The Measure of a Measurement

TL;DR

This paper investigates fractal scales in probability measures related to quantum measurement, wavelet analysis, and iterated function systems, revealing a common fractal structure characterized by specific scale parameters.

Contribution

It identifies and characterizes a fractal scale $s$ in probability measures arising from quantum information and wavelet analysis, linking these fields through measure asymptotics.

Findings

Existence of a fractal scale $s$ in probability measures on the unit interval.

The scale $s$ can be less than, equal to, or greater than 1, with specific asymptotic properties.

Connection between quantum measurement, wavelet analysis, and fractal measure properties.

Abstract

While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space and its recursive iterations to further iterated subdivisions. This paper explores some implications for associated probability measures (in the classical sense of measure theory), specifically their fractal components. We identify a fractal scale $s$ in a family of Borel probability measures $\mu$ on the unit interval which arises independently in quantum information theory and in wavelet analysis. The scales $s$ we find satisfy $s\in \mathbb{R}_{+}$ and $s\not =1$, some $s <1$ and some $s>1$. We identify these scales $s$ by considering the asymptotic properties of $\mu(J) /| J| ^{s}$ where $J$ are dyadic subintervals, and $| J| \to0$.