Posted

Frederik vom Ende, Sumeet Khatri, Sergey Denisov (Sep 01 2025).
Abstract: We study kk-positive linear maps on matrix algebras and address two problems, (i) characterizations of kk-positivity and (ii) generation of non-decomposable kk-positive maps. On the characterization side, we derive optimization-based conditions equivalent to kk-positivity that (a) reduce to a simple check when k=dk=d, (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for k<dk<d), and (c) recast kk-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single kk-positive map, produces one-parameter families that remain kk-positive and non-decomposable for small enough times. We illustrate this by generating such families for d=3d=3 and d=4d=4. We also formulate a semi-definite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying kk-positivity and a systematic way to sample kk-positive non-decomposable maps.

Order by:

Want to join this discussion?

Join our community today and start discussing with our members by participating in exciting events, competitions, and challenges. Sign up now to engage with quantum experts!