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

Tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state