Physicists Trace the Rise in Entropy to Quantum Information

The idea that the measure of disorder will always stay the same or increase is arguably the most sacrosanct principle in physical law. If your theory is found to be against the second law of thermodynamics, there is nothing you can do except to collapse in humiliation.

Physicists are troubled by something about the second law. Some people don't think we understand it or that its foundations are strong. Although it's called a law, it's usually thought of as just probabilistic, meaning that the outcome of any process will be the most likely one.

Physicists want laws of physics to be exact, and they don't just want descriptions of what will happen. Can the second law be tightened up to include more than a statement of likelihoods?

A number of groups have done that. The second law may have been woven out of the fundamental principles of quantum mechanics, which may have had directionality and irreversibility built into them at the deepest level. The second law comes about because of quantum effects, according to this view. It comes from the way in which quantum systems share information and the way in which cornerstone quantum principles dictate what is allowed and what is not. The most likely outcome of change is not an increase in entropy. It is a logical consequence of the quantum resource of information.

Quantum Inevitability

In the early 19th century, thermodynamics was created to describe the flow of heat and work. As steam power drove the Industrial Revolution, engineers wanted to make their devices as efficient as possible.

It wasn't much help in making better engines and machinery. Criteria that govern all processes of change became one of the central pillars of modern physics.

The first and second are the most fundamental laws of classical thermodynamics. The first law says that energy is always used, and the second says that heat always flows from hot to cold. In any process of change, this must increase in terms of entropy. The Austrian physicist Ludwig Boltzmann formulated entropy as a quantity related to the total number of microstates a system has: how many equivalent ways its particles can be arranged.

The second law shows why change happens. The classical laws of motion can be reversed at the level of individual particles. The second law states that change must happen in a way that increases the number of people. An arrow of time is imposed by this directionality. Time seems to flow from past to future because the universe began in a low-entropy state and is heading toward one of ever higher entropy. The scientists of the mid-19th century called the heat death of the universe because of the implication that eventually heat will be spread completely uniformly and there will be no driving force for further change.

Boltzmann's description of entropy seems to explain directionality. Many-particle systems that are more disordered and have higher entropy are more likely to end up producing them. The second law is a law of large numbers. In this view, there is no fundamental reason why entropy can decrease or increase. It is extremely unlikely.

Some questions are left hanging by this statistical physics. It forces us to be content with taking averages across the entire ensemble and directs us toward the most probable microstates.

The laws of classical physics allow only one outcome for any point in time. If only one outcome is ever possible, can that hypothetical ensemble of states enter the picture?

A physicist at Oxford has been trying to avoid this dilemma by developing a theory of a world in which probability and randomness are completely absent from physical processes. It wants to establish which processes are possible and which are forbidden.

The aim of constructor theory is to express all of physics in terms of possible and impossible transformations. It is similar to the way thermodynamics began, in that it considers change in the world to be something produced by machines that work in a pattern similar to the Carnot cycle. The constructor is a catalyst, facilitating a process and being returned to its original state at the end.

You can think of a number of different machines that can do this. When the house is built, the machines return to their original state.

It doesn't mean that a machine can reverse a task. A machine that builds a house might not be able to dismantle it. The operation of the constructor is different from the operation of the laws of motion.

The reason for the irreversibility is that a constructor is geared to a given environment. It requires information from the environment to complete that task. The machine is specific to the environment it is working in.

Even though quantum mechanical laws are themselves perfectly reversible, the quantum theorist Vlatko Vedral and his colleagues at Oxford and Italy showed that the processes that are irreversible in this sense are identified by the theory of constructor theory.

The researchers considered a transformation involving the states of quantum bits, which can exist in one of two states or in a combination of both. A single qubit may be transformed into a target state B 2 when it interacts with other qubits by moving past a row of them one qubit at a time. The interactionentangles the qubits so that you can't fully understand one of them unless you look at all the others.

It's possible to bring B into state B 2 as you please as the row gets larger. A machine that transforms B 1 to B 2 is constituted by the sequential interactions of B with the row of qubits. You can turn B 2 back to B 1 by sending B back along the row.

What if you reuse the array of qubits for the same process with a fresh B? The array becomes less and less able to produce the transformation from B 1 to B 2 if the number of qubits in the row is not very large. The theory predicts that the row becomes less able to do the reverse transformation from B 2 to B 1 The researchers have confirmed this prediction by using a row of three qubits.

You can approximate the constructor in one direction, but not the other. The second law imposed an asymmetric transformation. The transformation takes the system from a pure quantum state to a mixed one, which is entangled with the row. A pure state is one that we know very little about. You can't fully specify one of the entangled objects without knowing everything about the other. It's easier to switch from a pure quantum state to a mixed one than it is to switch from a pure quantum state to a mixed one. It is similar to trying to re-form a droplet of ink once it has dispersed in water, a process in which the irreversibility is imposed by the second law.

