2013/08/03 by Călin-Grigore Ambrozie, Ambrozie, Calin-Grigore, Aurelian Gheondea +1
Computer Science · Mathematics · #15A72 #15B48 #46L07 #81P45 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1308.0667
openalex publication_date 2013/08/03 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We present certain existence criteria and parameterisations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on *-subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson's Hahn-Banach type Theorem. We perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. A necessary and sufficient condition for the existence of solutions and a parametrisation of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states.