  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.
