Positivity of linear maps under tensor powers

TL;DR

This paper studies linear maps on matrix algebras that remain positive under tensor powers, revealing their existence, limitations in low dimensions, and implications for quantum entanglement and channel capacities.

Contribution

It demonstrates the existence of non-trivial tensor-stable positive maps, reduces their existence problem to a parameterized family, and links these maps to bounds on quantum channel capacities.

Findings

Existence of non-trivial tensor-stable positive maps for all n

No non-trivial maps in 2D spaces for all n

Tensor-stable positive maps provide bounds on quantum channel capacities

Abstract

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 $n\in\mathbb{N}$ 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.