Halos and undecidability of tensor stable positive maps

TL;DR

This paper proves the existence of essential tensor stable positive maps on hypercomplex numbers, links them to bound entangled states with negative partial transpose, and shows that tensor stable positivity is undecidable, suggesting deep implications for quantum entanglement theory.

Contribution

It demonstrates the existence of essential tsp maps on hypercomplex numbers and establishes the undecidability of tensor stable positivity, advancing understanding of quantum entanglement and positivity maps.

Findings

Existence of essential tsp maps on hypercomplex numbers.

Existence of NPT bound entangled states in the quantum state halo.

Tensor stable positivity on the matrix multiplication tensor is undecidable.

Abstract

A map $\mathcal{P}$ is tensor stable positive (tsp) if $\mathcal{P}^{\otimes n}$ 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.