POVM Quantization

TL;DR

This paper introduces a formal framework for quantizing measure spaces using POVMs, linking measure theory with quantum measurement and inference, and illustrating with various geometric examples.

Contribution

It develops a general formalism for POVM-based quantization of measure spaces, connecting measure theory with quantum measurement and statistical inference.

Findings

Framework applicable to various geometric measure spaces

Illustrative examples include circle, sphere, plane, and half-plane

Links established between POVM quantization and quantum measurement theory

Abstract

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on normalized positive operator-valued measure. The latter are built from families of density operators labelled by points of the measure space. We specially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the procedure: circle, 2-sphere, plane, half-plane. Links with POVM quantum measurement and quantum statistical inference are sketched.