Volume 57, Issue 1, January 2016
Index of content:
- SPECIAL ISSUE: OPERATOR ALGEBRAS AND QUANTUM INFORMATION THEORY
57(2016); http://dx.doi.org/10.1063/1.4926977View Description Hide Description
In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space , when X is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of 2 × 2 operator matrices and their off-diagonal parts. One of our main results states that if (X, (O n )) is an operator space, then for all .
57(2016); http://dx.doi.org/10.1063/1.4927070View Description Hide Description
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 non-positive partial transpose 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 local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.