On the definition of entropy production in Markovian open quantum systems

Entropy production for open quantum systems is defined as the sum of the increment of the entropy of the system and the entropy flow from the system into the environment. However, the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation does not provide information about the environment and how it is coupled to the system. So, the entropy flow to the environment and, hence, the entropy production cannot be determined from the GKSL equation alone. We address the following question: what minimal additional information should be supplemented to the GKSL equation that is sufficient to determine the entropy production. We approach this problem using the concept of complementary quantum channel from quantum information theory.