# 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.