We propose a conceptually simple method for proving entropic inequalities by means of spectrum methods. Our method relies on tight upper and lower bounds on the relative entropy between two quantum states in terms of the so-called BKM variance of associated L_1 densities. As an application of this bound, we answer three important problems in quantum information theory and dissipative quantum systems that were left open until now: first, we provide universal bounds on the strong data processing inequality for a wide class of quantum channels that satisfy a notion of detailed balance, independently of the size of the environment. Second, we solve a conjecture about the existence of the complete modified logarithmic Sobolev constant which controls the exponential entropic convergence of a quantum Markov semigroup towards its equilibrium. Finally, we prove a new generalisation of the strong sub-additivity of the entropy for the relative entropy distance to a von Neumann algebra under a specific gap condition.
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