Continuity of the Maximum-Entropy Inference

We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non-commutative observables and that a geodesic closure of a Gibbsian family equals the set of maximum-entropy states. We discuss eight descriptions of the set of maximum-entropy states with proofs of accuracy and an analysis of deviations.Comment: 34 pages, 1 figur

The Parade of Sovereignties: Establishing the Vocabulary of the New Russian Federalism

On the basis of extensive on-site interviews and documentary sources, the author interprets the dynamics of the collapse of the Soviet Union by analyzing the cascade of sovereignty declarations issued by republics of the USSR as well as by autonomous republics and other subunits of the Russian republic, in 1990-1991. Interrelationships among the declarations, and other putative causes of their content and timing, are explored. A case study of Tatarstan is provided. The study also analyzes the impact of the process on subsequent Russian approaches to federalism