Abstract
If a is a quantum effect and ρ is a state, we define the ρ-entropy S_a(ρ ) which gives the amount of uncertainty that a measurement of a provides about ρ. The smaller S_a(ρ ) is, the more information a measurement of a gives about ρ. In Section~2, we provide bounds on S_a(ρ ) and show that if a+b is an effect, then S_a+b(ρ )≥ S_a(ρ )+S_b(ρ ). We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their ρ-entropies. In Section~3, we employ S_a(ρ ) to define the ρ-entropy S_A(ρ ) for an observable A. We show that S_A(ρ ) directly provides the ρ-entropy S (ρ ) for an instrument . We establish bounds for S_A(ρ ) and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider ρ-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.