Given a function g on the exrtremal points of a convex compact set K, one may ask for the largest convex extension $g^{\cup}$ of $g$ on $K$. Similar the smallest concave extension $g^{\cap}$ ist defined. Entanglement of formation, entanglement of assistance, some concurrences belong to these classes of functions. If the extreme boundary of K is also compact and g is continuous, then a remarkable phenomenon takes place: $K$ is covered by convex subsets on each of which the extension, say $g^{\cup}$, is affine, and every one of these subsets is convexly generated by some extremal points of $K$ . Because of this I have called $g^{\cup}$ and $g^{\cap}$ "roofs" . I try to explain how to use this as a tool and I show the structure of some of the known examples, with and without bifurcations.
We introduce some recent results on quantum entanglement (which plays key roles in quantum information processing): in particular, the derivation of the measure of entanglement: Entanglement of Formation and Concurrence for mixed bipartite quantum states; the classification of quantum states under local unitary transformations: nonlocal properties and local invariants, geometry and dimensions of the orbits.
The quantum adiabatic approximation has a long history. Recently, the realization that the adiabatic approximation could be used as the basis for a method of quantum computing has generated a resurgence of interest in this topic. We present a simple proof of the theorem with explicit error bounds and sketch the application of the bounds to Grovers algorithm.
The randomness of finite words can be measured in terms of Kolmogorov complexity and algorithmic probability. Nevertheless, both notions depend on the arbitrary choice of an underlying universal computer U. We propose a machine-independent approach to algorithmic probability, resulting from a Markov process that describes computers that randomly simulate each other.
The state space of two bits contains the exponential family of product states. This family forms a surface of second order and is also a ruled surface. For the latter, there exists a geometrically motivated parametrization that implies an easy curvature formula. Which of these structures may be extended to exponential families of a more general type?
A natural geometric approach to dual structures will be presented. In particular exponential families will be discussed within the context of Stephan Weis' talk.
