Quantum Systems Obey Thermodynamics’ Second Law Even When Appearing Perfectly Ordered

Scientists are increasingly investigating the applicability of thermodynamics to quantum systems, and a new study by Yuuya Chiba (RIKEN Hakubi Research Team), Yasushi Yoneta (RIKEN, Center for Quantum Computing), and Ryusuke Hamazaki et al. from RIKEN and The University of Tokyo/Osaka addresses a long-standing problem reconciling the second law of thermodynamics with quantum mechanics in closed systems. The research demonstrates that the second law can emerge even from pure quantum states by defining a novel framework based on infinite-observable macroscopic thermal equilibrium and macroscopic operations. This work is significant because it establishes two distinct forms of the second law, utilising reasonable macroscopic criteria for observables, equilibrium states and operations, and offers insights into the timescales governing these quantum thermodynamic processes.

Recent research demonstrated that even pure quantum states can accurately represent thermal equilibrium, yet these states traditionally violate the second law due to their potential to generate work under arbitrary conditions.

This work addresses the emergence of the second law for adiabatic operations, constrained state transitions in closed systems, by introducing the concept of infinite-observable macroscopic thermal equilibrium (iMATE). A quantum state is considered to be in iMATE if its expectation values for all additive observables align with their established equilibrium values.
Researchers developed a macroscopic operation defined as unitary evolution driven by a time-dependent additive Hamiltonian, mirroring adiabatic operations in thermodynamics. Employing these concepts, the study proves that no extensive work can be extracted from any state in iMATE through these macroscopic operations, provided the operation times are independent of system size.

Furthermore, a quantum-mechanical form of entropy density was introduced, aligning with thermodynamic entropy density for any state representing iMATE. This allowed for the demonstration of the law of increasing entropy, proving that the entropy density of any initial state in iMATE cannot be decreased by macroscopic operations, followed by a time-independent relaxation process.

The theory establishes two distinct forms of the second law, both derived from macroscopically reasonable definitions of observables, equilibrium states, and operations. This work demonstrates how thermodynamics emerges from quantum mechanics, offering a new framework for understanding irreversible processes at the quantum level.

These findings have implications for the development of quantum technologies and a deeper understanding of the fundamental laws governing complex systems. The research provides a crucial step towards reconciling quantum mechanics with the established principles of thermodynamics, paving the way for future investigations into the nature of equilibrium and irreversibility.

Defining infinite-observable macroscopic thermal equilibrium and macroscopic operations

A central technique in this work involves calculating the quantum macroscopic entropy density to establish connections between mechanics and thermodynamics. The research begins by introducing infinite-observable macroscopic thermal equilibrium, or iMATE, defining a state, including pure states, as being in iMATE if the expectation values of all additive observables match their equilibrium values.

Macroscopic operations are then defined as unitary evolution generated by time-dependent additive Hamiltonians, representing adiabatic operations within the system. Employing these concepts, the study demonstrates that no extensive work can be extracted from any state within iMATE through these macroscopic operations.

Furthermore, a novel form of entropy density is introduced, designed to align with the standard thermodynamic entropy density for any state in iMATE. The core theoretical result proves that for any initial state in iMATE, this newly defined entropy density cannot decrease under macroscopic operations followed by a time-independent relaxation process.

This establishes two distinct formulations of the second law, grounded in macroscopically reasonable choices of observables, equilibrium states, and operational procedures. The study also considers the timescales relevant to these macroscopic operations, providing a comprehensive framework for understanding the emergence of the second law in the limit of large system sizes.

To verify the agreement between the quantum macroscopic entropy density and the thermodynamic entropy density, the researchers focus on representations of iMATE. They demonstrate that the quantum macroscopic entropy density extracts only macroscopic properties from the density matrices, ensuring its capture of relevant thermodynamic properties when applied to thermal equilibrium states.

