An operator-valued Lyapunov theorem

TL;DR

This paper extends Lyapunov's convexity theorem from classical scalar measures to quantum operator-valued measures, establishing conditions under which the range of a nonatomic quantum measure is a convex, weak*-closed set of quantum effects.

Contribution

It introduces an operator-valued Lyapunov theorem, defining key concepts for quantum random variables and proving convexity of the measure's range under specific conditions.

Findings

Range of nonatomic quantum measures is convex and weak*-closed.

Provides conditions for non-injectivity of quantum integration.

Defines essential boundedness and support for quantum random variables.

Abstract

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak*-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).