The Measure Aspect of Quantum Uncertainty, of Entanglement, and Respective Entropies.

Quantum uncertainty, namely the indeterminacy associated with probing of quantum state, is commonly characterized in terms of spectral distances (metric) featured in the outcomes of repeated experiments. Here we express it as an abundance (measure) of these outcomes. The concept of such $\mu$-uncertainties is governed by the theory of effective numbers [1], whose properties lead us to conclude the existence of state's intrinsic (minimal) $\mu$-uncertainty. The respective formulas involving arbitrary set of commuting operators are derived, and the associated entropy-like characteristics of quantum state, its $\mu$-entropies, are proposed. The latter, among other things, facilitate the concept of equivalent degrees of freedom, which is of particular interest in many-body settings. We introduce quantum effective numbers in order to analyze the state content of density matrices. This leads to a measure-like characterization of entanglement.