Theory of Generalized Measures and Applications Generalized Measures on Quantum Structures and Entropy

M. Portesi, Federico Holik, Pedro W. Lamberti +3 more

William Blacoe

Information-theoretic measures such as relative entropy and correlation are extremely useful when modeling or analyzing the interaction of probabilistic systems. We survey the quantum generalization of 5 such measures and point out some of their commonalities and interpretations. In particular we find the application of information theory to distributional semantics useful. By modeling the distributional meaning of words as density operators rather than vectors, more of their semantic structure may be exploited. Furthermore, properties of and interactions between words such as ambiguity, similarity and entailment can be simulated more richly and intuitively when using methods from quantum information theory.

Marius Krumm

This thesis investigates the connection between quantum theory, thermodynamics and information theory. Theories with structure similar to that of quantum theory are considered, mathematically described by the framework of "Generalized Probabilistic Theories". For these theories, a thought experiment by von Neumann is adapted to obtain a natural thermodynamic entropy definition, following a proposal by J. Barrett. Mathematical properties of this entropy are compared to physical consequences of the thought experiment. The validity of the second law of thermodynamics is investigated. In that context, observables and projective measurements are generalized to prove an entropy increase for projective measurements of ensembles. Information-theoretically motivated definitions of the entropy are compared to the entropy from the thermodynamic thought experiment. The conditions for the thermodynamic entropy to be well-defined are considered in greater detail. Several further properties of the theories under consideration (e.g. whether there is higher order interference, Pfister's state discrimination principle) and their relation to entropy are investigated.

Anatolij Dvureĕenskij

Guy Jumarie

The quantum mechanical entropy of density matrices defined by Von Neumann in quantum physics has been generalized in the form of quantum entropy of non-probabilistic matrices, and it is shown that the latter can be of some use in automatic control. This quantum entropy can be thought of as a measure of structure complexity and its application is quite relevant and meaningful in the following topics: stability of linear and non-linear systems, robustness and order reduction. Quantum entropy appears to be a new weighted combination of eigenvalues which would allow us to re-examine some problems with new points of view relating to information theory without probability.

Ilya Prigogine