We show that the maximum extractable work (ergotropy) from a quantum many-body system is constrained by "local athermality" of an initial state and "local entropy decrease" brought about by quantum operations. The obtained universal upper bound on ergotropy implies that the eigenstate thermalization hypothesis prohibits work extraction from energy eigenstates by means of finite-time unitary operations. This no-go property implies that Planck's principle, a form of the second law of thermodynamics, holds even for pure quantum states. Our result bridges two independently studied concepts of quantum thermodynamics, the second law and thermalization, via "intrasystem" correlations in many-body systems as a resource for work extraction.
We show that the maximum extractable work (ergotropy) from a quantum many-body system is constrained by "local athermality" of an initial state and "local entropy decrease" brought about by quantum operations. The obtained universal upper bound on ergotropy implies that the eigenstate thermalization hypothesis prohibits work extraction from energy eigenstates by means of finite-time unitary operations. This no-go property implies that Planck's principle, a form of the second law of thermodynamics, holds even for pure quantum states. Our result bridges two independently studied concepts of quantum thermodynamics, the second law and thermalization, via "intrasystem" correlations in many-body systems as a resource for work extraction.
We study the energy extraction from and charging to a finite-dimensional quantum system by general quantum operations. We prove that the changes in energy induced by unital quantum operations are limited by the ergotropy and charging bounds for unitary quantum operations. This implies that, in order to break the ergotropy bound for unitary quantum operations, one needs to perform a quantum operation with feedback control. We also show that the ergotropy bound for unital quantum operations, applied to initial thermal equilibrium states, is tighter than the inequality representing the standard second law of thermodynamics without feedback control.
We examine the relationship between the second law of thermodynamics and the energy eigenstates of quantum many-body systems that undergo cyclic unitary evolution. Using a numerically optimized control protocol, we analyze how the work extractability is affected by the integrability of the system. Our findings reveal that, in nonintegrable systems the number of work-extractable energy eigenstates converges to zero, even when the local control operations are optimized. In contrast, in integrable systems, there are exponentially many eigenstates from which positive work can be extracted, regardless of the locality of the control operations. We numerically demonstrate that such a strikingly different behavior can be attributed to the number of athermal energy eigenstates. Our results provide insights into the foundations of the second law of thermodynamics in isolated quantum many-body systems, which are expected to contribute to the development of quantum many-body heat engines.
Work extraction from the Gibbs ensemble by a cyclic operation is impossible, as represented by the second law of thermodynamics. On the other hand, the eigenstate thermalization hypothesis (ETH) states that just a single energy eigenstate can describe a thermal equilibrium state. Here, we attempt to unify these two perspectives and investigate the second law at the level of individual energy eigenstates, by examining the possibility of extracting work from a single energy eigenstate. Specifically, we performed numerical exact diagonalization of a quench protocol of local Hamiltonians and evaluated the number of work-extractable energy eigenstates. We found that it becomes exactly zero in a finite system size, implying that a positive amount of work cannot be extracted from any energy eigenstate, if one or both of the prequench and the postquench Hamiltonians are nonintegrable. We argue that the mechanism behind this numerical observation is based on the ETH for a nonlocal observable. Our result implies that quantum chaos, characterized by nonintegrability, leads to a stronger version of the second law than the conventional formulation based on the statistical ensembles.
The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.
Thermodynamics is operational in the sense that the very concept of thermal equilibrium depends crucially on the choice of observables, while the amount of extractable work is defined relative to the allowed operations. Here we show that isolated quantum many-body states admit a thermodynamic hierarchical structure of observables and allowed operations, where the same state can be thermal at one level of the hierarchy and athermal at another. Enlarging the set of allowed operations therefore renders such hidden athermality a potential resource for work extraction; the resulting work gain is bounded in terms of the difference between the entropy densities of the two levels. This entropy difference takes the form of the mutual-information density. We establish the bounds for three extensions of operational access: increased spatial resolution, increased duration of control, and nonlocal connectivity, which provide access to position-state Holevo information, correlations between neighboring regions and spatially nonlocal correlations, respectively. Thus the difference between entropies at different levels of the thermodynamic hierarchy governs the bound on the work gain associated with moving to a less restricted level.
We prove the second law of thermodynamics and the nonequilibrium fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy eigenstate that satisfies the eigenstate-thermalization hypothesis. Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can be experimentally tested by artificial isolated quantum systems such as ultracold atoms.