Counting the Identities of a Quantum State

Abstract

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a "total" that takes into account their varied relevance so acquired. For example, such effective count of position states available to lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of quantum computation relates to the efficiency of quantum algorithm. Despite a broad need for effective counting, well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of participation number or exponentiated entropies, are used in some areas. Here we develop and solve the theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of measure, our results may find applications not only in quantum physics but also in other quantitative sciences.