The proof relies on establishing the existence of a thermodynamic limit for free energy and energy density expectation values, building upon prior research. This theorem is specifically proven for the canonical Gibbs state, demonstrating that the limit of the quantum macroscopic entropy density converges to the thermodynamic entropy density. This result contrasts with the von Neumann entropy density, which does not necessarily agree with the thermodynamic entropy density, even for pure states representing iMATE or locally uniform states.

Defining infinite-observable macroscopic thermal equilibrium via macroscopic equivalence of quantum states

Macroscopic equivalence and infinite-observable macroscopic thermal equilibrium are central to understanding the second law of thermodynamics in quantum systems. The research introduces a new characterization of quantum states, defining two states ρL and σL as macroscopically equivalent if the limits of expectation values of all additive observables, composed of l-local observables, agree as the system size L approaches infinity.

This equivalence is evaluated for any positive integer l, taken arbitrarily large but independent of L, and forms the basis for defining infinite-observable macroscopic thermal equilibrium, or iMATE. A state ρL represents an iMATE if it is macroscopically equivalent to the canonical Gibbs state ρcan L (β|HL) = e−βHL/Z, where β is the inverse temperature and Z is the partition function.

The study focuses on additive observables, defined as the sum of translated l-local observables across a hypercubic lattice ΛL with N = Ld sites. For instance, in spin-1/2 chains, σx 0 and σz −1σx 0σz 2 serve as examples of 1- and 3-local observables, respectively, forming additive observables when summed over the lattice.

The number of independent additive observables increases with increasing l, providing a more refined characterization of macroscopic states. This approach generalizes previous notions of thermal equilibrium, such as MATE, aiming for greater consistency with thermodynamic principles as detailed in Table I.

Furthermore, the work defines macroscopic operations as unitary evolutions generated by time-dependent additive Hamiltonians, representing adiabatic operations. These operations are controlled by m external fields, f 1(t) to f m(t), which couple to m additive observables B1 to Bm. The research demonstrates that no extensive work can be extracted from any state in iMATE through these macroscopic operations, establishing a macroscopic version of passivity.

A quantum mechanical form of entropy density is also introduced, aligning with the standard entropy density for any state within iMATE. The core finding is the proof of two forms of the second law. For any initial state in iMATE, the newly defined entropy density cannot be decreased by any macroscopic operation followed by a time-independent relaxation process.

This confirms that the quantum macroscopic entropy density does not decrease under macroscopic operations, demonstrating the emergence of the second law of thermodynamics in closed quantum many-body systems given macroscopically reasonable definitions of equilibrium states, operations, and entropy. The study also addresses the time scales of macroscopic operations relevant to these results.

Infinite Observables and Macroscopic Thermal Equilibrium Define Second Law Consistency

Researchers have established a refined understanding of the second law of thermodynamics, reconciling it with quantum mechanics through the introduction of infinite-observable macroscopic thermal equilibrium (iMATE). This framework defines thermal equilibrium not by finite measurements, but by considering the expectation values of an infinite number of additive observables, effectively capturing macroscopic properties.

The work demonstrates that for systems in iMATE, no extensive work can be extracted via macroscopic operations, thereby upholding a key tenet of the second law. Furthermore, a corresponding entropy density has been formulated that aligns with thermodynamic entropy for states within iMATE, and it is proven that this entropy density cannot decrease under macroscopic operations followed by a relaxation process.

This provides two distinct formulations of the second law, grounded in reasonable assumptions about observable quantities, equilibrium states, and the nature of operations performed on the system. The authors also highlight the importance of considering appropriate timescales for macroscopic operations to ensure the validity of these results.

The authors acknowledge limitations related to the divergence of operation times in the thermodynamic limit, which can lead to violations of the second law. This underscores the optimality of their approach, which carefully defines macroscopic operations to avoid such scenarios. Future research could explore extending these results to more complex systems and investigating alternative definitions of macroscopic operations, potentially broadening the applicability of this framework and further solidifying the connection between quantum mechanics and thermodynamics.