Data processing inequality and open quantum systems: Beyond Markov states

Using a tripartite framework consisting of an open quantum system, its environment, and a passive reference system, in this study we discuss quantum mutual information (QMI) and data processing inequality (DPI). Without any restriction on the initial correlations, the necessary and sufficient conditions for the decrease or increase as well as for the conservation of QMI are obtained for any joint unitary evolution of the open quantum system and its environment. In the special case of the input Markov states, it has been shown that as long as the tripartite input state is a Markov state then DPI holds for every joint unitary evolution even in the presence of initial correlations encoded in the input. We also point out that the converse of the last statement, that is, sufficiency of the existence of a local quantum channel or of the fulfillment of DPI for the input Markov state does not hold in general and this fact is exhibited by counterexamples.