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Correspodence between quantoids and matroids
Miklín, Vojtěch ; Matúš, František (vedoucí práce) ; Mohammadi, Fatemeh (oponent)
Kvantoid je analogií pojmu matroid v kontextu kvantové informace. Tato práce shrnuje základní vlastnosti kvantoidů a korespondenci mezi kvantoidy a samoduálními matroidy. Nová sada axiomů je odvozena jako alternativa k sadě axiomů, která byla použita jako původní definice kvantoidu. Katalog všech kvantoidů na množině velikosti 1-5 je v apendixu a elektronická příloha této práce obsahuje větší databázi kvantoidů - kvantoidy na množině velikosti 1-7.
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Polymatroids and polyquantoids
Matúš, František
When studying entropy functions of multivariate probability distributions, polymatroids and matroids emerge. Entropy functions of pure multiparty quantum states give rise to analogous notions, called here polyquantoids and quantoids. Polymatroids and polyquantoids are related via linear mappings and duality. Quantum secret sharing schemes that are ideal are described by selfdual matroids. Expansions of integer polyquantoids to quantoids are studied and linked to that of polymatroids.
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Conditional probability spaces and closures of exponential families
Matúš, František
A set of conditional probabilities is introduced by conditioning in the probability measures from an exponential family. A closure of the set is found, using previous results on the closure of another exponential family in the variational distance. The conditioning in the exponential family of all positive probabilities on a finite space is discussed and related to the permutahedra.
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On maximization of the information divergence from an exponential family
Matúš, František ; Ay, N.
The information divergence of a probability measure P from an exponential family E over a finite set is defined as infimum of the divergences of P from Q subject to Q in E. For convex exponential families the local maximizers of this function of P are found. General exponential family E of dimension d is enlarged to an exponential family E* of the dimension at most 3d+2 such that the local maximizers are of zero divergence from E*.
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