diabe-250 is backed by a US patent can be traced most directly to US8163312B2 for “Herbal formulation for prevention and treatment of diabetes and associated complications”. Its inventor is listed as a G. Geetha Krishnan and Innoveda Biological Solutions (P) Ltd. is listed as the current assignee.

  The patent text describes combinations of Indian herbs for diabetes and some complications. Of course no patent is proof of efficacy for any specific branded product or dose.

  The ingredients in Diabe-250 vary by retailer and there’s no consistent, quantitative per-tablet composition on public pages. This said, multiple listings name the following ingredients:

  • “Vidanga” (Embelia ribes)
  • “Gorakh buti” (Aerva lanata)
  • “Raj patha” (Cyclea peltata)
  • “Vairi” or “salacia” (often Salacia oblonga), and
  • “Lajalu” (Biophytum sensitivum)

  The brand page also asserts a “unique combination of 16 herbs” and describes additional “Ayurveda” staples such as berberine source, turmeric, and jamun. However, there doesn’t appear to be a full label image or a quantitative breakdown of the composition of Diabe-250.

  Retail and brand pages also claim Diabe-250 “helps maintain healthy sugar levels”, “improves lipid profile/reduces cholesterol”, and “reduces diabetic complications”, sometimes also including non-glycaemic effects such as “better sleep” and “regular bowel movement”. Several pages also include the caveat that it’s a “dietary nutritional supplement” and that it’s “not for medicinal use”. However, none of these source cite a peer-reviewed clinical trial of Diabe-250 itself.

  In fact, there appear to be no peer-reviewed, product-specific clinical trials of Diabe-250 or Daiba-250 in humans; there are also no clinical trial registry records that were specific to this brand. If such a trial exists and its results were published in a peer-reviewed journal, it hasn’t been cited on the sellers’ or brand pages or in accessible databases.

  ---

  Some ingredient classes in Diabe-250 are interesting even if they don’t validate Diabe-250 as a finished product. For instance, Salacia spp., especially S. reticulata, S. oblonga, and S. chinensis have been known to be α-glucosidase inhibitors. In vitro studies and chemistry reviews have also described Salacia spp. can be potent inhibitors of maltase, sucrase, and isomaltase.

  In one triple-blind, randomised crossover trial in 2023, biscuits fortified with S. reticulata extract reduced HbA1c levels by around 0.25% (2.7 mmol/mol) over three months versus the placebo, with an acceptable safety profile.In post-prandial studies involving healthy volunteers and type 2 diabetes, several randomised crossover designs had lower post-meal glucose and insulin area under the curve when Salacia extract was co-ingested along with carbohydrate.

  Similarly, berberine-based neutraceuticals (such as those including Berberis aristata) have shown glycaemic improvements in the clinical literature (at large, not specific to Diabe-250) in people with type 2 diabetes. However, these effects were often reported in combination with other compounds and which researchers also indicated depended strongly on formulation and dose.

  Finally, a 2022 systematic review of “Ayurvedic” medicines in people with type 2 diabetes reported heterogeneous evidence, including some promising signals, but also emphasised methodological limitations and the need for randomised controlled trials of higher quality.

  Right now, if Diabe-250 works as advertised, there’s no scientific proof in the public domain, especially in the form of product-specific clinical trials that define its composition, dosage, and endpoints.

  ---

  In India, Ayurvedic drugs come under the Drugs & Cosmetics Rules 1945. Labelling provisions under Section 161 require details such as the manufacturer’s address, batch, and manufacturing and expiry dates while practice guides also note the product license number on the label for “Ayurvedic” drugs. However, several retail pages for Diabe-250 display it as a “dietary nutritional supplement” and add that it’s “not for medicinal use”, implying that it’s being marketed with supplement-style claims rather than as an Ayurvedic “medicine” in the narrow regulatory sense — which runs against the claim in the CSIR post on X.com. Public pages also didn’t display an AYUSH license number for Diabe-250. I haven’t checked a physical pack.

