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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.

##### MSC:
 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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