Nontrivial quantum observables can always be optimized via some form of coherence

TL;DR

This paper demonstrates that optimizing the expectation value of any nontrivial quantum observable can be achieved through coherence measures, linking quantum resource theory to observable maximization.

Contribution

It establishes the equivalence between observable optimization and coherence maximization, providing a computationally efficient method and operational interpretation for coherence measures.

Findings

Maximizing quantum observables is equivalent to maximizing coherence.

A family of coherence measures can be efficiently computed via semidefinite programming.

Hierarchy of coherence measures relates to operational significance in optimization.

Abstract

In this paper we consider 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. As such, for any nontrivial observable, there exists a set of preferred basis states where the superposition between such states is always useful for optimizing 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.