Sequential Product and Jordan Product of Quantum Effects

The quantum effects for a physical system can be described by the set E(H)\mathcal{E(H)} of positive operators on a complex Hilbert space H\mathcal{H} that are bounded above by the identity operator I. We denote the set of sharp effects by P(H){\mathcal{P(H) }}. For A,B ? E(H)A,B\in\mathcal{E(H)}, the operation of sequential product A°B=A^\frac12BA^\frac12A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}} was proposed as a model for sequential quantum measurements. Denote by A*B=\fracAB+BA2A\ast B=\frac{AB+BA}{2} the Jordan product of A,B ? E(H)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*B ? P(H)A\ast B\in{\mathcal{P(H) }} if and only if A°B ? P(H)A\circ B\in{\mathcal{P(H)}}.