Sequential Product and Jordan Product of Quantum Effects

Abstract

The quantum effects for a physical system can be described by the set \(\mathcal{E(H)}\) of positive operators on a complex Hilbert space \(\mathcal{H}\) that are bounded above by the identity operator I. We denote the set of sharp effects by \({\mathcal{P(H) }}\). For \(A,B\in\mathcal{E(H)}\), the operation of sequential product \(A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}\) was proposed as a model for sequential quantum measurements. Denote by \(A\ast B=\frac{AB+BA}{2}\) the Jordan product of \(A,B\in\mathcal{E(H)}\). The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on A∗B imply that A and B commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if A∗B satisfies certain associative laws, then AB=BA. Moreover, \(A\ast B\in{\mathcal{P(H) }}\) if and only if \(A\circ B\in{\mathcal{P(H)}}\).