Full-separability and bi-separability of qubits using Bell operators, partial transpose, witnesses and explicit (full/bi) separability

TL;DR

This paper introduces new Bell operators and entanglement witnesses for multi-qubit systems using Hilbert-Schmidt decomposition, providing bounds for (full/bi) separability and explicit forms for certain states, with some optimality conditions.

Contribution

It proposes novel Bell operators and entanglement witnesses based on HS decomposition for n-qubit systems, offering bounds and explicit separable forms, and explores optimality conditions.

Findings

Derived upper bounds for (full/bi) separability.

Explicitly found (full/bi) separable forms for specific states.

Identified conditions for optimal entanglement witnesses.

Abstract

Using the Hilbert-Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EW ) for n (>2) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we give results for the GHZ(3),W(3) , and cluster CL(4)states.