Different Types of Conditional Expectation and the Lueders - von Neumann Quantum Measurement

TL;DR

This paper explores new types of conditional expectations and extends the Lueders - von Neumann measurement to continuous spectra within Jordan operator algebras, providing criteria for their existence and analyzing their relationships.

Contribution

It introduces a novel type of conditional expectation and extends quantum measurement theory to observables with continuous spectra in a general algebraic framework.

Findings

Criteria for existence of different conditional expectations

Extension of Lueders - von Neumann measurement to continuous spectra

No-go result for certain observables satisfying canonical commutator

Abstract

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lueders - von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lueders - von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.