Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

TL;DR

This paper explores bi-linear positivity notions in quantum maps, linking them to tri-partite entanglement measures like Schmidt numbers, and provides concrete examples of entanglement witnesses for three-qubit states.

Contribution

It introduces bi-linear analogues of positivity, connects them to tri-partite entanglement measures, and constructs explicit entanglement witnesses for three-qubit systems.

Findings

Bi-linear positivity notions relate to tri-partite Schmidt numbers.

Tri-partite Schmidt numbers classify different bi-separability types.

Concrete witnesses for three-qubit entanglement are provided.

Abstract

We consider bi-linear analogues of $s$-positivity for linear maps. The dual objects of these notions can be described in terms of Schimdt ranks for tri-tensor products and Schmidt numbers for tri-partite quantum states. These tri-partite versions of Schmidt numbers cover various kinds of bi-separability, and so we may interpret witnesses for those in terms of bi-linear maps. We give concrete examples of witnesses for various kinds of three qubit entanglement.