The equations of quantum mechanics, which govern the microscopic world, are reversible in principle: time evolution can be run backward. Yet lukewarm water made by mixing hot and cold water never spontaneously separates back into hot and cold. In the macroscopic world we live in, some changes proceed in only one direction, and the second law of thermodynamics explains them. How can the irreversible laws of thermodynamics arise from the reversible laws of quantum mechanics? This has long been a fundamental problem in physics.
A joint research team from RIKEN, the University of Tokyo and Osaka University has now built a new theory that offers a mathematically rigorous answer. The participants include Yuuya Chiba, Visiting Researcher at RIKEN; Eiko Hamazaki (Ritsu Hamazaki), RIKEN Hakubi Team Leader; Yasushi Yoneda, Visiting Researcher; and Akira Shimizu, Project Researcher and Professor Emeritus at the University of Tokyo. According to research results announced by RIKEN on November 4, 2026, the team defined thermal equilibrium states and external operations within the framework of quantum mechanics and proved that the second law of thermodynamics holds under certain conditions. The paper was published in the journal PRX Quantum on November 3 (November 4 Japan time) (Vol. 7, 043023, DOI: 10.1103/3y8f-yrp8).
This result is a mathematical proof in theoretical physics, not a verification by experiment or numerical simulation. What matters is that the team did not modify quantum mechanics itself. Instead, it clarified the range of physical quantities and operations that thermodynamics deals with. This explains how reversible quantum mechanics and irreversible thermodynamics can coexist without contradiction in the same physical system.
The Contradiction Between Reversible Quantum Mechanics and Irreversible Thermodynamics
The time evolution of a quantum system isolated from its surroundings follows the Schrödinger equation. This evolution is described by a mathematical operation called a "unitary transformation," and applying the inverse transformation restores the original state. In other words, the fundamental equations of an isolated quantum system lose no information, and time evolution itself is reversible.
A problem arises, however, if we assume that every unitary operation mathematically allowed in quantum mechanics can also be applied to a thermodynamic system. Recent research has shown that even a "pure quantum state," one that is not a probabilistic mixture, can in some cases be indistinguishable from a thermal equilibrium state in macroscopic observation. But if arbitrary unitary operations are allowed, energy can be extracted even from such a pure state, changing it into another pure state.
This contradicts "Planck's principle," one formulation of the second law of thermodynamics. Planck's principle states that for a closed system in thermal equilibrium, no net work can be extracted by an operation that ultimately returns the external control conditions to their original values. An example is expanding and compressing a gas in an adiabatic container with a piston and then returning the piston to its original position.
Another problem lies in the definition of entropy. The "von Neumann entropy," commonly used in quantum information theory, measures how probabilistically mixed a quantum state is, and it is always zero when the whole system is in a pure quantum state. But when a pure quantum state macroscopically represents thermal equilibrium, its thermodynamic entropy is generally not zero. The entropy defined in quantum mechanics and the entropy used in thermodynamics do not match.
The team traced the cause of these contradictions to treating every operation permitted in quantum mechanics as a thermodynamic operation. Mathematically, one can imagine operations that precisely control the states of roughly 10²³ particles one by one. But the operations actually considered in thermodynamics act on the system as a whole, through magnetic fields or the volume of a container, and do not freely control individual particles. Defining this difference rigorously was the starting point of the research.
"iMATE" and Macro Operations: Defining Thermal Equilibrium
The team first focused on physical quantities that are meaningful in thermodynamics. Energy, particle number and total magnetization can be obtained for the whole system by summing the contributions of its parts when the system is divided into several pieces. Quantities with this property are called "additive observables."
Using this property, the team introduced a new notion of thermal equilibrium, "iMATE (infinite-observable macroscopic thermal equilibrium)." iMATE refers to a quantum state in which, for the whole system or for subsystems of a certain size, the macroscopic values of all additive observables match those of the corresponding thermal equilibrium state. This definition does not depend on whether the quantum state is pure or mixed. Even a microscopically pure quantum state can be treated as a thermodynamic equilibrium state if its macroscopic observables show equilibrium values.
Next, the team formulated the quantum-mechanical counterpart of thermodynamic operations as "macro operations." These are operations in which a finite number of external control variables, such as an external magnetic field or the piston position, are varied over time, and the quantum state is changed by a Hamiltonian in which those variables couple to additive observables. The Hamiltonian is the mathematical quantity that determines a system's energy and time evolution. Because macro operations also follow unitary time evolution, the reversibility of quantum mechanics is preserved.
The key point is that the externally controllable variables are limited to a finite number. Changing a magnetic field or the boundary conditions of a container is allowed, but manipulating the vast number of internal degrees of freedom of a material one by one is not. This extracts only the thermodynamically meaningful operations from among all mathematically possible quantum operations.
Note that the "adiabatic operation" here means an operation that exchanges work with the outside without exchanging heat with the surroundings. It differs from the operation in the quantum adiabatic theorem, in which parameters are changed extremely slowly, and it includes abrupt changes of the control variables.
Proving Planck's Principle From Quantum Mechanics
Having defined iMATE and macro operations, the team first derived Planck's principle mathematically. The target was a "periodic macro operation" that starts from a quantum state in iMATE and ultimately returns the external control variables to their initial values. Examples include moving a piston and then returning it to its original position, or changing a magnetic field and then restoring its original strength.
The team proved that, in the thermodynamic limit where the system size becomes infinite, if the operation time does not depend on the system size, no macroscopic amount of energy can be extracted by a periodic macro operation. In a closed system, a decrease in energy corresponds to work extracted to the outside, so this is precisely Planck's principle. The result holds for any quantum state satisfying iMATE, including pure quantum states.