  A well-known study in JAMA in 2008, of “Ayurvedic” products purchased over the internet, found that around 20% of them contained lead, mercury or arsenic, and public-health advisories and case reports that have appeared since have echoed these concerns. This isn’t a claim about Diabe-250 specifically but a category-level risk of “Ayurvedic” products that are available to buy online and which are compounded by the unclear composition of Diabe-250. The inconsistent naming also opens the door to counterfeit products that are also more likely to be contaminated.

  Materials published by the Indian and state governments, including the Ministry of AYUSH, have framed “Ayurveda” as complementary to allopathic medicine. For example, if a person with diabetes chooses to try “Ayurvedic” support, the standard advice is to not discontinue prescribed therapy and to monitor one’s glucose, especially if the individual is using α-glucosidase-like agents that alter the post-prandial response.

  In sum, Diabe-250 is a multi-herb “Ayurvedic” tablet marketed by Innoveda for glycaemic support and has often been promoted with a related US patent owned by the company. However, patents are not clinical trials and patent offices don’t clinically evaluate drugs described in patent applications. That information can only come from clinical trials, especially when a drug is being touted as “science-led”, as the CSIR has vis-à-vis Diabe-250. But there are no published clinical trials of the product. And while there’s some evidence for some of its constituents, particularly Salacia, to reduce postprandial glucose and to effect small changes in the HbA1c levels over a few months, there’s no product-specific proof.

  2025.09.14

• A danger of GST 2.0

  Since Union finance minister Nirmala Sitharaman’s announcement last week that India’s Goods and Services Tax (GST) rates will be rationalised anew from September 22, I’ve been seeing a flood of pieces all in praise — and why not?

  The GST regime has been somewhat controversial since its launch because, despite simplifying compliance for businesses and industry, it increased the costs for consumers. The Indian government exacerbated that pain point by undermining the fiscal federalism of the Union, increasing its revenues at the expense of states’ as well as cutting allocations.

  While there is (informed) speculation that the next Finance Commission will further undercut the devolution of funds to the states, GST 2.0 offers some relief to consumers in the form of making various products more affordable. Populism is popular, after all.

  However, increasing affordability isn’t always a good thing even if your sole goal is to increase consumption. This is particularly borne out in the food and nutrition domain.

  For example, under the new tax regime, from September 22, the GST on pizza bread will slip from 5% to zero. This means both sourdough pizza bread and maida (refined flour) pizza bread will go from 5% to zero. However, because there is more awareness of maida as an ingredient in the populace and less so of sourdough, and because maida as a result enjoys a higher economy of scale and is thus less expensive (before tax), the demand for maida bread is likely to increase more than the demand for sourdough bread.

  This is unfortunate: ideally, sourdough bread should be more affordable — or, alternatively, the two breads should be equally affordable as well as have threshold-based front-of-pack labelling. That is to say, liberating consumers to be able to buy new food products or more of the old ones without simultaneously empowering consumers to make more informed choices could tilt demand in favour of unhealthier foods.

  Ultimately, the burden of non-communicable diseases in the population will increase, as will consumers’ expenses on healthcare, dietary interventions, and so on. I explained this issue in The Hindu on September 9, 2025, and set out solutions that the Indian government must implement in its food regulation apparatus posthaste.

  Without these measures, GST 2.0 will likely be bad news for India’s dietary and nutritional ambitions.

  2025.09.10

• Lighting the way with Parrondo’s paradox

  In science, paradoxes often appear when familiar rules are pushed into unfamiliar territory. One of them is Parrondo’s paradox, a curious mathematical result showing that when two losing strategies are combined, they can produce a winning outcome. This might sound like trickery but the paradox has deep connections to how randomness and asymmetry interact in the physical world. In fact its roots can be traced back to a famous thought experiment explored by the US physicist Richard Feynman, who analysed whether one could extract useful work from random thermal motion. The link between Feynman’s thought experiment and Parrondo’s paradox demonstrates how chance can be turned into order when the conditions are right.

  Imagine two games. Each game, when played on its own, is stacked against you. In one, the odds are slightly less than fair, e.g. you win 49% of the time and lose 51%. In another, the rules are even more complex, with the chances of winning and losing depending on your current position or capital. If you keep playing either game alone, the statistics say you will eventually go broke.

