Work Extraction from a Single Energy Eigenstate

Work extraction from the Gibbs ensemble by a cyclic operation is impossible, as represented by the second law of thermodynamics. On the other hand, recent studies revealed that a thermal equilibrium state can be represented not only by the Gibbs state, but also by a single energy eigenstate. This is referred to as the eigenstate thermalization hypothesis (ETH). We attempt to unify these two perspectives 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 pre- and the post-quench Hamiltonians are non-integrable. This result suggests that the second law of thermodynamics is true even at the level of individual energy eigenstates if the system is non-integrable (i.e., quantum chaotic), which is analogous to the ETH.
Reference: K. Kaneko, E. Iyoda, T. Sagawa, arXiv:1809.01946.