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

Abstract

We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state $|\Psi ⟩$ 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.

We added detail descriptions for several proofs. Also, to explain the fundamental properties of the lattice and non-lattice cases, we newly added Lemma 37 in Appendix C

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• Locally-Measured Rényi Divergences
• Semi-Finite Length Analysis for Information Theoretic Tasks
• Quantum hypothesis testing between qubit states with parity