  But then there’s a twist. If you alternate the games — sometimes playing one, sometimes the other — your fortune can actually grow. This is Parrondo’s paradox, proposed in 1996 by the Spanish physicist Juan Parrondo.

  The answer to how combining losing games can result in a winning streak lies in how randomness interacts with structure. In Parrondo’s games, the rules are not simply fair or unfair in isolation; they have hidden patterns. When the games are alternated, these patterns line up in such a way that random losses become rectified into net gains.

  Say there’s a perfectly flat surface in front of you. You place a small bead on it and then you constantly jiggle the surface. The bead jitters back and forth. Because the noise you’re applying to the bead’s position is unbiased, the bead simply wanders around in different directions on the surface. Now, say you introduce a switch that alternates the surface between two states. When the switch is ON, an ice-tray shape appears on the surface. When the switch is OFF, it becomes flat again. This ice-tray shape is special: the cups are slightly lopsided because there’s a gentle downward slope from left to right in each cup. At the right end, there’s a steep wall. If you’re jiggling the surface when the switch is OFF, the bead diffuses a little towards the left, a little towards the right, and so on. When you throw the switch to ON, the bead falls into the nearest cup. Because each cup is slightly tilted towards the right, the bead eventually settles near the steep wall there. Then you move the switch to OFF again.

  As you repeat these steps with more and more beads over time, you’ll see they end up a little to the right of where they started. This is Parrando’s paradox. The jittering motion you applied to the surface caused each bead to move randomly. The switch you used to alter the shape of the surface allowed you to expend some energy in order to rectify the beads’ randomness.

  The reason why Parrondo’s paradox isn’t just a mathematical trick lies in physics. At the microscopic scale, particles of matter are in constant, jittery motion because of heat. This restless behaviour is known as Brownian motion, named after the botanist Robert Brown, who observed pollen grains dancing erratically in water under a microscope in 1827. At this scale, randomness is unavoidable: molecules collide, rebound, and scatter endlessly.

  Scientists have long wondered whether such random motion could be tapped to extract useful work, perhaps to drive a microscopic machine. This was Feynman’s thought experiment as well, involving a device called the Brownian ratchet, a.k.a. the Feynman-Smoluchowski ratchet. The Polish physicist Marian Smoluchowski dreamt up the idea in 1912 and which Feynman popularised in a lecture 50 years later, in 1962.

  Picture a set of paddles immersed in a fluid, constantly jolted by Brownian motion. A ratchet and pawl mechanism is attached to the paddles (see video below). The ratchet allows the paddles to rotate in one direction but not the other. It seems plausible that the random kicks from molecules would turn the paddles, which the ratchet would then lock into forward motion. Over time, this could spin a wheel or lift a weight.

  In one of his physics famous lectures in 1962, Feynman analysed the ratchet. He showed that the pawl itself would also be subject to Brownian motion. It would  jiggle, slip, and release under the same thermal agitation as the paddles. When everything is at the same temperature, the forward and backward slips would cancel out and no net motion would occur.

  This insight was crucial: it preserved the rule that free energy can’t be extracted from randomness at equilibrium. If motion is to be biased in only one direction, there needs to be a temperature difference between different parts of the ratchet. In other words, random noise alone isn’t enough: you also need an asymmetry, or what physicists call nonequilibrium conditions, to turn randomness into work.

  Let’s return to Parrondo’s paradox now. The paradoxical games are essentially a discrete-time abstraction of Feynman’s ratchet. The losing games are like unbiased random motion: fluctuations that on their own can’t produce net gain because the gains become cancelled out. But when they’re alternated cleverly, they mimic the effect of adding asymmetry. The combination rectifies the randomness, just as a physical ratchet can rectify the molecular jostling when a gradient is present.

  This is why Parrondo explicitly acknowledged his inspiration from Feynman’s analysis of the Brownian ratchet. Where Feynman used a wheel and pawl to show how equilibrium noise can’t be exploited without a bias, Parrondo created games whose hidden rules provided the bias when they were combined. Both cases highlight a universal theme: randomness can be guided to produce order.

