2015/02/28 by Alexander Müller-Hermes, David Reeb, Michael M. Wolf · 1 citation
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP #math.OA
paper · pdf · doi:10.1063/1.4927070
published as J. Math. Phys. 57, 015202 (2016) · 25 pages, no figures
arxiv created 2015/12/20 · arxiv updated 2015/12/22
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n∈ℕ there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions we reduce the existence question of such non-trivial "tensor-stable positive maps" to a one-parameter family of maps and show that an affirmative answer would imply the existence of NPPT bound entanglement. As an application we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We furthermore show that the latter is an upper bound even for the LOCC-assisted quantum capacity, and that moreover it is a strong converse rate for this task.