von Neumann entropy

A superconductor is a material in which electricity flows without loss, like water through a perfectly smooth pipe. But if you squeeze it slightly, like a spring, its properties change. Scientists have long wanted to measure how much heat such a compressed material absorbs, but instruments couldn't fit into the cramped setup. Now they've devised a way around this obstacle.

A quantum state is like mountain weather: it changes with every glance. To estimate its uncertainty, physicists make copies and take measurements. Each copy is a thermometer that breaks after use.

A fussy thermometer shows different temperatures depending on who is looking at it. A black hole behaves much the same way: its thermal properties, entropy and temperature, depend on how we describe the information hidden inside.

A hunter sets snares in the woods. They know the animal's habits but not whether it will ever show up. In much the same way, physicists set up detectors in hopes of catching particles of dark matter — the invisible substance thought to make up most of the Universe's mass. A new study offers hunters an unusual tool: not a trap, but a measuring tape.

At the heart of a dying star, as its core collapses into a neutron sphere, an invisible storm is born. Trillions of neutrinos, particles that barely interact with matter, race through the scorching plasma. There are so many of them that they begin to influence each other, changing their 'flavors' (electron, muon, tau) like chameleons.

The quantum vacuum is not empty: it is permeated by entanglement linking distant regions. But when physicists tried to compute this entanglement, it turned out to be infinite. The culprit: particles with very low energy that can smear out over arbitrarily large volumes. They are like waves with no shores to contain them.

The quantum world is full of strange connections. Two particles can become “entangled”: the state of one instantly affects the other, even if they are kilometers apart. But how do you know if they are truly entangled? It's like a ball of yarn: at first glance, you can't tell whether it's a single tangled ball or just a mess of loose threads.

A delicate watermark on wet paper, and a fogged-up glass before your eyes. That's roughly what reading quantum states of light looks like: fragile photon patterns get distorted by losses, and detectors catch only a small fraction of the signal.

A translator gets distracted by every rustle in the hall, and a one-hour meeting stretches into two. That's roughly how thermal noise disrupts quantum teleportation: the carrier particles lose synchronization, and the message gets corrupted.

If you accelerate an observer to near-light speed, the emptiness around them stops being cold. A thermal glow emerges — an effect predicted in 1976 and named after William Unruh. For earthly accelerations, this glow is vanishingly faint, like a whisper at a rock concert.

A search party combs a foggy field. At first, people spread out wide: a high 'temperature' of search means a great variety of attempts. Upon finding a trace, they close in. But the speed of convergence is critical — narrow the circle too early, and the best find stays aside. The coordinator assesses the spread of searchers and commands when to speed up or slow down.

The power grid is a huge restaurant where thousands of 'dishes'—potential accidents — threaten to be served every minute. Checking every plate fully would take hours, and a quick glance sometimes misses danger, like invisible mold on cheese.

In the microscopic world, noise, random fluctuations, usually destroys fragile quantum states, much like ripples on water break the reflection in a puddle. However, in some systems, collective, coordinated noise from many particles produces the opposite effect: a stable unidirectional flow emerges.

The hunt for non-Abelian anyons is a quest for a quantum ghost that leaves only one clue: entropy. These particles live in two-dimensional systems at ultralow temperatures. Their exchange doesn't just flip the sign of the wavefunction — it entangles states, creating topological order.

Eight hundred participants, firms and workers, enter the market. Each holds a strict preference list in mind. The challenge: pair up so that no one wants a do-over. In mathematics, this is called a game theory with a ban on repeat alliances, and its roots trace back to the work of John von Neumann and Oskar Morgenstern.

A topological superconducting nanowire is like a bridge, with two inseparable sentinels — Majorana modes — stationed at its ends. These exotic quasiparticles, predicted back in the 1930s, promise to become the foundation of an error-proof quantum computer.

When a building collapses, steel rebar turns into a chaotic magnetic compass. Entropy, a measure of disorder, increases, but it's this very disorder that points to voids where people might be.

We're used to thinking that noticing only confirmations of our guesses is a mistake. But recent calculations, inspired by wave behavior, flip that view. It turns out that the habit of seeking the familiar is the best way to reduce information noise (scientists call it entropy).

A translator hears not just words, but intonation, pauses, the tremor in a voice. Ordinary entropy analysis of cosmic perturbations is like meeting minutes: it records facts but loses the nuances. Pseudoentropy entropy is a new tool that captures the hidden "music" of the early universe.

Creating an alloy from several metals is like cooking a soup from mysterious ingredients: guessing the final texture without trial and error is nearly impossible. Conventional models often get it wrong when there are more than three different kinds of atoms.

