The work reports a general way to evaluate the encoding optimum of a quantum key distribution model. The evaluation is based on the concept of mutual information and compression. A method for evaluating the encoding optimum is proposed. The original point-to-point quantum key distribution model is subjected to this method. It is shown that this model (and probably any existing model) does not reach its encoding optimum.

This paper points out that, in a four-dimensional spherically symmetric spacetime manifold, one can consider coordinate transformations expressed by fractional linear maps which give rise to isometries and make it possible to bring infinity down to a finite distance. Schwarzschild and Nariai spacetimes are then described in projectively transformed coordinates.

In this paper, we study statistical Cauchy-Riemannian maximal submanifolds in the statistical Sasakian space form which naturally inherit their Sasakian structure from the ambient. We show that there exists a Cauchy-Riemann maximal submanifold in statistical Sasakian space form where the $\psi$-holomorphic sectional curvature of the ambient space is bounded. Moreover, a Cauchy-Riemannian maximal submanifold in the statistical Sasakian space form has at most four principal curvatures under some properties of second fundamental form $C$ and its dual.

Lin, Lu, and Yau formulated the Ricci curvature of edges in simple undirected graphs. Using their formulation, we calculate the Ricci curvatures of Cayley graphs for the dihedral groups, the general quaternion groups, and the cyclic groups with some generating sets chosen so that their cardinal numbers are less than or equal to four. For the dihedral group and the general quaternion group, we obtained the Ricci curvatures of all edges of the Cayley graph with generator sets consisting of the four elements that are the two generators defining each group and their inverses elements. We deal with Cayley graphs where two different edge directions are determined for each vertex. The introduction of two types of edges is intended to discretize the fiber bundle, separating the edges in the horizontal and vertical directions.

Considering the $B$-branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to $B$-branes. We construct a family of coherent sheaves on the complex projective space, which generates the corresponding bounded derived category and such that the supports of the elements of this family are two by two disjoint. Using that family, we prove that the cardinal of the set of holomorphic gauge fields on any $B$-brane over the projective space is less than two.