Nonlinear entanglement witnesses for four qubits in mutually unbiased bases

TL;DR

This paper develops nonlinear entanglement witnesses for four-qubit states in mutually unbiased bases, enhancing detection of genuine multipartite entanglement, including thermal states at all temperatures.

Contribution

It introduces a method to construct nonlinear entanglement witnesses for four qubits in mutually unbiased bases, expanding detection capabilities beyond linear witnesses.

Findings

Nonlinear witnesses detect entanglement in four-qubit states more broadly.

The method applies to thermal states across all temperatures.

The approach involves convex feasible regions and envelope construction.

Abstract

Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.