Halos and undecidability of tensor stable positive maps

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 entang...

Full description

Saved in:
Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2022-07, Vol.55 (26), p.264006
Main Authors: van der Eyden, Mirte, Netzer, Tim, De las Cuevas, Gemma
Format: Article
Language:English
Subjects:
bound entanglement
nonstandard numbers
positive maps
tensor stable positivity
undecidability
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/ac726e