The Measure Aspect of Quantum Uncertainty, of Entanglement, and the Associated Entropies

Indeterminacy associated with probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes repeated experiments. Here we express it as an effective amount (measure) distinct instead. The resulting $\mu$-uncertainties are described by number theory [1] whose central result, existence minimal amount, leads to well-defined notion intrinsic irremovable uncertainty. We derive $\mu$-uncertainty formulas for arbitrary set commuting operators, including cases continuous spectra. entropy-like characteristics, $\mu$-entropies, convey how many degrees freedom effectively involved given measurement process. In order construct led numbers designed count independent, mutually orthogonal states comprising density matrix. This concept basis-independent and measure-based characterization entanglement.