Sufficiency in Quantum Statistical Inference |
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Authors: | Anna Jen?ová Dénes Petz |
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Institution: | (1) Mathematical Institute of the Slovak Academy of Sciences, Stefanikova 49, Bratislava, Slovakia;(2) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary |
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Abstract: | This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in
classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual
relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways
and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families.
Our factorization theorem also implies two further important results, previously known only in finite Hilbert space dimension,
but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization
of the general form of states in the equality case of strong subadditivity.
Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology
and Center of Excellence SAS Physics of Information I/2/2005.
Supported by the Hungarian grant OTKA T032662 |
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