Tight Asymptotic Bounds on Local Hypothesis Testing Between a Pure Bipartite State and the White Noise State
Masahito Hayashi, Masaki Owari · IEEE Transactions on Information Theory · 2017
We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state |Ψ| and the white noise state (the completely mixed state) under one-way local operations and classical communications (LOCC), two-way LOCC, and separable Positive Operator Valued Measures (POVMs). As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive 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 results clarify the relationship between the entanglement of Renyi entropy and the hypothesis testing under LOCC, since the entanglement of Renyi entropy appears in the formula of both the Hoeffding bounds and Stein's lemma type of error exponents. This paper 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 with the one-way LOCC.