  Information-theoretic approaches provide a promising avenue for extending the
laws of thermodynamics to the nanoscale. Here, we provide a general fundamental
lower limit, valid for systems with an arbitrary Hamiltonian and in contact
with any thermodynamic bath, on the work cost for the implementation of any
logical process. This limit is given by a new information measure---the
coherent relative entropy---which accounts for the Gibbs weight of each
microstate. The coherent relative entropy enjoys a collection of natural
properties justifying its interpretation as a measure of information, and can
be understood as a generalization of a quantum relative entropy difference. As
an application, we show that the standard first and second laws of
thermodynamics emerge from our microscopic picture in the macroscopic limit.
Finally, our results have an impact on understanding the role of the observer
in thermodynamics: Our approach may be applied at any level of knowledge---for
instance at the microscopic, mesoscopic or macroscopic scales---thus providing
a formulation of thermodynamics that is inherently relative to the observer. We
obtain a precise criterion for when the laws of thermodynamics can be applied,
thus making a step forward in determining the exact extent of the universality
of thermodynamics and enabling a systematic treatment of Maxwell-demon-like
situations.

Quantum Physics