Hilbert Space Representations of Decoherence Functionals and Quantum Measures

TL;DR

This paper demonstrates that decoherence functionals and quantum measures can be represented within Hilbert spaces, establishing conditions for uniqueness and spanning properties, and linking strong positivity to such representations.

Contribution

It introduces a Hilbert space representation for decoherence functionals and quantum measures, clarifying conditions for uniqueness and spanning, and connecting strong positivity with representability.

Findings

Decoherence functionals can be represented by vector-valued measures on Hilbert spaces.

The representation is unique up to isomorphism in finite systems.

Quantum measures have Hilbert space representations if and only if they are strongly positive.

Abstract

We show that any decoherence functional $D$ can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map $U$ from the history Hilbert space $K$ to the standard Hilbert space $H$ of the usual quantum formulation. We show that $U$ is an isomorphism from $K$ onto a closed subspace of $H$ and that $U$ is an isomorphism from $K$ onto $H$ if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.