Halos and undecidability of tensor stable positive maps

Abstract

A map P is tensor stable positive (tsp) if P⊗n is positive for all n, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

Many bounded versions of undecidable problems are NP-hard

“…Let us now review the reduction R : MPO → TSP of [14], which proves that TSP is RE-hard. The same reduction also yields that BTSP is NP-hard.…”

“…Many problems in quantum information and quantum many-body physics are undecidable. This includes the spectral gap of physical systems [1,2], membership problems for quantum correlations [3][4][5][6][7], properties of tensor networks [8][9][10], measurement occurrence and reachability problems [11,12], and many more [13][14][15][16][17]. In addition, other problems are believed to be undecidable, such as detecting quantum capacity [18], distillability of entanglement [12], or tensor-stable positivity [14].…”