# 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 of *g* on *K*. Similar the smallest concave extension 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 , is affine, and every one of these subsets is convexly generated by some extremal points of *K* . Because of this I have called and "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