Quantum statistics: decision problems and entropic functionals on state spaces

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The general objective is to provide concepts and methods solving basic state identification anddecision problems within the field of quantum statistics. These naturally appear in various scenarioswhile quantum information processing tasks are addressed. Following an operational viewpoint adoptedin quantum statistics we assume that the missing information about the quantum state should be inferredfrom outcomes of measurements performed on the considered system. This is in correspondence tothe approach of mathematical statistics where statistical decision, inference or estimation problems aretypically formulated with respect to observed data considered as actual values of a random variablewith unknown distribution.We use a *-algebraic formalism which allows to treat classical random variables and stochasticprocesses, as well as quantum states -with their intrinsic randomness- in a mathematically unified way.The emphasis is on the notion of state space of a C*-algebra which models the algebra of observablesassociated to a physical system, either classical or quantum. Assuming that there is an arbitrary largenumber of copies of the system of interest available for observation, solutions of classical statisticalproblems are typically related to some entropic quantities. This in general turns out to be true as wellin quantum mechanics. Although, it is non-trivial to find quantum counterparts of classical results andtheir proofs usually require new mathematical methods.We focus on problems of asymptotic (multiple) state discrimination, testing compound quantumhypotheses and quantum maximum-entropy inference, and corresponding entropic distances on statespaces being Umegaki relative entropy and generalizations of quantum Chernoff distance. Our mathematical methods mainly combine results of matrix analysis and mathematical statistics as well as(non-commutative) ergodic theory.