  The implications of these ideas extend well beyond thought experiments. Inside living cells, molecular motors like kinesin and myosin actually function like Brownian ratchets. These proteins move along cellular tracks by drawing energy from random thermal kicks with the aid of a chemical energy gradient. They demonstrate that life itself has evolved ways to turn thermal noise into directed motion by operating out of equilibrium.

  Parrondo’s paradox also has applications in economics, evolutionary biology, and computer algorithms. For example, alternating between two investment strategies, each of which is poor on its own, may yield better long-term outcomes if the fluctuations in markets interact in the right way. Similarly, in genetics, when harmful mutations alternate in certain conditions, they can produce beneficial effects for populations. The paradox provides a framework to describe how losing at one level can add up to winning at another.

  Feynman’s role in this story is historical as well as philosophical. By dissecting the Brownian ratchet, he demonstrated how deeply the laws of thermodynamics constrain what’s possible. His analysis reminded physicists that intuition about randomness can be misleading and that only careful reasoning could reveal the real rules.

  In 2021, a group of scientists from Australia, Canada, France, and Germany wrote in Cancers that the mathematics of Parrondo’s paradox could also illuminate the biology of cancerous tumours. Their starting point was the observation that cancer cells behave in ways that often seem self-defeating: they accumulate genetic and epigenetic instability, devolve into abnormal states, sometimes stop dividing altogether, and often migrate away from their original location and perish. Each of these traits looks like a “losing strategy” — yet cancers that use these ‘strategies’ together are often persistent.

  The group suggested that the paradox arises because cancers grow in unstable, hostile environments. Tumour cells deal with low oxygen, intermittent blood supply, attacks by the immune system, and toxic drugs. In these circumstances, no single survival strategy is reliable. A population of only stable tumour cells would be wiped out when the conditions change. Likewise a population of only unstable cells would collapse under its own chaos. But by maintaining a mix, the group contended, cancers achieve resilience. Stable, specialised cells can exploit resources efficiently while unstable cells with high plasticity constantly generate new variations, some of which could respond better to future challenges. Together, the team continued, the cancer can alternate between the two sets of cells so that it can win.

  The scientists also interpreted dormancy and metastasis of cancers through this lens. Dormant cells are inactive and can lie hidden for years, escaping chemotherapy drugs that are aimed at cells that divide. Once the drugs have faded, they restart growth. While a migrating cancer cell has a high chance of dying off, even one success can seed a tumor in a new tissue.

  On the flip side, the scientists argued that cancer therapy can also be improved by embracing Parrondo’s paradox. In conventional chemotherapy, doctors repeatedly administer strong drugs, creating a strategy that often backfires: the therapy kills off the weak, leaving the strong behind — but in this case the strong are the very cells you least want to survive. By contrast, adaptive approaches that alternate periods of treatment with rest or that mix real drugs with harmless lookalikes could harness evolutionary trade-offs inside the tumor and keep it in check. Just as cancer may use Parrondo’s paradox to outwit the body, doctors may one day use the same paradox to outwit cancer.

  On August 6, physicists from Lanzhou University in China published a paper in Physical Review E discussing just such a possibility. They focused on chemotherapy, which is usually delivered in one of two main ways. The first, called the maximum tolerated dose (MTD), uses strong doses given at intervals. The second, called low-dose metronomic (LDM), uses weaker doses applied continuously over time. Each method has been widely tested in clinics and each one has drawbacks.

  MTD often succeeds at first by rapidly killing off drug-sensitive cancer cells. In the process, however, it also paves the way for the most resistant cancer cells to expand, leading to relapse. LDM on the other hand keeps steady pressure on a tumor but can end up either failing to control sensitive cells if the dose is too low or clearing them so thoroughly that resistant cells again dominate if the dose is too strong. In other words, both strategies can be losing games in the long run.

