Complexity of separability under stronger distance promises

Determine the computational difficulty of deciding whether a bipartite quantum state ρ ∈ L(C^n ⊗ C^m) is separable under the stronger promise that either ρ is separable or ρ lies at an inverse-logarithmic (or constant) distance from the boundary of the set of separable states, measured using standard norms on density operators.

Background

The authors compare mixed-unitary detection with the well-studied separability problem, for which NP-hardness is known under certain promises. They explicitly note that the analogous question of separability under tighter (inverse-logarithmic or constant) distance promises remains unresolved.

This highlights the sensitivity of complexity to the choice of distance scale and metric when moving beyond inverse-polynomial promises.