Sequential product on standard effect algebra ${\cal E} (H)$

Abstract

A quantum effect is an operator $A$ on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$, ${\cal E} (H)$ is the set of all quantum effects on $H$. In 2001, Professor Gudder and Nagy studied the sequential product $A\circ B=A^{{1/2}}BA^{{1/2}}$ of $A, B\in {\cal E}(H)$. In 2005, Professor Gudder asked: Is $A\circ B=A^{{1/2}}BA^{{1/2}}$ the only sequential product on ${\cal E} (H)$? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on ${\cal E} (H)$; secondly, we present a general method for constructing sequential products on ${\cal E} (H)$; finally, we study some properties of the sequential products constructed by the method