Hilbert space representations of decoherence functionals and quantum measures

Rok, strany: 2012, 1209 - 1230

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.

ISO 690:

Gudder, S. 2012. Hilbert space representations of decoherence functionals and quantum measures. In Mathematica Slovaca, vol. 62, no.6, pp. 1209-1230. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0074-1

APA:

Gudder, S. (2012). Hilbert space representations of decoherence functionals and quantum measures. Mathematica Slovaca, 62(6), 1209-1230. 0139-9918. DOI: https://doi.org/10.2478/s12175-012-0074-1