Optimizing nontrivial quantum observables using coherence

Abstract

In this paper we consider the quantum resources required to maximize the mean values of any nontrivial quantum observable. We show that the task of maximizing the mean value of an observable is equivalent to maximizing some form of coherence, up to the application of an incoherent operation. For any nontrivial observable, there always exists a set of preferred basis states where the superposition between such states is always useful for optimizing the mean value of a quantum observable. The usefulness of such states is expressed in terms of an infinitely large family of valid coherence measures which is then shown to be efficiently computable via a semidefinite program. We also show that these coherence measures respect a hierarchy that gives the robustness of coherence and the l(1) norm of coherence additional operational significance in terms of such optimization tasks.