Classical complexity and quantum entanglement

Generalizing a decision problem for bipartite perfect matching, Edmonds (J. Res. Natl. Bur. Standards 718(4) (1967) 242) introduced the problem (now known as the Edmonds Problem) of deciding if a given linear subspace ofM(N) contains a non-singular matrix, whereM(N) stands for the linear space of complexN ×N matrices. This problem led to many fundamental developments in matroid theory, etc. Classical matching theory can be defined in terms of matrices with non-negative entries. The notion of Positive operator, central in QuantumTheory, is a natural generalization ofmatrices with non-negative entries. (Here operator refers tomaps frommatrices tomatrices.) First, we reformulate theEdmonds Problem in terms of completely positive operators, or equivalently, in terms of bipartite density matrices. It turns out that one of the most important cases when Edmonds’ problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable) bipartite density matrices. We introduce a very general class (or promise) of linear subspaces ofM(N) on which there exists a polynomial deterministic time algorithm to solve Edmonds’ problem. The algorithm is a thoroughgoing generalization of algorithms in Linial, Samorodnitsky and Wigderson, Proceedings of the 30th ACM Symposium on Theory of Computing, ACM, NewYork, 1998; Gurvits andYianilos, and its analysis benefits from an operator analog of permanents, so-called Quantum Permanents. Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD. © 2004 Elsevier Inc. All rights reserved.