Extremal extensions of entanglement witnesses and their connection with UPB

TL;DR

This paper establishes a novel link between unextendable product bases (UPB) and extremal positive maps, enhancing entanglement detection methods through automorphisms of the Choi map and related extensions.

Contribution

It introduces a new connection between UPB and non-CP positive maps, and demonstrates how automorphisms of the Choi map produce extremal extensions that improve entanglement detection.

Findings

Automorphisms of P maps produce extremal extensions aiding entanglement detection.

Extensions of the Choi map can detect PPT entangled states.

New connection between UPB and extremal positive maps is established.

Abstract

In this paper we describe a new connection between UPB (unextendable product bases) and P (positive) maps which are not CP (completely positive). We show that inner automorphisms of the set of P maps which are not CP, produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth PPT (positive under partial transpose) entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orthogonal complement of certain UPB. This brings out a hitherto undiscovered connection between the Choi map and UPB. We also show that certain other recently considered extremal extensions are obtainable by such extensions of the Choi map.