  The question the study’s authors asked was whether combining these two flawed strategies in a specific sequence could achieve better results than deploying either strategy on its own. This is the sort of situation Parrondo’s paradox describes, even if not exactly. While the paradox is concerned with combining outright losing strategies, the study has discussed combining two ineffective strategies.

  To investigate, the researchers used mathematical models that treated tumors as ecosystems containing three interacting populations: healthy cells, drug-sensitive cancer cells, and drug-resistant cancer cells. They applied equations from evolutionary game theory that tracked how the fractions of these groups shifted in different conditions.

  The models showed that in a purely MTD strategy, the resistant cells soon took over, and in a purely LDM strategy, the outcomes depended strongly on drug strength but still ended badly. But when the two schedules were alternated, the tumor behaved differently. The more sensitive cells were suppressed but not eliminated while their persistence prevented the resistant cells from proliferating quickly. The team also found that the healthy cells survived longer.

  Of course, tumours are not well-mixed soups of cells; in reality they have spatial structure. To account for this, the team put together computer simulations where individual cells occupied positions on a grid; grew, divided or died according to fixed rules; and interacted with their neighbours. This agent-based approach allowed the team to examine how pockets of sensitive and resistant cells might compete in more realistic tissue settings.

  Their simulations only confirmed the previous set of results. A therapeutic strategy that alternated between MTD and LDM schedules extended the amount of time before the resistant cells took over and while the healthy cells dominated. When the model started with the LDM phase in particular, the  sensitive cancer cells were found to compete with the resistant cancer cells and the arrival of the MTD phase next applied even more pressure on the latter.

  This is an interesting finding because it suggests that the goal of therapy may not always be to eliminate every sensitive cancer cell as quickly as possible but, paradoxically, that sometimes it may be wiser to preserve some sensitive cells so that they can compete directly with resistant cells and prevent them from monopolising the tumor. In clinical terms, alternating between high- and low-dose regimens may delay resistance and keep tumours tractable for longer periods.

  Then again this is cancer — the “emperor of all maladies” — and in silico evidence from a physics-based model is only the start. Researchers will have to test it in real, live tissue in animal models (or organoids) and subsequently in human trials. They will also have to assess whether certain cancers, followed by a specific combination of drugs for those cancers, will benefit more (or less) from taking the Parrando’s paradox way.

  As Physics reported on August 6:

  [University of London mathematical oncologist Robert] Noble … says that the method outlined in the new study may not be ripe for a real-world clinical setting. “The alternating strategy fails much faster, and the tumor bounces back, if you slightly change the initial conditions,” adds Noble. Liu and colleagues, however, plan to conduct in vitro experiments to test their mathematical model and to select regimen parameters that would make their strategy more robust in a realistic setting.

  2025.09.07

• GST 2.0 + WordPress.com

  Union finance minister Nirmala Sitharaman announced sweeping changes to the GST rates on September 3. However, I think the rate for software services (HSN 99831) will remain unchanged at 18%. This is a bummer because every time I renew my WordPress.com site or purchase software over the internet in rupees, the total cost increases by almost a fifth.

  The disappointment is compounded by the fact that WordPress.com and many other software service providers provide adjusted rates for users in India in order to offset the country’s lower purchasing power per capita. For example, the lowest WordPress and Ghost plans by WordPress.com and MagicPages.co, respectively, cost $4 and $12 a month. But for users in India, the WordPress.com plan costs Rs 200 a month while MagicPages.co offers a Rs 450 per month plan, both with the same feature set — a big difference. The 18% GST however wipes out some, not all, of these gains.

  Paying for software services over the internet when they’re billed in dollars rather than rupees isn’t much different. While GST doesn’t apply, the rupee-to-dollar rate has become abysmal. [Checks] Rs 88.14 to the dollar at 11 am. Ugh.

  I also hoped for a GST rate cut on software services because if content management software in particular becomes more affordable, more people would be able to publish on the internet.

