  This paper continues research initiated in quant-ph/0201022 . The main
subject here is the so-called Edmonds' problem of deciding if a given linear
subspace of square matrices contains a nonsingular matrix . We present a
deterministic polynomial time algorithm to solve this problem for linear
subspaces satisfying a special matroids motivated property, called in the paper
the Edmonds-Rado property . This property is shown to be very closely related
to the separability of bipartite mixed states . One of the main tools used in
the paper is the Quantum Permanent introduced in quant-ph/0201022 .
