Entropy of Quantum Measurements

If a is a quantum effect and [Formula: see text] is a state, we define the [Formula: see text]-entropy [Formula: see text] which gives the amount of uncertainty that a measurement of a provides about [Formula: see text]. The smaller [Formula: see text] is, the more information a measurement of a gives about [Formula: see text]. In Entropy for Effects, we provide bounds on [Formula: see text] and show that if [Formula: see text] is an effect, then [Formula: see text]. We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their [Formula: see text]-entropies. In Entropy of Observables and Instruments, we employ [Formula: see text] to define the [Formula: see text]-entropy [Formula: see text] for an observable A. We show that [Formula: see text] directly provides the [Formula: see text]-entropy [Formula: see text] for an instrument [Formula: see text]. We establish bounds for [Formula: see text] and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider [Formula: see text]-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.