On the quantum f-relative entropy and generalized data processing inequalities

We study the fundamental properties of the quantum f-relative entropy, where f(.) is an operator convex function. We give the equality conditions under monotonicity and joint convexity, and these conditions are more general than, since they hold for a class of operator convex functions, and different for f(t) = -ln(t) from, the previously known conditions. The quantum f-entropy is defined in terms of the quantum f-relative entropy and we study its properties giving the equality conditions in some cases. We then show that the f-generalizations of the Holevo information, the entanglement-assisted capacity, and the coherent information also satisfy the data processing inequality, and give the equality conditions for the f-coherent information.