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.

中文翻译：

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相互无偏基中四个量子位的非线性纠缠见证

纠缠见证是一种厄米算子，可用于检测混合状态的真正多部分纠缠。非线性纠缠见证具有纠缠区域检测范围更广的优点。我们在相互无偏的基础上为四个量子位密度矩阵构建真正的纠缠见证。首先，我们找到具有正部分转置状态的凸可行区域。然后，为了揭示纠缠区域，我们提出了一些适当的线性纠缠见证，我们发现这个线性见证家族的包络是非线性见证。最后，我们研究了热纠缠，我们向一些哈密顿人展示了目击者可以在所有温度下检测到纠缠。