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Quantum physics frequently involves a need to count the states, subspaces,measurement outcomes, and other elements of quantum dynamics. However, withquantum mechanics assigning probabilities to such objects, it is oftendesirable to work with the notion of a "total" that takes into account theirvaried relevance. For example, such an effective count of position statesavailable to a lattice electron could characterize its localization properties.Similarly, the effective total of outcomes in the measurement step of a quantumcomputation relates to the efficiency of the quantum algorithm. Despite a broadneed for effective counting, a well-founded prescription has not beenformulated. Instead, the assignments that do not respect the measure-likenature of the concept, such as versions of the participation number orexponentiated entropies, are used in some areas. Here, we develop the additivetheory of effective number functions (ENFs), namely functions assigningconsistent totals to collections of objects endowed with probability weights.Our analysis reveals the existence of a minimal total, realized by the uniqueENF, which leads to effective counting with absolute meaning. Touching upon thenature of the measure, our results may find applications not only in quantumphysics, but also in other quantitative sciences.

Effective Number Theory: Counting the Identities of a Quantum State