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Sequential product and Jordan product of quantum effects. (English) Zbl 1215.81009
Summary: The quantum effects for a physical system can be described by the set $${\mathcal E}({\mathcal H})$$ 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 $${\mathcal P}({\mathcal H})$$. For $$A,B \in{\mathcal E}({\mathcal H})$$, 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 $$A*B=\frac{AB+BA}{2}$$ the Jordan product of $$A,B\in{\mathcal E}({\mathcal H})$$. 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 $$A \ast B$$ satisfies certain associative laws, then $$AB=BA$$. Moreover, $$A*B\in{\mathcal P}({\mathcal H})$$ if and only if $$A\circ B\in{\mathcal P}({\mathcal H})$$.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 17A15 Noncommutative Jordan algebras 81P15 Quantum measurement theory, state operations, state preparations 62L86 Fuzziness and sequential statistical methods
##### Keywords:
quantum effect; sequential product; Jordan product
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##### References:
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