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Sequential quantum measurements. (English) Zbl 1018.81005

Summary: A quantum effect is an operator \(A\) on a complex Hilbert space \(H\) that satisfies \(0\leq A\leq I\). We denote the set of quantum effects by \({\mathcal E}(H)\). The set of self-adjoint projection operators on \(H\) corresponds to sharp effects and is denoted by \({\mathcal P}(H)\). We define the sequential product of \(A,B\in{\mathcal E}(H)\) by \(A\circ B= A^{1/2} BA^{1/2}\). The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on \(A\circ B\) imply that \(A\) and \(B\) commute for the usual operator product. For example, if \(A\circ B\) satisfies certain distributive or associative laws, then \(AB= BA\). Moreover, if \(A\circ B\in{\mathcal P}(H)\), then \(AB= BA\) and \(A\circ B= B\) or \(B\circ A= B\) if and only if \(AB= BA= B\). A natural definition of stochastic independence is introduced and briefly studied.

MSC:

81P15 Quantum measurement theory, state operations, state preparations
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References:

[1] Gudder S., J. Math. Phys. 39 pp 5772– (1998) · Zbl 0935.81005 · doi:10.1063/1.532592
[2] Busch P., Phys. Lett. A 249 pp 10– (1998) · doi:10.1016/S0375-9601(98)00704-X
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