Hilbert Space Representations of Decoherence Functionals and Quantum Measures

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.

25 pages

References in corpus (4)

• An introduction to measurement based quantum computation
• Quantum Dynamics without the Wave Function
• Dynamical Wave Function Collapse Models in Quantum Measure Theory
• Partial Decoherence of Histories and the Diosi Test

Cited by in corpus (3)

• Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
• Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics
• Phenomenological quantum mechanics: I. Phenomenology of quantum observables