Existence of non-entanglement-breaking ∞-locally entanglement annihilating channels

Determine whether there exist quantum channels T that are ∞-locally entanglement annihilating—meaning that for every k ≥ 1, T^{⊗k} maps any input state to a fully separable k-partite state—but are not entanglement breaking.

Background

Entanglement-annihilating channels are those that, when applied in parallel to multiple subsystems, output only separable states. Entanglement-breaking channels are a strict subset that always destroy entanglement even in a single use.

The authors note examples of 2-locally entanglement annihilating channels that are not entanglement breaking, but whether non-entanglement-breaking channels can be ∞-locally entanglement annihilating is unknown. They further show that the existence of such channels would imply the existence of non-trivial tensor-stable positive maps.