Topology of the cone of positive maps on qubit systems

Abstract

An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra M 2 (C) has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of M 2 (C) have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of M n (C). Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.

Extremal Positive Maps on M3(ℂ) and Idempotent Matrices

“…Obviously, S 1 8 = I. The structure of the set analogous to Λ, but representing maps on the algebra M 2 , has been studied using geometrical methods in [11]. Because there is no such geometrical identification for n = 3, this time we exploit the semigroup aspect of the set Λ.…”

“…Here, we go one step further and explore the relation between those stable subspaces and the idempotent real matrices that represent the conditional expectations projecting onto the spaces. To this end, we employ mostly geometrical techniques that allowed us before to establish the structure theorem for maps on the algebra M 2 [11], as well as the methods from the theory of compact semigroups [13,4].…”

Two proofs of Størmer's theorem

“…The above scheme of the proof of Størmer's theorem was apparently folklore for some time; it appears explicitly in [12]. However, its value was limited by the fact that the proof of Proposition 6 given in [11] was itself long and computational.…”