The entanglement witness is an important tool to detect entanglement. In 2017 an idea considering a pair of Hermitian operators in product form was published, which is called ultrafine entanglement witnessing. In 2018 some rigorous results were given. Here we improve their work. First we point this idea can be directly derived from an earlier concept named joint separable numerical range and explain how it works as a series of witnesses. Second by a simple method we present a sufficient condition for an effective pair. Finally we prove this condition is necessary for optimization. [M Gachechiladze et al 2018 J. Phys. A : Math. Theor . 51 36].

中文翻译：

---

联合可分离数值范围和二分法超精细纠缠见证

纠缠见证人是检测纠缠的重要工具。2017年，发布了一种以产品形式考虑一对Hermitian算子的想法，称为超精细纠缠见证。在2018年给出了一些严格的结果。在这里，我们改善了他们的工作。首先，我们指出，这个想法可以直接从较早的概念（称为联合可分离数值范围）中得出，并解释它是如何作为一系列证人工作的。其次，通过一种简单的方法，我们为有效对提供了充分的条件。最后，我们证明此条件对于优化是必要的。[M Gachechiladze等人，2018 J. Phys。A：数学。理论。51 36]。