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A strengthened monotonicity inequality of quantum relative entropy: A unifying approach via Rényi relative entropy

We derive a strengthened monotonicity inequality for quantum relative entropy by employing properties of $α$-Rényi relative entropy. We develop a unifying treatment towards the improvement of some quantum entropy inequalities. In particular, an emphasis is put on a lower bound of quantum conditional mutual information (QCMI) as it gives a Pinsker-like lower bound for the QCMI. We also give some improved entropy inequalities based on Rényi relative entropy. The inequalities obtained, thus, extends some well-known ones. We also obtain a condition under which a tripartite operator becomes a Markov state. As a by-product we provide some trace inequalities of operators, which are of independent interest.