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 of M(N) contains a non-singular matrix, where M(N) stands for the linear space of complex N × 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 Quantum Theory, is a natural generalization of matrices with non-negative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds 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 of M(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, New York, 1998; Gurvits and Yianilos, 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.

Generalizing a decision problem for bipartite perfect matching , J. Edmonds introduced in [15] the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M (N ) contains a nonsingular matrix, where M (N ) stands for the linear space of complex N × N matrices. This problem led to many fundamental developments in matroid theory etc. 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. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of 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 of M (N ) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem . The algorithm is a thoroughgoing generalization of algorithms in [29], [38], 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. Introduction and Main DefinitionsLet M (N ) be the linear space of N × N complex matrices . The following fundamental problem has been posed by J. Edmonds in [15]: Problem 1.1: Given a linear subspace V ⊂ M (N ) to decide if there exists a nonsingular matrix A ∈ V .We will assume throughout the paper that the subspace V is presented as a finite spanning k-tuple of rational matrices S(V ) = {A 1 , ..., A k }(k ≤ N 2 ), i.e . the linear space generated by them is equal to V . As usual, the complexity parameter of the input < S(V ) > is equal to ( N + "number of bits of entries of matrices A i , 1 ≤ i ≤ k" ). Thus Edmonds' problem is equivalent to checking if the following determinantal polynomial P A (x 1 , ..., x k ) = det(is not identically equal to zero. This determinantal polynomial can be efficiently evaluated, hence randomized poly-time algorithms, based on Schwartz's lemma or its recent improvements, are readily available (notice that our problem is defined over infinite field with infinite characteristic). But for general linear subspaces of M(N), i.e. without an extra assumption (promise), poly-time deterministic algorithms are not known and the problem is believed to be "HARD" . Like any other homogeneous polynomial, P A (x 1 , ..., x k ) is a weighted sum of monomials of degree N , i.e.where I k,N stands for a set of vectors r = (r 1 , ..., r k ) with nonnegative integer components and 1≤i≤k r i = N . We will make substantial use of the following (Hilbert) norm of determinantal polynomial P (.) : P 2 G =:It is easy to show tha...

For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral lp norms for 1 ≤ p ≤ ∞, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity tr ρ 2 not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by NMR quantum computation implementations or other experimental situations.

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Very Long Baseline Interferometry (VLBI) allows us to study a core of AGN with a sub-parsec resolution. We analyze the dependencies "apparent angular size -redshift" and "apparent motion -redshift" which contain an imprint of the source's properties and cosmology. We present data on the "angular size -redshift" relation obtained with VLBI at 5 GHz on a sample of 300 AGN distributed over the widest available range of redshifts 0.016 < ζ < 4.5. The sample exceeds those used in similar studies earlier by Kellermann (1993, 79 sources) and Wilkinson et al. (1997, 160 sources). Unlike extended source, the angular size-redshift for compact radio sources appears consistent with the predictions of standard FViedmann world models with q 0 ~ 0.5 without taking into account evolutionary effects or selection effects due to a "linear size -luminosity" or "linear sizespectral index" dependences. We discuss different approaches allowing us to disentangle intrinsic evolutionary properties of sources and parameters of the cosmological model. Recent estimates of parameters of the cosmological model are given. We also discuss a perspective of conclusive cosmological tests using the VLBI technique. LIG acknowledges support from the EU under contract CHGECT 920011 and a travel grant from the Leids Kerkhoven Boscha Fond.