Detection of entanglement via moments of positive maps

TL;DR

This paper investigates the effectiveness of moments of positive maps, especially the reduction map, in detecting various forms of entanglement across different quantum systems, revealing their equivalences and detection capabilities.

Contribution

It demonstrates the equivalence of reduction map and partial transpose in certain cases and shows the ability of moments of positive maps to detect bound entangled states in multiple systems.

Findings

Reduction map is equivalent to partial transpose for two qubits.

Moments of reduction map can detect bound entangled states in $2 imes 4$ and qutrit-qutrit systems.

Moments of positive maps can detect a broader range of entanglement than previous criteria.

Abstract

We have reexamined the moments of positive maps and the criterion based on these moments to detect entanglement. For two qubits, we observed that reduction map is equivalent to partial transpose map as the resulting matrices have the same set of eigenvalues although both matrices look different in same computational basis. Consequently, the detection power of both maps is same. For $2 \otimes 4$ systems, we find that moments of reduction map are capable to detect a family of bound entangled states. For qutrit-qutrit systems, we show that moments of reduction map can detect two well known families of bound entangled states. The moments of another positive map can detect the complete range of entanglement for a specific family of quantum states, whereas the earlier criterion fails to detect a small range of entangled states. For three qubits system, we find that applying reduction map to one of the qubit is equivalent to partial transpose operation. In particularly, for GHZ state and W state mixed with white noise, all the moments of a reduction map are exactly the same as the moments of partial transpose map.