**Abstract**
We analyze the difference in the local distinguishability among the following three restrictions; (i) Local operations and only one-way classical communications (one-way LOCC) are permitted. (ii) Local operations and two-way classical communications (two-way LOCC) are permitted. (iii) All separable operations are permitted. We obtain two main results concerning the discrimination between a given bipartite pure state and the completely mixed state with the condition that the given state should be detected perfectly. As the first result, we derive the optimal discrimination protocol for a bipartite pure state in the cases (i) and (iii). As the second result, by constructing a concrete two-way local discrimination protocol, it is proven that the case (ii) is much better than the case (i), i.e., two-way classical communication remarkably improves the local distinguishability in comparison with one-way classical communication at least for a low-dimensional bipartite pure state.