Joint separable numerical range and bipartite ultrafine entanglement witnessing

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]. (paper)