On extremal quantum states of composite systems with fixed marginals
Oliver Rudolph · Journal of Mathematical Physics · 2004
We study the convex set C(ρ1,ρ2) of all bipartite quantum states with fixed marginal states ρ1 and ρ2. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182]. Here we present an alternative necessary and sufficient condition for a state in C(ρ1,ρ2) 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 2×2 we give a simple new proof for the fact that all extremal states in C(121,121) 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 in C(131,131) that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.