Stochastic thermodynamics of quantum maps with and without equilibrium. (arXiv:1704.06029v1 [quant-ph])

We study stochastic thermodynamics for a quantum system of interest whose
dynamics are described by a completely positive trace preserving (CPTP) map due
to its interaction with a thermal bath. We define CPTP maps with equilibrium as
CPTP maps with an invariant state such that the entropy production due to the
action of the map on the invariant state vanishes. Thermal maps are a subgroup
of CPTP maps with equilibrium. In general, for CPTP maps the thermodynamic
quantities such as the entropy production or work performed on the system
depend on the combined state of the system plus its environment. We show that
these quantities can be written in term of system properties for maps with
equilibrium. The relations we obtain are valid for arbitrary strength of the
coupling between the system and the thermal bath. The fluctuations of
thermodynamic quantities are considered in the framework of a two-point
measurement scheme. We derive the fluctuation theorem for the entropy
production for general maps and a fluctuation relation for the stochastic work
on a system that starts in the Gibbs state. Some simplifications for the
distributions in the case of maps with equilibrium are given. We illustrate our
results considering spin 1/2 systems under thermal maps, non-thermal maps with
equilibrium, maps with non-equilibrium steady states and concatenations of
them.