A Representation of Quantum Measurement in Order-Unit Spaces

TL;DR

This paper generalizes quantum measurement formalism using order-unit spaces, replacing operator algebras, and explores conditions under which these spaces form Jordan algebras, providing insights into quantum probability and measurement structures.

Contribution

It introduces a generalized framework for quantum measurement using order-unit spaces and characterizes when these spaces become Jordan algebras, extending traditional operator algebra approaches.

Findings

Operator algebras are replaced by order-unit spaces with specific properties.

Conditions identified under which order-unit spaces become Jordan algebras.

Provides a characterization of projection lattices in operator algebras.

Abstract

A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lueders - von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by order-unit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these order-unit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras.