The Classification of Decoherence Functionals: An Analogue of Gleason's Theorem

TL;DR

This paper classifies decoherence functionals in finite-dimensional quantum histories, showing they can be represented via trace operations with operators on tensor product spaces, extending Gleason's theorem.

Contribution

It provides a complete classification of decoherence functionals for finite-dimensional Hilbert spaces, analogous to Gleason's theorem in standard quantum mechanics.

Findings

Decoherence functionals can be expressed as trace operations with operators on tensor product spaces.

The classification applies specifically to finite-dimensional Hilbert spaces.

The result extends Gleason's theorem to the context of quantum histories.

Abstract

Gell-Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are `histories' and decoherence functionals. Within this scheme it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where $\UP$ is the lattice of projectors $\PV$ in some Hilbert space $\V$; in effect we seek the history analogue of Gleason's famous theorem in standard quantum theory. In the present paper we present the solution to this problem for the case where $\V$ is finite-dimensional. In particular, we show that every decoherence functional $d(\a,\b)$, $\a,\b\in\PV$ can be written in the form $d(\a,\b)=\tr_{\V\otimes\V}(\a\otimes\b X)$ for some operator $X$ on the tensor product space $\V\otimes\V$.