The irreversibility is a consequence of the way the system changes. There is no statistical aspect to it. The inevitable outcome of irreversibility is governed by the quantum interactions of the components.

Demon in the Machine

James Clerk Maxwell, the Scottish scientist who pioneered the statistical view of thermodynamics along with Boltzmann, created another way of thinking about the second law. The issue of information was connected to the thermodynamic law without realizing it.

The implications of a heat death and rule of change seemed to undermine free will. He wanted to pick a hole in the second law. In his hypothetical scenario, a demon would turn heat back into a resource for doing work. There is a distribution of energy in a gas. Some molecules are moving faster and have more energy than others. They are all mixed at random, so there is no way to make use of the differences.

Enter the demon. The compartment of gas is divided into two and a trapdoor is installed between them. The demon allows the hot molecule to pass through the trapdoor in one direction but not the other. The demon has a hot gas on one side and a cooler on the other, and it can exploit the temperature difference to drive a machine.

The second law was apparently undermined by the demon's use of information about the motions of molecules. Information can be used to do work just like a barrel of oil. We can exploit this information because it is hidden at the scale. It's this lack of knowledge of the microstates that makes classical thermodynamics speak of averages and ensembles.

Physicists proved that the demon doesn't subvert the second law in the long term because it gathers enough information to make room for more. The physicist Rolf Landauer showed in 1961 that this can never be done without raising the heat in the surroundings. The second law is not broken.

The second law is being seen as a quantum problem. The demon treats the gas particles as classical billiard balls because of the perception that quantum mechanics is a more fundamental description. It shows the growing interest in quantum information theory. We can use quantum principles to do things that we can't do classically. Information about particles can be spread around and manipulated in a variety of ways.

The approach of quantum information suggests a way to get rid of the statistical picture that is problematic in the classical view of thermodynamics.

He said that taking recourse in an ensemble shows the fact that we have only partial information about the state, and so we have to average over a probability distribution. There are states of partial information that can be generated through quantum theory. When a quantum system gets entangled with its environment, it ends up in a mixed state, where you can't know everything, even in principle.

You are forced to speak in terms of probabilities because some of the information is fundamentally unknown. Scandolo said that probabilities arise naturally from entanglement.

The ideas have been made more precise. Scandolo and Chiribella have proposed four axioms about quantum information that are not based on probabilities. There are constraints on the information in a quantum system. Everything that happens to the system plus environment is in principle irreversible, just as it is implied by the standard mathematical formula of how a quantum system changes over time.

Scandolo and Chiribella show that uncorrelated systems always grow more correlated through repeated interactions. The properties of objects are correlated with each other. A constraint on how correlations can change is also a constraint on entropy. If the system's entropy goes down, the environment's will have to increase in order for the two entropies to stay the same. Scandolo said that their approach derives the existence of entropy from the underlying axioms.

Redefining Thermodynamics

A resource theory is a model for any situation in which the actions are not possible. Resource theories have been incorporated into his work.

The picture of the physical world suggested by quantum information theory has fundamental limitations on which physical processes are possible. It is fundamentally impossible to say that it is not possible in quantum information theory.

There are a few main ingredients. The operations that are allowed are called free operations. An agent can access a pile of coal to fire up a furnace and power a steam engine if they so choose. Extra memory could allow a demon to subvert the second law for a while.

A kind of zooming in on the details of the classical second law can be done with quantum resource theories. We don't need to think about a lot of particles; we can make statements about what is allowed. When we do this, it becomes clear that the classical second law is just a coarse-grained sum of a whole family of relationships. The second law says that you can transform a nonequilibrium state into one that is closer to thermal equilibrium. Asking which of the states is closer to thermal is not a simple question. To answer it, we have to check a lot of inequalities.

There are a lot of mini-second laws in resource theories. She says that sometimes she feels like everyone in this field has their own second law.

The physicist from the University of Vienna said that the resource-theory approach has no conceptual or mathematical loose ends. It is still about information. The inability to keep track of information is the reason why the second law holds.

Hilbert’s Problem

The challenge laid down by the German mathematician David Hilbert is what inspired these efforts to rebuild thermodynamics. In 1900, he posed 23 mathematics problems that he wanted to see solved. The sixth item on the list was to treat the physical sciences in which mathematics plays an important part.

Physicists are trying to reformulate quantum mechanics and its more abstract version, quantum field theory, using axioms that are simpler and more physically transparent than the traditional ones. The theory of probabilities is one of the aspects of physics that is ripe for reinvention.

I think that Hilbert's sixth problem is far from being completely solved, and I find it very intriguing and important research.

Maybe, though, the real value of re-deriving the second law is not in satisfying the ghost of Hilbert but in deeper understanding of the law itself. Einstein said that a theory is more impressive the more simple it is.