Fred Shultz, Wellesley College (USA)

Abstract: A state on the tensor product of m x m and n x n matrices
is separable if it can be written as a convex combination of pure
product states.  These are precisely the states that are not
entangled.  The set S of all separable states is compact and convex.

Given such a state there may be more than one decomposition into pure
product states.  This is related to the facial structure of S.  In
general faces of S are a combination of faces like those in the usual
state space, together with faces that are simplicial in nature.
Uniqueness of decompositions is related to simplicial structure.

Given any convex set, one wants to know its affine automorphisms.
These also are important in physical applications, since time
evolution of state spaces is represented by a one parameter family of
affine automorphisms. We give a description of all affine
automorphisms of the space S of separable states, and describe their
relationship to time evolution of the set of all states that preserves
separability and entanglement.