  2025.09.04

• Towards KD45

  On the subject of belief, I’m instinctively drawn to logical systems that demand consistency, closure, and introspection. And the KD45 system among them exerts a special pull. It consists of the following axioms:

  • K (closure): If you believe an implication and you believe the antecedent, then you believe the consequent. E.g. if you believe “if X then Y” and you believe X, then you also believe Y.
  • D (consistency): If you believe X, you don’t also believe not-X (i.e. X’s negation).
  • 4 (positive introspection): If you believe X, then you also believe that you believe X, i.e. you’re aware of your own beliefs.
  • 5 (negative introspection): If you don’t believe X, then you believe that you don’t believe X, i.e. you know what you don’t believe.

  Thus, KD45 pictures a believer who never embraces contradictions, who always sees the consequences of what they believe, and who is perfectly aware of their own commitments. It’s the portrait of a mind that’s transparent to itself, free from error in structure, and entirely coherent. There’s something admirable in this picture. In moments of near-perfect clarity, it seems to me to describe the kind of believer I’d like to be.

  Yet the attraction itself throws up a paradox. KD45 is appealing precisely because it abstracts away from the conditions in which real human beings actually think. In other words, its consistency is pristine because it’s idealised. It eliminates the compromises, distractions, and biases that animate everyday life. To aspire to KD45 is therefore to aspire to something constantly unattainable: a mind that’s rational at every step, free of contradiction, and immune to the fog of human psychology.

  My attraction to KD45 is tempered by an equal admiration for Bayesian belief systems. The Bayesian approach allows for degrees of confidence and recognises that belief is often graded rather than binary. To me, this reflects the world as we encounter it — a realm of incomplete evidence, partial understanding, and evolving perspectives.

  I admire Bayesianism because it doesn’t demand that we ignore uncertainty. It compels us to face it directly. Where KD45 insists on consistency, Bayesian thinking insists on responsiveness. I update beliefs not because they were previously incoherent but because new evidence has altered the balance of probabilities. This system thus embodies humility, my admission that no matter how strongly I believe today, tomorrow may bring evidence that forces me to change my mind.

  The world, however, isn’t simply uncertain: it’s often contradictory. People hold opposing views, traditions preserve inconsistencies, and institutions are riddled with tensions. This is why I’m also drawn to paraconsistent logics, which allow contradictions to exist without collapsing. If I stick to classical logic, I’ll have to accept everything if I also accept a contradiction. One inconsistency causes the entire system to explode. Paraconsistent theories reject that explosion and instead allow me to live with contradictions without being consumed by them.

  This isn’t an endorsement of confusion for its own sake but a recognition that practical thought must often proceed even when the data is messy. I can accept, provisionally, both “this practice is harmful” and “this practice is necessary”, and work through the tension without pretending I can neatly resolve the contradiction in advance. To deny myself this capacity is not to be rational — it’s to risk paralysis.

  Finally, if Bayesianism teaches humility and paraconsistency teaches tolerance, the AGM theory of belief revision teaches discipline. Its core idea is that beliefs must be revised when confronted by new evidence, and that there are rational ways of choosing what to retract, what to retain, and what to alter. AGM speaks to me because it bridges the gap between the ideal and the real. It allows me to acknowledge that belief systems can be disrupted by facts while also maintaining that I can manage disruptions in a principled way.

  That is to say, I don’t aspire to avoid the shock of revision but to absorb it intelligently.

  Taken together, my position isn’t a choice of one system over another. It’s an attempt to weave their virtues together while recognising their limits. KD45 represents the ideal that belief should be consistent, closed under reasoning, and introspectively clear. Bayesianism represents the reality that belief is probabilistic and always open to revision. Paraconsistent logic represents the need to live with contradictions without succumbing to incoherence. AGM represents the discipline of revising beliefs rationally when evidence compels change.

  A final point about aspiration itself. To aspire to KD45 isn’t to believe I will ever achieve it. In fact, I acknowledge I’m unlikely to desire complete consistency at every turn. There are cases where contradictions are useful, where I’ll need to tolerate ambiguity, and where the cost of absolute closure is too high. If I deny this, I’ll only end up misrepresenting myself.

