On extremal quantum states of composite systems with fixed marginals

We study the convex set of all bipartite quantum states with fixed marginal states. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincar\'e (to appear), quant-ph/0307182, [1]]. Here we present an alternative necessary and sufficient condition for a state with given marginals to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2 x 2 we give a simple new proof for the fact that all extremal states with maximally mixed marginals are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state with maximally mixed marginals in dimension 3 x 3 that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.Comment: 6 pages, to appear in J.Math.Phy

A graph theoretical Poincare-Hopf Theorem

We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It is the exit set of the gradient vector field. We prove that the sum of i(v) over V is always is equal to the Euler characteristic X(G) of the graph G. This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows to compute X(G) for large graphs for which other methods become impractical.Comment: 9 figure

Children’s views and experiences of their contact with social workers: a focused review of the evidence

"The review is ... aimed at exploring children’s reported experiences in terms of their relationship with social workers, as well as their views about social work tasks, and their perceptions concerning what makes a ‘good social worker’. The review also aimed to discuss the implications for initial qualification and continuing professional development in social work and to identify potential areas for further research." - Exec. summar