Nonlinear entanglement witnesses for four qubits in mutually unbiased bases

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.

Closing loopholes of measurement-device-independent nonlinear entanglement witnesses

“…Consider the witness operator W φ, an arbitrary state |ψ acting on H A ⊗ H B , and a corresponding operator X = | φ ψ|. Then using these operators and states, a nonlinear operator can be constructed as [52]…”

“…But the second term in F φ will always have a non-positive value, so that F φ(ρ) ≤ W φ. Therefore, F φ(σ) can be defined as a nonlinear entanglement witness that can detect more entangled states than W φ [52].…”

“…However, one can construct better entanglement witnesses than the linear ones by "bending" the linear EWs towards negativity so that it still has semi-positive values for all separable states. Such entanglement witnesses, due to their nonlinear nature, are called nonlinear entanglement witnesses (NEWs) [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67]. Nonlinear entanglement witnesses, due to their construction, can detect more entangled states than their linear counterparts.…”