The average value inequality in sequential effect algebras

TL;DR

This paper investigates an inequality in sequential effect algebras related to quantum measurement theory and provides a counterexample to a question posed by Gudder in 2005.

Contribution

The paper constructs a specific example demonstrating that the proposed inequality does not always hold in sequential effect algebras.

Findings

Counterexample to Gudder's inequality question

Shows the inequality does not universally hold in sequential effect algebras

Contributes to understanding the structure of sequential effect algebras

Abstract

A sequential effect algebra $(E,0,1, \oplus, \circ)$ is an effect algebra on which a sequential product $\circ$ with certain physics properties is defined, in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If $a, b\in (E,0,1,\oplus, \circ)$ and $a\bot b$ and $a\circ b\bot a\circ b$, is it the case that $2(a\circ b)\leq a^2\oplus b^2$ ? In this paper, we construct an example to answer the problem negatively.