  However, I’m not going to be complacent either. I believe it’s important to aspire even if what I’m trying to achieve is going to be perpetually out of reach. By holding KD45 as a guiding ideal, I hope to give shape to my desire for rationality even as I expect to deviate from it. The value lies in the direction, not the destination.

  Therefore, I state plainly (he said pompously):

  • I admire the clarity of KD45 and treat it as the horizon of rational belief
  • I embrace the flexibility of Bayesianism as the method of navigating uncertainty
  • I acknowledge the need for paraconsistency as the condition of living in a world of contradictions
  • I uphold the discipline of AGM belief revision as the art of managing disruption
  • I aspire to coherence but accept that my path will involve noise, contradiction, and compromise

  In the end, the point isn’t to model myself after one system but to recognise the world demands several. KD45 will always represent the perfection of rational belief but I doubt I’ll ever get there in practice — not because I think I can’t but because I know I will choose not to in many matters. To be rational is not to be pure. It is to balance ideals with realities, to aspire without illusion, and to reason without denying the contradictions of life.

  2025.08.24

• On the PixxelSpace constellation

  The announcement that a consortium led by PixxelSpace India will design, build, and operate a constellation of 12 earth-observation satellites marks a sharp shift in how India approaches large space projects. The Indian National Space Promotion and Authorisation Centre (IN-SPACe) awarded the project after a competitive process.

  What made headlines was that the winning bid asked for no money from the government. Instead, the group — which includes Piersight Space, SatSure Analytics India, and Dhruva Space — has committed to invest more than Rs 1,200 crore of its own resources over the next four to five years. The constellation will carry a mix of advanced sensors, from multispectral and hyperspectral imagers to synthetic aperture radar, and it will be owned and operated entirely by the private side of the partnership.

  PixxelSpace has said the zero-rupee bid is a conscious decision to support the vision of building an advanced earth-observation system for India and the world. The companies have also expressed belief they will recover their investment over time by selling high-value geospatial data and services in India and abroad. IN-SPACe’s chairman has called this a major endorsement of the future of India’s space economy.

  Of course the benefits for India are clear. Once operational, the constellation should reduce the country’s reliance on foreign sources of satellite imagery. That will matter in areas like disaster management, agriculture planning, and national security, where delays or restrictions on outside data can have serious consequences. Having multiple companies in the consortium brings together strengths in hardware, analytics, and services, which could create a more complete space industry ecosystem. The phased rollout will also mean technology upgrades can be built in as the system grows, without heavy public spending.

  Still, the arrangement raises difficult questions. In practice, this is less a public–private partnership than a joint venture. I assume the state will provide its seal of approval, policy support, and access to launch and ground facilities. If it does share policy support, it will have to explain why that’s vouchsafed for the collaboration isn’t of being expanded to the industry as a whole. I also heard IN-SPACe will ‘collate’ demand within the government for the constellation’s products and help meet them.

  Without assuming a fiscal stake, however, the government is left with less leverage to set terms or enforce priorities, especially if the consortium’s commercial goals don’t always align with national needs. It’s worth asking why the government issued an official request-for-proposal if didn’t intend to assume a stake, and whether the Rs-350-crore soft loan IN-SPACe originally offered for the project will still be available, repurposed or quietly withdrawn.

  I think the pitch will also test public oversight. IN-SPACe will need stronger technical capacity, legal authority, procedural clarity, and better public communication to monitor compliance without frustrating innovation. Regulations on remote sensing and data-sharing will probably have to be updated to cover a fully commercial system that sells services worldwide. Provisions that guarantee government priority access in emergencies and that protect sensitive imagery will have to be written clearly into law and contracts. Infrastructure access, from integration facilities to launch slots, must be managed transparently to avoid bottlenecks or perceived bias.

  The government’s minimal financial involvement saves public money but it also reduces long-term control. If India repeats this model, it should put in place new laws and safeguards that define how sovereignty, security, and public interest are to be protected when critical space assets are run by private companies. Without such steps, the promise of cost-free expansion could instead lead to new dependencies that are even harder to manage in future.

  Featured image credit: Carl Wang/Unsplash.

  2025.08.22