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

TL;DR

This paper introduces a measure-based approach to quantum uncertainty and entanglement, defining effective numbers and entropies that quantify the degrees of freedom involved in quantum measurements.

Contribution

It develops a novel measure-theoretic framework for quantum uncertainties and entanglement, extending effective number theory to quantum operators and states.

Findings

Derived $$-uncertainty formulas for commuting operators.

Introduced basis-independent quantum effective numbers.

Provided a measure-based characterization of entanglement.

Abstract

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