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On the factorization criterion for quantum statistics. (English) Zbl 1117.81315
Summary: We discuss some fundamental statistical notions for a quantum system in the simplest case when only a finite number of different states is possible and all states are pure. Thus our non-commutative statistical space is described a finite system of unit vectors \((\varphi _\theta ,\, \theta =1,\dots , n)\) in a Hilbert space \(H\). Each statistics is described by a self-adjoint operator \(S\) in \(H\). We define the sufficiency of \(S\) in some natural way motivated by the classical factorization criterion. We show that a generalization of the Blackwell-Rao theorem cannot be obtained. Moreover, some linear space containing all non-biased estimators with minimal variance is described. In fact, this space is huge, compared to the commutative case.

81P15 Quantum measurement theory, state operations, state preparations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
62B05 Sufficient statistics and fields
81P68 Quantum computation
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