A boiling kettle with the lid on doesn’t show the bubbling, but the sound and vibration betray the intensity of the water boiling.

An invisible trap made of laser light held 11,000 atoms — a record for quantum engineering. Previously, physicists wrestled with a few dozen of these particles; now the count runs into the thousands. Each atom becomes a qubit — a quantum bit that is both zero and one at once, like a spinning coin balanced on its edge.

Imagine a library where instead of books, you have every conceivable combination of a few dozen quantum particles. The number of combinations grows like a snowball: add one particle, and the volume doubles. For fifty particles, there are more shelves than atoms in the visible universe.

Entropy is a measure of disorder. Two shuffled decks of cards, dumped into one pile, won’t become more chaotic than the sum of their individual disorders. This is the rule of subadditivity: the whole is no more chaotic than its parts.

An eternal Universe where the measure of disorder (энтропия) grows is, for most of its time, a homogeneous void. In it, complex structures occasionally assemble by chance, such as fleeting consciousnesses — ‘мозги Больцмана’.

A soap bubble hangs in the air, and its shape already contains everything needed to calculate the internal tension. A similar principle worked in a new study on quantum gravity. Physicists took a closed surface, an analog of such a bubble, and specified a few numbers on it: for example, the electric field strength and the curvature.

A busy intersection without traffic lights. Cars drive, brake, and change lanes, and from the outside, it looks like complete chaos. But if you watch for a while, you can notice a rhythm: the flow thickens and thins. In physics, this rhythm of disorder is called entropy: it shows how far a system is from calm equilibrium, like water in a boiling kettle.

If you accelerate one of four entangled particles to enormous speed, their quantum entanglement doesn't just melt away: it first weakens, then strengthens again. Physicists discovered this unexpected twist while studying the behavior of a special state called the Dicke state in a world governed by the laws of relativity.

Ordinary water freezes as expected. But sometimes hot water turns to ice faster than cold — the Mpemba effect. Recently, a similar rule-breaking was found in quantum entanglement — the invisible link between particles discovered by Erwin Schrödinger.

The fifth coordinate, curled into a tiny loop, could be hiding right under our noses. In such a universe, everything we see would be stuck to a four-dimensional film, the brane, like water droplets on a window pane. But why doesn't matter slip into the fifth dimension? The answer may lie in the twisting of spacetime itself.

Spacetime isn't a flat sheet, but a crumpled fabric: stars and planets leave creases on it. Information is written into every crease, and to erase it, heat would have to be released — that's how Landauer's principle works. Thus, the 'crumpledness' of space can be measured through entropy — the universal measure of disorder.

The quantum world is usually drawn with points, but real instruments only see blurry spots. Heisenberg and von Neumann long suspected: precision is unattainable.

Two simultaneous interpreters sit in the same booth. The first translates from English into Russian but swallows half the words. The second goes from Russian into French, and also loses half. If you put them together, you'd expect a mess: the losses would add up, and only a quarter of the meaning would reach the listener. But what if they start correcting each other?

A voice recorder captures a city square. You hear the hum of a crowd, footsteps, snatches of conversation, the rumble of traffic. Somewhere in that sonic stew, a street musician plays a simple tune on a flute. Hearing it isn't easy: there's just too much mixed together.

Quantum entanglement is a fragile thing. Two friends move to different cities, each carrying a box with one of a pair of gloves: if one box has the left glove, the other has the right. Entangled particles behave similarly, except that until measurement, no one knows which is where.

Two dice always land on the same number, no matter how far apart you toss them. In quantum physics, this behavior is called entanglement, and it's the key resource for future computers and secure communication. The trouble is, not all entanglement is easy to spot.

If you squeeze an ordinary piece of metal, it heats up — even those who have pumped a bicycle tire know that. But what happens if you squeeze a magnet in which atomic arrows refuse to freeze even at absolute zero?

A coin tossed with the right hand always lands heads, while the left hand always lands tails. That's roughly how the lambda-b particle behaves: its intrinsic rotation (spin) is inextricably linked to which D-meson is born in its decay. This link is quantum entanglement, but not between two particles — rather between two properties of the same particle.

Four tiny puddles, and in them ripple droplets of a superfluid. These droplets are atoms cooled to near absolute zero and paired up like little magnets. When the whole setup rotates, the fluid flows between the puddles. Simply counting how many droplets end up in each reveals the angle and angular velocity.

Spread a handful of coins on the table. Each coin is a possible outcome: heads, tails, or something else. If we know how chaotic this set is (its entropy), which combinations of probabilities will we encounter more often? It turns out the answer depends on how much chaos we allow.