Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including $f$-divergences like Hellinger and $χ^2$-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.

This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Koßmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.