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

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.

28 pages

References in corpus (2)

• Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory
• Three-qubit entanglement witnesses with the full spanning properties

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• Separability criterion for three-qubit states with a four dimensional norm
• On the convex cones arising from classifications of partial entanglement in the three qubit system
• There exist infinitely many kinds of partial separability/entanglement