In the Hilbert space description of quantum theory one has two major inputs: Firstly its linearity, expressing the superposition principle, and, secondly, the scalar product, allowing to compute transition probabilities. The scalar product defines an Euclidean geometry. One may ask for the physical meaning in quantum physics of geometric constructs in this setting. As an important example we consider the length of a curve in Hilbert space and the “velocity”, i. e. the length of the tangents, with which the vector runs through the Hilbert space. The Hilbert spaces are generically complex in quantum physics: There is a multiplication with complex numbers: Two linear dependent vectors represent the same state. By restricting to unit vectors one can diminish this arbitrariness to phase factors. As a consequence, two curves of unit vectors represent the same curve of states if they differ only in phase. They are physically equivalent. Thus, considering a given curve — for instance a piece of a solution of a Schrödinger equation – one can ask for an equivalent curve of minimal length. This minimal length is the “Fubini-Study length”. The geometry, induced by the minimal length requirement in the set of vector states, is the “Fubini-Study metric”. There is a simple condition from which one can read off whether all pieces of a curve in Hilbert space fulfill the minimal length condition, so that their Euclidean and their Study-Fubini length coincide piecewise: It is the parallel transport condition, defining the geometric (or Berry) phase of closed curves by the following mechanism: We replace the closed curve by changing its phases to a minimal length curve. Generically, the latter will not close. Its initial and its final point will differ by a phase factor, called the geometric phase (factor). We only touch these aspects in our essay and advice the reading of [6] instead. We discuss, as quite another application, the Tam-Mandelstamm inequalities. The set of vector states associated to a Hilbert space can also be described as the set of its 1-dimensional subspaces or, equivalently, as the set of all rank one projection operators. Geometrically it is the “projective space ” given by

This talk reviews some results contained in Refs.[1]-[7], and the present extended abstract mainly recalls the open problems posed during the talk. The quantum convex structures that will be considered are those of Quantum States, Quantum Operations (in particular trace-preserving, i. e. channels) and POVM’s (Positive Operator Valued Measures). More than focusing only on the convex structures themselves, I will analyze some physically meaningful interrelations that link them each other: 1) one-to-one maps between States and Quantum Operations, and between States and POVM’s, corresponding to Quantum Calibration; 2) dilation maps from the convex set of States to those of Quantum Operations and of POVM’s, corresponding to Quantum Programmability; 3) mapping POVM’s to POVM’s via channels, corresponding to pre-processing of POVM’s. Quantum Calibration. The convex Quantum Operations and that of bipartite states are connected each-other (apart from a normalization) by the Choi-Jamiolkowski isomorphism between CP-maps and positive bipartite operators. Such correspondence can be extended to the following one: R = M ⊗ I(F), describing the output state R of the local action of the map M on the input state F (I denotes the identity map). One calls the state F tomographically

Experimentally feasible measures of distance between

We show a continuity theorem for Stinespring’s dilation: two completely positive maps between arbitrary C ∗-algebras are close in cb-norm iff we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory. Key words: completely positive maps, dilation theorems, Stinespring representation, Bures distance, completely bounded norms, quantum information theory