Entropy of Quantum Measurements

Abstract

If a is a quantum effect and ρ is a state, we define the ρ-entropy Sa(ρ) which gives the amount of uncertainty that a measurement of a provides about ρ. The smaller Sa(ρ) is, the more information a measurement of a gives about ρ. In Entropy for Effects, we provide bounds on Sa(ρ) and show that if a+b is an effect, then Sa+b(ρ)≥Sa(ρ)+Sb(ρ). We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their ρ-entropies. In Entropy of Observables and Instruments, we employ Sa(ρ) to define the ρ-entropy SA(ρ) for an observable A. We show that SA(ρ) directly provides the ρ-entropy SI(ρ) for an instrument I. We establish bounds for SA(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.