  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.
