  We consider asymptotic hypothesis testing (or state discrimination with
asymmetric treatment of errors) between an arbitrary fixed bipartite pure state
$\ket{\Psi}$ and the completely mixed state under one-way LOCC (local
operations and classical communications), two-way LOCC, and separable POVMs. As
a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable
POVMs. Further, we derive a Stein's lemma type of optimal error exponents under
one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which
clarifies the difference between one-way and two-way LOCC POVM. Our study gives
a very rare example in which the optimal performance under the infinite-round
two-way LOCC is also equal to that under separable operations and can be
attained with two-round communication, but not attained with the one-way LOCC.
