de Bruijn-type identity for systems with flux

We show that an information-theoretic relation called the de Bruijn-type identity can be reformulated in a physical context with probability currents. The time derivatives of relative entropies under the continuity equation are presented, which shows that the conservation of distance between a pair of distributions is generally not guaranteed. As an important implication of these results, we discuss and present a possible conceptual framework for the classical no-cloning (deleting) theorem and qualitatively assert that we can attribute the perfect performance of the operating machine to the openness (non-vanishing flow at boundaries between the processing machine and the system) during the process.