The n-th root of sequential effect algebras

TL;DR

This paper investigates the uniqueness of n-th roots in sequential effect algebras, providing an affirmative answer to a strengthened version of a longstanding open problem in quantum measurement theory.

Contribution

It demonstrates that for any positive integer n>1, there exist sequential effect algebras where the n-th root of an element is not unique and not a k-th root for any k<n.

Findings

Existence of non-unique n-th roots in sequential effect algebras.

Construction of examples where roots are not k-th roots for any smaller k.

Affirmative solution to a strengthened open problem from 2005.

Abstract

Sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Professor Gudder presented 25 open problems to motivate its study. The 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique ? We can strengthen the problem as following: For each given positive integer $n>1$, is there a sequential effect algebra such that the n-th root of its some element $c$ is not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ? Recently, we answered the strengthened problem affirmatively.