  Classical matching theory can be defined in terms of matrices with
nonnegative entries. The notion of Positive operator, central in Quantum
Theory, is a natural generalization of matrices with nonnegative entries. Based
on this point of view, we introduce a definition of perfect Quantum (operator)
matching . We show that the new notion inherits many "classical" properties,
but not all of them . This new notion goes somewhere beyound matroids . For
separable bipartite quantum states this new notion coinsides with the full rank
property of the intersection of two corresponding geometric matroids . In the
classical situation, permanents are naturally associated with perfects
matchings. We introduce an analog of permanents for positive operators, called
Quantum Permanent and show how this generalization of the permanent is related
to the Quantum Entanglement. Besides many other things, Quantum Permanents
provide new rational inequalities necessary for the separability of bipartite
quantum states . Using Quantum Permanents, we give deterministic poly-time
algorithm to solve Hidden Matroids Intersection Problem and indicate some
"classical" complexity difficulties associated with the Quantum Entanglement.
Finally, we prove that the weak membership problem for the convex set of
separable bipartite density matrices is NP-HARD.
