Bures Geometry on C*-algebraic State Spaces

Abstract: The inner geometry induced by the Bures distance function on the state space of a unital C*-algebra, or on parts of it, is considered in detail. As a key result the local dilation function is calculated at a state when the state argument is bound to vary along certain parameterized curves passing through this state. The parameterized curves considered are those possessing differentiable local implementations as vector states relative to some unital *-representation in the vicinity of the state in question. The space of all tangent forms at a given state correspondent to this category of curves is specified as normed linear space with a quadratic norm which is depending from the state in a characteristic manner. In analyzing the structure of the local dilation function special emphasis is laid on describing such parts of the state space, in restriction to which the line element would be of Finslerian type, and which were geodesically convex subsets. All results obtained are illustrated by examples, and the implications obtained are compared to those known from the investigations of Uhlmann et al in the finite dimensional case.