A class of exposed indecomposable positive maps

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2n x 2n complex matrices with n>1. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

Entanglement in SO(3)-invariant bipartite quantum systems

The structure of the state spaces of bipartite (N tensor N) quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin j which transform according to an irreducible representation of the rotation group. A positive map theta is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation theta_2 = (I tensor theta) acting on the composite system can be expressed in terms of the invariant 6-j symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of (4 tensor 4) systems. The separable states are shown to form a three-dimensional prism and a three-dimensional manifold of bound entangled states is identified. A positive maps is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of (4 tensor 4) systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.Comment: 15 pages, 3 figure