A resource theory of superposition

The superposition principle lies at the heart of many non-classical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find an elementary connection to unambiguous state discrimination, and propose several general quantitative superposition measures. We show that several main results from resource theories of coherence still hold in our more general setting. Of special importance are two results about the free completion of trace decreasing operations and the free probabilistic transformation between pure states that are also valid for the special case of coherence.