  In 2017 an idea considering a pair of Hermitian operators of 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.
