Complexity of separability testing with stronger boundary-distance promises

Determine the computational complexity of deciding whether a bipartite quantum state is separable under the promise that either the state is separable or it lies at least inverse-logarithmic (or constant) trace-norm distance from the boundary of the set of separable states. This problem asks for the complexity classification (e.g., NP-hardness or tractability) when the margin from the separable set’s boundary is substantially larger than an inverse-polynomial bound.

Background

The paper proves strong NP-hardness (via polynomial-time Turing reductions) of deciding whether a channel is mixed-unitary under an inverse-polynomial margin from the mixed-unitary boundary. It then raises analogous margin-strengthening questions for channels and explicitly notes that the corresponding question for separable states remains open.

The authors emphasize that such stronger promises (inverse-logarithmic or constant) are sensitive to the chosen distance measure, unlike the inverse-polynomial regime where multiple norms are equivalent up to polynomial factors.

References

What is the computational difficulty of deciding if a given channel is mixed-unitary, given more restrictive promises on the channel's distance from the boundary of the set of mixed-unitary channels? For example, one may consider the problem in which a given channel is promised either to be mixed-unitary or to be at an inverse logarithmic (or even constant) distance from the boundary of the mixed-unitary channels. We note that the analogous problem for separable states is also open.

— Detecting mixed-unitary quantum channels is NP-hard  (1902.03164 - Lee et al., 2019) in Section 5 (Conclusion), Item 2