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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
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