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Correspodence between quantoids and matroids
Miklín, Vojtěch ; Matúš, František (advisor) ; Mohammadi, Fatemeh (referee)
The notion of quantoid is an analogy to the notion of matroid in the context of quantum realm. This thesis summarizes basic properties of quantoids and the correspondence between quantoids and selfdual matroids. A new set of axioms is derived as an alternative to the set which was used as the original definition of quantoid. A catalog enumerating all quantoids with the size of their ground set up to 5 elements is attached in the appendix and a larger database of quantoids (up to 7 elements in the ground set) is enclosed as an attachment of this thesis.
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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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Optimální podmínky pro maximalizaci informační divergence exponenciální rodiny
Matúš, František
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. All directional derivatives of the divergence from E are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for P to be a maximizer of the divergence from E are presented, including new ones when P is not projectable to E.
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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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