NP-hardness of the quantum separability problem under polynomial-time mapping reductions

Determine whether the quantum separability problem—deciding if a bipartite density operator can be expressed as a convex combination of unentangled pure states—is NP-hard with respect to polynomial-time mapping reductions, rather than only under polynomial-time Turing reductions or other weaker notions.

Background

The main result establishes NP-hardness of mixed-unitary detection via polynomial-time Turing reductions. The authors ask whether NP-hardness can be strengthened to polynomial-time mapping reductions for mixed-unitary detection, and explicitly note that the analogous strengthening for the separability problem remains open.

This reflects a broader question in quantum information complexity: for key state-membership problems (like separability), whether known hardness via Turing reductions can be elevated to mapping reductions, which are strictly stronger in the standard complexity framework.