Workshop on Geometry and Computation

Abstracts

On the affine structure of exponential families

Nihat Ay  (MPI MiS)
A natural geometric approach to dual structures will be presented. In particular exponential families will be discussed within the context of Stephan Weis' talk.

Theory of Quantum Entanglement: Measure and Local Equivalence

Shao-Ming Fei  (MPI MiS)
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 adiabatic theorem and quantum adiabatic computing

Sabine Jansen  (Technische Universität Berlin)
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.

Stationary algorithmic probability

Markus Müller  (Technische Universität Berlin)
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.

On convex (concave) roofs

Armin Uhlmann  (Universität Leipzig)
Given a function g on the exrtremal points of a convex compact set K, one may ask for the largest convex extension formula8 of g on K. Similar the smallest concave extension formula14 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 formula8, is affine, and every one of these subsets is convexly generated by some extremal points of K . Because of this I have called formula8 and formula14 "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.

Curvature of exponential families and ruled surfaces

Stephan Weis  (Universität Erlangen)
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?

Date and Location

January 21, 2006
Max Planck Institute for Mathematics in the Sciences
Inselstraße 22
04103 Leipzig
Germany
see travel instructions

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
Contact by Email

Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
Contact by Email

05.04.2017, 12:42