A Note on Sequential Product of Quantum Effects

Abstract

The quantum effects for a physical system can be described by the set $ℰ(\mathscr{H})$ of positive operators on a complex Hilbert space $\mathscr{H}$ that are bounded above by the identity operator $I$. For $A,B\in ℰ(\mathscr{H})$, let $A\circ B=A^{1/2}B{A}^{1/2}$ be the sequential product and let $A\mathrm{*}B=(AB+BA)/\mathrm{2}$ be the Jordan product of $A,B\in ℰ(\mathscr{H})$. The main purpose of this note is to study some of the algebraic properties of effects. Many of our results show that algebraic conditions on $A\circ B$ and $A\mathrm{*}B$ imply that $A$ and $B$ have $\mathrm{3}\times \mathrm{3}$ diagonal operator matrix forms with $I_{\overline{ℛ(A)}\cap \overline{ℛ(B)}}$ as an orthogonal projection on closed subspace $\overline{ℛ(A)}\cap \overline{ℛ(B)}$ being the common part of $A$ and $B$. Moreover, some generalizations of results known in the literature and a number of new results for bounded operators are derived.