The Uniqueness Problem of Sequence Product on Operator Effect Algebra $\varepsilon (H)$

TL;DR

This paper investigates the structure of sequential products on quantum effects in Hilbert spaces, revealing the existence of non-Lüders form products and answering an open problem in the field.

Contribution

It proves the existence of sequential products on quantum effects that are not of the generalized Lüders form, addressing an open problem posed by Gudder.

Findings

Existence of sequential products not of Lüders form

Negative answer to Gudder's open problem

Advances understanding of quantum effect algebra structures

Abstract

A quantum effect is an operator on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$. We denote the set of all quantum effects by ${\cal E}(H)$. In this paper we prove, Theorem 4.3, on the theory of sequential product on ${\cal E}(H)$ which shows, in fact, that there are sequential products on ${\cal E}(H)$ which are not of the generalized L\"{u}ders form. This result answers a Gudder's open problem negatively.