Optimization via Quantum Information Combinatorics and Its Applications to Extend Fundamentals of Quantum Information Science and Technology

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This research project first investigates quantum nonlocality and maximum quantum violation of generalized Bell inequalities from the viewpoint of combinatorial optimization, including semidefinite programming. Also, quantum computational geometry is explored in the space of quantum states, which leads to a new algorithm to compute the quantum channel capacity. Some related optimization problems are also studied in view of algebraic-geometric structures.Generalized Bell inequalities reveal quantum nonlocality from various standpoints. Those Bell inequalities are shown to correspond to facet inequalities of the cut polytope of a certain tripartite graph. We derive a unified method of generating such Bell inequalities by applying triangular eliminations to facets of the cut polytope of a complete graph. These Bell inequalities are compared theoretically and numerically with respect to its strength to reveal quantum nonlocality. The problem of finding the maximum violation of a generalized Bell inequality is first investigated from the point of view of optimization, and semidefinite programming is applied to derive bounds. Also, new interesting relation is shown between this problem and the 2-prover 1-round interactive proof in computational complexity.Quantum computational geometry is demonstrated to have nice structure as in the classical information-theoretic case. The Voronoi diagram with respect to quantum divergence in the space of quantum information geometry is characterized via the von Neumann entropy and its Legendre transformation. Voronoi diagrams for pure quantum states are also investigated, and relations among these diagrams are shown. These structures are applied in computing the quantum channel capacity by applying the minimum enclosing sphere algorithm for quantum divergence. This algorithm can produce an almost optimum solution with some guaranteed bound.