If arbitrary unitary operations are allowed, work can be extracted from a pure state. But when operations are restricted to macro operations with a finite number of external variables, extracting macroscopic work, which thermodynamics forbids, becomes impossible. Without changing the fundamental equations of quantum mechanics, and simply by specifying the range of permitted operations, the team showed consistency with thermodynamics.
However, what this principle forbids is the extraction of net work through periodic operations that return the control variables to their original values. If the control variables are not returned to their initial values, for example when the piston is left expanded, work can be extracted even in thermodynamics. The present proof does not prohibit such non-periodic operations.
Deriving the Law of Increasing Entropy With a New Definition of Entropy
The team then turned to proving the "law of increasing entropy," another formulation of the second law of thermodynamics. The challenge was how to define entropy for pure quantum states. As long as the von Neumann entropy of the whole system is used, the value for a pure state is always zero and cannot properly express changes in thermodynamic entropy.
The team therefore introduced the "quantum macro entropy density." It is defined by spatially averaging the quantum states of local regions of a certain size and taking the von Neumann entropy of the resulting state. Rather than looking only at the microscopic state of the whole system, averaging the states of local regions captures the entropy that corresponds to macroscopic thermodynamics.
The team mathematically proved that this quantum macro entropy density matches the thermodynamic entropy density for all quantum states satisfying iMATE. The relation holds even when the entire system is in a pure quantum state. In other words, a state that is microscopically pure yet macroscopically has thermodynamic entropy can now be represented within quantum mechanics without contradiction.
The team further analyzed a process in which a macro operation is applied to a system in iMATE and the system is then allowed to relax with the external control variables held fixed. Even if a macro operation drives the system out of thermal equilibrium, once external conditions are fixed, the state continues to change through internal interactions. By imposing physically natural conditions on this relaxation process, the team proved that the quantum macro entropy density does not decrease across the operation and relaxation.
In ordinary thermodynamic systems that return to thermal equilibrium after relaxation, this result corresponds to the law of increasing entropy: thermodynamic entropy does not decrease under adiabatic operations. The team thus derived Planck's principle, that macroscopic work cannot be extracted from periodic operations, and the law of increasing entropy, that entropy does not decrease, as separate, independent theorems. They are different propositions in quantum mechanical terms, but by properly defining macroscopic observables and operations, both can be explained within the same quantum-mechanical framework.
The differences between this study and the conventional theoretical setting that allows arbitrary microscopic operations are summarized below.
| Item | Setting allowing arbitrary microscopic operations | This study |
|---|---|---|
| Thermal equilibrium state | Pure states can also represent equilibrium, but consistency with the second law becomes a problem | Defined by iMATE, based on additive observables |
| Permitted operations | Arbitrary unitary operations | Macro operations with a finite number of external control variables |
| Entropy | Von Neumann entropy of the whole system, etc. | Quantum macro entropy density |
| Entropy of pure states | Von Neumann entropy of the whole system is zero | Matches the thermodynamic entropy density in iMATE |
| Work extraction | Operations that extract work from a pure state are possible | Extraction of macroscopic work by periodic macro operations is forbidden under certain time conditions |
| Law of increasing entropy | Von Neumann entropy of the whole system cannot express thermodynamic increase | Non-decrease proved for macro operations and a specified relaxation process |
The theory assumes no new irreversibility in the quantum state itself. By defining the observed physical quantities, thermal equilibrium states and external operations in line with thermodynamics, it makes reversible quantum time evolution compatible with the second law of thermodynamics.
Long-Duration Operations Can Violate the Second Law: Remaining Challenges
The result comes with an important condition. What the team rigorously proved applies in the thermodynamic limit, as the system is made larger, when the time required for a macro operation does not depend on system size and stays within a finite range. The operation time cannot be made arbitrarily long.
In real thermodynamics, quasi-static processes, in which a piston is moved extremely slowly and the system is changed while kept nearly in equilibrium, play an important role. As a system grows, the time needed for changes in heat and pressure to propagate through the whole system also grows, so the time required for a quasi-static operation can increase with system size. Such operations are not covered by the present proof.
The team also showed that if the theory is naively extended to cases where the operation time grows with system size, mathematical counterexamples that violate the second law can be constructed. This means that for arbitrary long-duration operations, the current definition of macro operations alone cannot guarantee that the second law holds. However, it has not been demonstrated that the second law is broken in real materials. Rather, it suggests that additional conditions on the quantum state or the external control are needed for the second law to hold under long-duration operations.
Identifying those additional conditions is the task ahead. For example, the chaotic or ergodic nature of quantum many-body systems and the properties of local energy transport and relaxation may be relevant, but this study has not determined which conditions are necessary. The team expects to use quantum simulators, which can mimic various quantum-mechanical time evolutions, to examine counterexamples under long-duration operations and the conditions under which the second law holds. Quantum simulation with cold atoms or superconducting circuits could serve as a research tool for this purpose, though the paper does not include completed experimental verification.
This study mathematically shows that the physical constraints on which operations can be performed from the outside are key to resolving the conflict between quantum mechanics and thermodynamics. While it demonstrates that the second law of thermodynamics can be derived even from pure quantum states, problems remain to be solved before the theory can be extended to long-duration operations. Whether the conditions under which macroscopic irreversibility emerges from reversible quantum mechanics can be made clearer will be the focus of future theoretical work and verification by quantum simulation.