Optimal bounds on functions of quantum states under quantum channels

Abstract

Let ρ1, ρ2 be quantum states and (ρ1, ρ2) 7→ D(ρ1, ρ2) be a scalar function such as the trace distance, the fidelity, and the relative entropy, etc. We determine optimal bounds for D(ρ1, Φ(ρ2)) for Φ blongs to S for different class of functions D(·, ·), where S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.